/
Author: Lee J.M.
Tags: mathematics differential geometry mathematical analysis tensor analysis
ISBN: 978-0-8218-4815-9
Year: 2009
Text
Manifolds
and Differential
Geometry
Jeffrey M. Lee
Graduate Studies
in Mathematics
Volume 107
American Mathematical Society
Manifolds
and Differential
Geometry
Jeffrey M. Lee
Graduate Studies
in Mathematics
Volume 107
^jn! American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
David Cox (Chair)
Steven G. Krantz
Rafe Mazzeo
Martin Scharlemann
2000 Mathematics Subject Classification. Primary 58A05, 58A10, 53C05, 22E15, 53C20,
53B30, 55R10, 53Z05.
For additional information and updates on this book, visit
www.ams.org/bookpages/gsm-107
Library of Congress Cataloging-in-Publication Data
Lee, Jeffrey M., 1956-
Manifolds and differential geometry / Jeffrey M. Lee.
p. cm. — (Graduate studies in mathematics : v. 107)
Includes bibliographical references and index.
ISBN 978-0-8218-4815-9 (alk. paper)
1. Geometry, Differential. 2. Topological manifolds. 3. Riemannian manifolds. I. Title.
QA641.L38 2009
516.3'6—dc22
2009012421
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10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09
Contents
Preface
Chapter 1. Differentiable Manifolds
§1.1. Preliminaries
§1.2. Topological Manifolds
§1.3. Charts, Atlases and Smooth Structures
§1.4. Smooth Maps and Diffeomorphisms
§1.5. Cut-off Functions and Partitions of Unity
§1.6. Coverings and Discrete Groups
§1.7. Regular Submanifolds
§1.8. Manifolds with Boundary
Problems
Chapter 2. The Tangent Structure
§2.1. The Tangent Space
§2.2. Interpretations
§2.3. The Tangent Map
§2.4. Tangents of Products
§2.5. Critical Points and Values
§2.6. Rank and Level Set
§2.7. The Tangent and Cotangent Bundles
§2.8. Vector Fields
§2.9. 1-Forms
§2.10. Line Integrals and Conservative Fields
VI
Contents
§2.11. Moving Frames 120
Problems 122
Chapter 3. Immersion and Submersion 127
§3.1. Immersions 127
§3.2. Immersed and Weakly Embedded Submanifolds 130
§3.3. Submersions 138
Problems 140
Chapter 4. Curves and Нуpersurfaces in Euclidean Space 143
§4.1. Curves 145
§4.2. Hypersurfaces 152
§4.3. The Levi-Civita Covariant Derivative 165
§4.4. Area and Mean Curvature 178
§4.5. More on Gauss Curvature 180
§4.6. Gauss Curvature Heuristics 184
Problems 187
Chapter 5. Lie Groups 189
§5.1. Definitions and Examples 189
§5.2. Linear Lie Groups 192
§5.3. Lie Group Homomorphisms 201
§5.4. Lie Algebras and Exponential Maps 204
§5.5. The Adjoint Representation of a Lie Group 220
§5.6. The Maurer-Cartan Form 224
§5.7. Lie Group Actions 228
§5.8. Homogeneous Spaces 240
§5.9. Combining Representations 249
Problems 253
Chapter 6. Fiber Bundles 257
§6.1. General Fiber Bundles 257
§6.2. Vector Bundles 270
§6.3. Tensor Products of Vector Bundles 282
§6.4. Smooth Functors 283
§6.5. Horn 285
§6.6. Algebra Bundles 287
§6.7. Sheaves 288
Contents
νιι
§6.8. Principal and Associated Bundles 291
Problems 303
Chapter 7. Tensors 307
§7.1. Some Multilinear Algebra 308
§7.2. Bottom-Up Approach to Tensor Fields 318
§7.3. Тор-Down Approach to Tensor Fields 323
§7.4. Matching the Two Approaches to Tensor Fields 324
§7.5. Tensor Derivations 327
§7.6. Metric Tensors 331
Problems 342
Chapter 8. Differential Forms 345
§8.1. More Multilinear Algebra 345
§8.2. Differential Forms 358
§8.3. Exterior Derivative 363
§8.4. Vector-Valued and Algebra-Valued Forms 367
§8.5. Bundle-Valued Forms 370
§8.6. Operator Interactions 373
§8.7. Orientation 375
§8.8. Invariant Forms 384
Problems 388
Chapter 9. Integration and Stokes' Theorem 391
§9.1. Stokes' Theorem 394
§9.2. Differentiating Integral Expressions; Divergence 397
§9.3. Stokes' Theorem for Chains 400
§9.4. Differential Forms and Metrics 404
§9.5. Integral Formulas 414
§9.6. The Hodge Decomposition 418
§9.7. Vector Analysis on R3 425
§9.8. Electromagnetism 429
§9.9. Surface Theory Redux 434
Problems 437
Chapter 10. De Rham Cohomology 441
§10.1. The Mayer-Vietoris Sequence 447
§10.2. Homotopy Invariance 449
Vlll
Contents
§10.3. Compactly Supported Cohomology 456
§10.4. Poincare Duality 460
Problems 465
Chapter 11. Distributions and Probenius' Theorem 467
§11.1. Definitions 468
§11.2. The Local Probenius Theorem 471
§11.3. Differential Forms and Integrability 473
§11.4. Global Probenius Theorem 478
§11.5. Applications to Lie Groups 484
§11.6. Fundamental Theorem of Surface Theory 486
§11.7. Local Fundamental Theorem of Calculus 494
Problems 498
Chapter 12. Connections and Covariant Derivatives 501
§12.1. Definitions 501
§12.2. Connection Forms 506
§12.3. Differentiation Along a Map 507
§12.4. Ehresmann Connections 509
§12.5. Curvature 525
§12.6. Connections on Tangent Bundles 530
§12.7. Comparing the Differential Operators 532
§12.8. Higher Covariant Derivatives 534
§12.9. Exterior Covariant Derivative 536
§12.10. Curvature Again 540
§12.11. The Bianchi Identity 541
§12.12. G-Connections 542
Problems 544
Chapter 13. Riemannian and Semi-Riemannian Geometry 547
§13.1. Levi-Civita Connection 550
§13.2. Riemann Curvature Tensor 553
§13.3. Semi-Riemannian Submanifolds 560
§13.4. Geodesies 567
§13.5. Riemannian Manifolds and Distance 585
§13.6. Lorentz Geometry 588
§13.7. Jacobi Fields 594
Contents
IX
§13.8. First and Second Variation of Arc Length 599
§13.9. More Riemannian Geometry 612
§13.10. Cut Locus 617
§13.11. Rauch's Comparison Theorem 619
§13.12. Weitzenbock Formulas 623
§13.13. Structure of General Relativity 627
Problems 634
Appendix A. The Language of Category Theory 637
Appendix B. Topology 643
§B.l. The Shrinking Lemma 643
§B.2. Locally Euclidean Spaces 645
Appendix C. Some Calculus Theorems 647
Appendix D. Modules and Multilinearity 649
§D.l. R-Algebras 660
Bibliography 663
Index 667
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Preface
Classical differential geometry is the approach to geometry that takes full
advantage of the introduction of numerical coordinates into a geometric
space. This use of coordinates in geometry was the essential insight of Rene
Descartes that allowed the invention of analytic geometry and paved the way
for modern differential geometry. The basic object in differential geometry
(and differential topology) is the smooth manifold. This is a topological
space on which a sufficiently nice family of coordinate systems or "charts"
is defined. The charts consist of locally defined η-tuples of functions. These
functions should be sufficiently independent of each other so as to allow
each point in their common domain to be specified by the values of these
functions. One may start with a topological space and add charts which
are compatible with the topology or the charts themselves can generate the
topology. We take the latter approach. The charts must also be compatible
with each other so that changes of coordinates are always smooth maps.
Depending on what type of geometry is to be studied, extra structure is
assumed such as a distinguished group of symmetries, a distinguished
"tensor" such as a metric tensor or symplectic form or the very basic geometric
object known as a connection. Often we find an interplay among many such
elements of structure.
Modern differential geometers have learned to present much of the
subject without constant direct reference to locally defined objects that depend
on a choice of coordinates. This is called the "invariant" or "coordinate
free" approach to differential geometry. The only way to really see exactly
what this all means is by diving in and learning the subject.
The relationship between geometry and the physical world is
fundamental on many levels. Geometry (especially differential geometry) clarifies,
XI
Xll
Preface
codifies and then generalizes ideas arising from our intuitions about certain
aspects of our world. Some of these aspects are those that we think of as
forming the spatiotemporal background of our activities, while other aspects
derive from our experience with objects that have "smooth'4 surfaces. The
Earth is both a surface and a ulived-in space", and so the prefix "geo" in the
word geometry is doubly appropriate. Differential geometry is also an
appropriate mathematical setting for the study of what we classically conceive of
as continuous physical phenomena such as fluids and electromagnetic fields.
Manifolds have dimension. The surface of the Earth is two-dimensional,
while the configuration space of a mechanical system is a manifold which
may easily have a very high dimension. Stretching the imagination further
we can conceive of each possible field configuration for some classical field
as being an abstract point in an infinite-dimensional manifold.
The physicists are interested in geometry because they want to
understand the way the physical world is in "actuality". But there is also a
discovered "logical world" of pure geometry that is in some sense a part
of reality too. This is the reality which Roger Penrose calls the Platonic
world.1 Thus the mathematicians are interested in the way worlds could be
in principle and geometers are interested in what might be called "possible
geometric worlds". Since the inspiration for what we find interesting has its
roots in our experience, even the abstract geometries that we study retain
a certain physicality. Prom this point of view, the intuition that guides the
pure geometer is fruitfully enhanced by an explicit familiarity with the way
geometry plays a role in physical theory.
Knowledge of differential geometry is common among physicists thanks
to the success of Einstein's highly geometric theory of gravitation and also
because of the discovery of the differential geometric underpinnings of
modern gauge theory2 and string theory. It is interesting to note that the gauge
field concept was introduced into physics within just a few years of the time
that the notion of a connection on a fiber bundle (of which a gauge field is a
special case) was making its appearance in mathematics. Perhaps the most
exciting, as well as challenging, piece of mathematical physics to come along
in a while is string theory mentioned above.
The usefulness of differential geometric ideas for physics is also apparent
in the conceptual payoff enjoyed when classical mechanics is reformulated
in the language of differential geometry. Mathematically, we are led to the
subjects of symplectic geometry and Poisson geometry. The applicability
of differential geometry is not limited to physics. Differential geometry is
1 Penrose seems to take this Platonic world rather literally giving it a great deal of ontological
weight as it were.
2The notion of a connection on a fiber bundle and the notion of a gauge field are essentially
identical concepts discovered independently by mathematicians and physicists.
Preface
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also of use in engineering. For example, there is the increasingly popular
differential geometric approach to control theory.
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There is a bit more material in this book than can be comfortably
covered in a two semester course. A course on manifold theory would include
Chapters 1, 2, 3, and then a selection of material from Chapters 5, 7, 8, 9,
10, and 11. A course in Riemannian geometry would review material from
the first three chapters and then cover at least Chapters 8 and 13. A more
leisurely course would also include Chapter 4 before getting into Chapter
13. The book need not be read in a strictly linear manner. We included
here a flow chart showing approximate chapter dependence. There are
exercises throughout the text and problems at the end of each chapter. The
reader should at least read and think about every exercise. Some exercises
are rather easy and only serve to keep the reader alert. Other exercises take
a bit more thought.
Differential geometry is a huge field, and even if we had restricted our
attention to just manifold theory or Riemannian geometry, only a small
fragment of what might be addressed at this level could possibly be included.
In choosing what to include in this book, I was guided by personal interest
and, more importantly, by the limitations of my own understanding. While
preparing this book I used too many books and papers to list here but a few
that stand out as having been especially useful include [A-M-R], [Hicks],
[LI], [Lee, John], [ONI], [ON2], and [Poor].
XIV
Preface
I would like to thank Lance Drager, Greg Friedman, Chris Monico, Mara
Neusel, Efton Park, Igor Prokhorenkov, Ken Richardson, Magdalena Toda,
and David Weinberg for proofreading various portions of the book. I am
especially grateful to Lance Drager for many detailed discussions concerning
some of the more difficult topics covered in the text, and also for the many
hours he spent helping me with an intensive final proofreading. It should
also be noted that certain nice ideas for improvements of a few proofs are
due to Lance. Finally, I would like to thank my wife Maria for her support
and patience and my father for his encouragement.
Chapter 1
D liferent iable
Manifolds
"Besides language and music, mathematics is one of the primary
manifestations of the free creative power of the human mind."
- Hermann Weyl
In this chapter we introduce differentiable manifolds and smooth maps.
A differentiable manifold is a topological space on which there are defined
coordinates allowing basic notions of differentiability. The theory of
differentiable manifolds is a natural result of extending and clarifying notions
already familiar from multivariable calculus. Consider the task of writing
out clearly, in terms of sets and maps, what is going on when one does
calculus in polar coordinates on the plane or in spherical coordinates on a sphere.
If this were done with sufficient care about domains and codomains, a good
deal of ambiguity in standard notation would be discovered and clarifying
the situation would almost inevitably lead to some of the very definitions
that we will see shortly. In some sense, a good part of manifold theory is just
multivariable calculus done carefully. Unfortunately, this care necessitates
an increased notational burden that can be intimidating at first. The reader
should always keep in mind the example of surfaces and the goal of doing
calculus on surfaces. Another point is that manifolds can have nontrivial
topology, and this is one reason the subject becomes so rich. In this chapter
we will make some connection with some basic ideas from topology such as
covering spaces and the fundamental group.
Manifold theory and differential geometry play a role in an increasingly
large amount of modern mathematics and have long played an important
1
2
1. Differentiable Manifolds
role in physics. In Einstein's general theory of relativity, spacetime is taken
to be a 4-dimensional manifold. Manifolds of arbitrarily high dimension play
a role in many physical theories. For example, in classical mechanics, the
set of all possible locations and orientations of a rigid body is a manifold of
dimension six, and the phase space of a system of N Newtonian particles
moving in 3-dimensional space is a manifold of dimension 6N.
1.1. Preliminaries
To understand the material to follow, it is necessary that the reader have a
good background in the following subjects.
1) Linear algebra. The reader should be familiar with the idea of the
dual space of a vector space and also with the notion of a quotient vector
space. A bit of the language of module theory will also be used and is
outlined in Appendix D.
2) Point set topology. We assume familiarity with the notions of sub-
space topology, compactness and connectedness. The reader should know
the definitions of Hausdorff topological spaces, regular spaces and normal
spaces. The reader should also have been exposed to quotient
topologies. Some of the needed concepts are reviewed in the online supplement
[Lee, Jeff].
Convention: A neighborhood of a point in a topological space is often
defined to be a set whose interior contains the point. Our convention in the
sequel is that the word "neighborhood" will always mean open neighborhood
unless otherwise indicated. Nevertheless, we will sometimes write "open
neighborhood" for emphasis.
3) Abstract algebra. The reader will need a familiarity with the basics
of abstract algebra at least to the level of the basic isomorphism theorems
for groups and rings.
4) Multivariable calculus. The reader should be familiar with the idea
that the derivative at a point ρ of a map between open sets of (normed)
vector spaces is a linear transformation between the vector spaces. Usually
the normed spaces are assumed to be the Euclidean coordinate spaces such
as M.n with the norm ||x|| = yjx · x. A reader who felt the need for a
review could do no better than to study roughly the first half of the classic
book "Calculus on Manifolds" by Michael Spivak. Also, the online
supplement ([Lee, Jeff]) gives a brief treatment of differential calculus on Banach
spaces. Here we simply review a few definitions and notations.
Notation 1.1. The elements of Шп are η-tuples of real numbers, and we
shall write the indices as superscripts, so ν = (ν1,... ,vn). Furthermore,
when using matrix algebra, elements of W1 are most often written as column
1.1. Preliminaries
3
vectors so in this context, υ = [г;1,... ,νη]1. On the other hand, elements
of the dual space of W1 are often written as row vectors and the indices are
written as subscripts. With this convention, an element of the dual space
(Mn)* acts on an element of W1 by matrix multiplication. We shall only
be careful to write elements of W1 as column vectors if necessary (as when
we write Av for some m χ η matrix A and υ G Rn). If / : U —> Rm then
/ = (/i?..., /m) for real-valued functions /\ ..., /m.
Definition 1.2. Let U be an open subset of W1. A map / : U -» Rm is
said to be differentiable at α G U if and only if there is a linear map
Aa : Шп -» Mm such that
Ит ||/(а + /г)-/(а)-Л(/г)|| = Q
ц/ιΙΗο ||/*||
The map Aa is uniquely determined by / and α and is denoted by Df(a)
or Af |e.
Notation 1.3. If L : V —> W is a linear transformation and г; G V, we shall
often denote L(v) by either L · ν or Li> depending on which is clearer in a
given context. This applies to D/(a), so we shall have occasion to write
things like Df(a)(v), Df(a) ·ι>, Df(a)v or Df\av. The space of linear maps
from V to W is denoted by L(V, W) and also by Hom(V, W).
With respect to standard bases, Df(a) is given by the m χ η matrix of
partial derivatives (the Jacobian matrix). Thus if w = Df(a)v, then
j
Note that for a differentiable real-valued function /, we will often denote
df/dx* by dj.
Recall that if U, V and W are vector spaces, then a map β : U χ
V —> W is called bilinear if for each fixed щ G U and fixed vo G V,
the maps υ ь-> β(η$,ν) and и \-> /3(гх, г;о) are linear. If β : V χ V —>
W is bilinear and β(η,ν) = β(υ,ύ) for all u, ν G V, then β is said to be
symmetric. Similarly, antisymmetry or skewsymmetry is defined by the
condition β(η,ν) = —β(ν,ν). If V is a real (resp. complex) vector space,
then a bilinear map V χ V —> Ш (resp. C) is called a bilinear form.
If Df(a) is defined for all α G /7, then we obtain a map Df : U —>
L(Mn, Mm), and since L(Mn, Rm) ^ Rmn, we can consider the differentiability
of Df. Thus if Df is differentiable at a, then we have a second derivative
D2f(a) : Rn -» L(Mn,IRm), and so if v, w G Mn, then (D2f(a)v) w G Mm.
This allows us to think of D2 f(a) as a bilinear map:
D2f(a)(v,w) := (D2f(a)v) w for v,w G Rn.
4
1. Differentiable Manifolds
If и = D2f(a)(v,w), then with respect to standard bases, we have
Higher derivatives Drf can be defined similarly as multilinear maps (see
D.13 of Appendix D), and if Drf exists and is continuous on /7, then we
say that / is r-times continuously differentiable, or Cr, on U. A map / is
Cr if and only if all partial derivatives of order less than or equal to r of the
component functions fl exist and are continuous on U. The vector space
of all such maps is denoted by Cr(f7, Mm), and Cr(f7, R) is abbreviated to
Cr(U). If / is Cr for all r > 0, then we say that / is smooth or C°°. The
vector space of all smooth maps from U to Rm is denoted by C°°({7, Mm)
and we thereby include oo as a possible value for r. Also, / is said to
be Cr at ρ if its restriction to some open neighborhood of ρ is of class
Cr. If / is C2 near α then D2f(a) is symmetric and this is reflected in
the fact that we have equality of mixed second order partial derivatives;
dP/dxWxk = дР/дхкдхК Similarly, if / is Cr near α then the order of
partial differentiations can be rearranged in any mixed partial derivative of
order less than or equal to r.
Definition 1.4. A bijection / between open sets i/cRn and V С Rm is
called a Cr diffeomorphism if and only if / and /_1 are both Cr
differentiable. If r — oo, then we simply call / a diffeomorphism.
Definition 1.5. Let U be open in W1. A map / : U —> W1 is called a local
Cr diffeomorphism if and only if for every ρ G U there is an open set
Up С U with ρ e Up such that f(Up) is open and /1^ : Up —> f{Up) is a Cr
diffeomorphism.
We will sometimes think of the derivative of a curve1 с : / С R —> Μ™ at
£o £ I, as a velocity vector and so we are identifying Dc\tQ G L(M, Mm) with
Dc\tQ -1 G Mm. Here the number 1 is playing the role of the unit vector in R.
Especially in this context, we write the velocity vector using the notation
c(t0) or c'(i0).
Let / : U С Шп -» Шт be a map and suppose that we write Rn = Rk χ Rl.
Let (x, y) denote a generic element of Rh χ Rl. For every (a, 6) G U С Rh χ Rl
the partial maps fa'-У^ f(a,y) and fb:x\-^ f(x,b) are defined in some
neighborhood of b (resp. a). We define the partial derivatives, when they
exist, by D2f(a, b) := Dfa(b) and D\f(a, b) :— Dfb(a). These are, of course,
х\Уе will often use the letter / to denote a generic (usually open) interval in the real line.
1.1. Preliminaries
5
linear maps.
Dif(a,b) :R*-»Rm,
D2f(a,b) :R'-»]Rm.
Remark 1.6. Notice that if we consider the maps ια : χ \-+ (а, я) and
Lb :x^ (ж, 6), then D2f(a,b) = D(f о ta)(b) and Dif(a,b) = D(foLb)(a).
Proposition 1.7. /// has continuous partial derivatives Dif(x,y), i = 1,2
near (ж, y) Glfcx R*, ί/ien Df(x,y) exists and is continuous. In this case,
we have for ν = (νχ, v2) Gl^x Rl,
Df(x, y) · (vi, v2) = D\f(x, y) · vi + D2f(x, y) · v2.
Clearly we can consider maps on several factors / : Rkl xRk<2 χ · · · xRkr —>
Rm and then we can define partial derivatives D{f : Шкг —> Mm for г =
1,... ,r in the obvious way. Notice that the meaning of Dif depends on
how we factor the domain. For example, we have both R3 = R2 χ R and
also R3 = RxRxR. Let U be an open subset of Rn and let / : U -» R
be a map. Note that for the factorization Rn = R χ · · · χ Μ, the linear map
(Dif) (a) is often identified with the number dif (a).
Theorem 1.8 (Chain Rule). Let U be an open subset of Rn and V an
open subset of Rm. ///:[/—> Rm and g : V —> Rd are maps such that
f is differentiable at a G U and g is differentiable at f(a), then g о f is
differentiable at a and
D(gof)(a) = Dg(f(a))oDf(a).
Furthermore, if f and g are Cr at a and f(a) respectively, then g о f is Cr
at a.
The chain rule may also be written Д (g ο /) (α) = Σ Djg(f(a))DiP(a)
where / = (Ζ1,...,/-).
Notation 1.9. Einstein Summation Convention. Summations such as
occur often in differential geometry. It is often convenient to employ a
convention whereby summation over repeated indices is implied. This
convention is attributed to Einstein and is called the Einstein summation
convention. Using this convention, the above equation would be written
The range of the indices is either determined by context or must be explicitly
mentioned. We shall use this convention in some later chapters.
6
1. Differentiable Manifolds
Finally, we will use the notion of a commutative diagram. The reader
unfamiliar with this notion should consult Appendix A.
1.2. Topological Manifolds
We recall a few concepts from point set topology. A cover of a topological
space X is a family of sets {ΙΙβ)β^Β such that X = IJ/?^· ^ a^ the sets ΙΙβ
are open, we call it an open cover. A refinement of a cover {ΙΙβ)β^Β of
a topological space X is another cover {Vi}iej such that every set from the
second cover is contained in at least one set from the original cover. This
means that if {ΙΙβ}β^Β is the given cover of X, then a refinement may be
described as a cover {Vi}i^j together with a set map г ь-> β(ϊ) of the indexing
sets I —> В such that Vi С ί/β^) for all i. Two covers {/7а}а€л and {ΙΙβ}β^Β
have a common refinement. Indeed, we simply let I = Α χ Β and then let
Ui = υαΓ\ΙΙβ if г = (а, /3). This common refinement will obviously be open
if the two original covers were open. We say that a cover {Vi}iej of X (by
not necessarily open sets) is a locally finite cover if every point of X has
a neighborhood that intersects only a finite number of sets from the cover.
A topological space X is called paracompact if every open cover of X has
a refinement which is a locally finite open cover.
Definition 1.10. Let X be a set. A collection 05 of subsets of X is called
a basis of subsets of X if the following conditions are satisfied:
(i) X = Ub€»b;
(ii) If Bi,B2 G 55 and χ G Βι Π Б2, then there exists a set В G 55
withx G Б С Б1ПБ2.
It is a fact of elementary point set topology that if 55 is a basis of subsets
of a set X, then the family Τ of all possible unions of members of 55 is a
topology on X. In this case, we say that 55 is a basis for the topology Τ
and Τ is said to be generated by the basis. In thinking about bases for
topologies, it is useful to introduce a certain technical notion as follows: If
55 is any family of subsets of X, then we say that a subset U С X satisfies
the basis criterion with respect to 55 if for any χ G /7, there is a B G 55
with χ G В С U. Then we have the following technical lemma which is
sometimes used without explicit mention.
Lemma 1.11. If 55 is α basis of subsets of X, then the topology generated
by 55 is exactly the family of all subsets of X which satisfy the basis criterion
with respect to the family 55. (See Problem 2.)
A topological space is called second countable if its topology has a
countable basis. The space M.n with the usual topology derived from the
Euclidean distance function is second countable since we have a basis for
1.2. Topological Manifolds
7
the topology consisting of open balls with rational radii centered at points
with rational coordinates.
Definition 1.12. An η-dimensional topological manifold is a paracom-
pact Hausdorff topological space, say M, such that every point ρ G Μ is
contained in some open set Up that is homeomorphic to an open subset of
the Euclidean space M.n. Thus we say that a topological manifold is "locally
Euclidean". The integer η is referred to as the dimension of M, and we
denote it by dim(M).
Note: At first it may seem that a locally Euclidean space must be
Hausdorff, but this is not the case.
Example 1.13. Rn is trivially a topological manifold of dimension n.
Example 1.14. The unit circle S1 := {(x,y) G Ш2 : x2 + y2 = 1} is a
1-dimensional topological manifold. Indeed, the map R —> S1 given by
θ \-> (cos 0, sin Θ) has restrictions to small open sets which are homeomor-
phisms. The boundary of a square in the plane is a topological manifold
homeomorphic to the circle, and so we say that it is a topological circle.
More generally, the n-sphere
Sn := {(x\ ..., xn+l) e Rn+1 : Σ{χ1)2 = l}
is a topological manifold.
If Mi and M2 are topological manifolds of dimensions щ and 712
respectively, then Μι χ Μ2, with the product topology, is a topological
manifold of dimension щ + П2- Such a manifold is called a product
manifold. Μι χ Μ2 is locally Euclidean. Indeed, the required homeomorphisms
are constructed in the obvious way from those defined on Μχ and M2: If
φρ : Up С Μι -> νφ(ρ) С Rni and фд : Uq С M2 -» V^(q) С ШП2 are
homeomorphisms, then Up χ Uq is a neighborhood of (p, q) and we have a
homeomorphism
ΦΡ x фд : Up x Uq -> νφ(ρ) x Уф{д) С Rni x МП2,
where (фр χ фд)(х,у) := (фр (χ) ,фд (у)). That Μχ χ Μ2 is Hausdorff is
an easy exercise. Then, by Proposition B.5 of Appendix Β, Μχ χ Μ2 is
paracompact since it is metrizable.
The product manifold construction can obviously be extended to
products of a finite number of manifolds.
Example 1.15. The n-torus
Tn := S1 x S1 χ · · · χ S1 (n factors)
is a topological manifold of dimension n.
8
1. Differentiable Manifolds
Recall that a topological space is connected if it is not the disjoint
union of nonempty open subsets. A subset of a topological space is said to
be connected if it is connected with the subspace topology. If a space is not
connected, then it can be decomposed into components:
Definition 1.16. Let X be a topological space and χ G X. The
component containing x, denoted C(x), is defined to be the union of all connected
subsets of X that contain x. A subset of a topological space X is a
(connected) component if it is C(x) for some χ G X.
A topological space is called locally connected if the topology has a
basis consisting of connected sets. If a topological space is locally connected,
then the connected components of each open set (considered with its sub-
space topology) are all open. In particular, each connected component of
a locally connected space is open (and also closed). This clearly applies to
manifolds.
A continuous curve 7 : [a, b] —> Μ is said to connect a point ρ to a point
q in Μ if 7(a) = ρ and 7(b) = q. We define an equivalence relation on a
topological space Μ by declaring ρ ~ q if and only if there is a continuous
curve connecting ρ to q. The equivalence classes are called path
components, and if there is only one path component, then we say that Μ is path
connected.
Exercise 1.17. The path components of a manifold Μ are exactly the
connected components of M. Thus, a manifold is connected if and only if it
is path connected.
We shall not give many examples of topological manifolds at this time
because our main concern is with smooth manifolds defined below. We give
plenty of examples of smooth manifolds, and every smooth manifold is also
a topological manifold.
Manifolds are often defined with the requirement of second countability
because, when the manifold is Hausdorff, this condition implies paracom-
pactness. Paracompactness is important in connection with the notion of
a "partition of unity" discussed later in this book. It is known that for a
locally Euclidean Hausdorff space, paracompactness is equivalent to the
property that each connected component is second countable. Thus if a locally
Euclidean Hausdorff space has at most a countable number of components,
then paracompactness implies second countability. Proposition B.5 of
Appendix В gives a list of conditions equivalent to paracompactness for a locally
Euclidean Hausdorff space. One theorem that fails if the manifold has an
uncountable number of components is Sard's Theorem 2.34. Our approach
will be to add in the requirement of second countability when needed.
1.2. Topological Manifolds
9
In defining a topological manifold, one could allow the dimension η of
the Euclidean space to depend on the homeomorphism φ and so on the
point ρ e M. However, it is a consequence of a result of Brouwer called
"invariance of domain" that η would be constant on connected components
of M. This result is rather easy to prove if the manifold has a differentiable
structure (defined below), but is more difficult in general. We shall simply
record Brouwer's theorem:
Theorem 1.18 (Invariance of Domain). The image of an open set U С
M.n under an injective continuous map f : U —> Rn is open and f is a
homeomorphism from U to f(U). It follows that ifUcM4 is homeomorphic
to V С Rm, then m = n.
Let us define η-dimensional closed Euclidean half-space to be Hn :=
R£n>0 ;= {(a1,... ,an) e Шп : an > 0}. The boundary of Hn is дШп =
R£n~0 =: {(a1,..., an) : an = 0}. The interior of Hn is int(Hn) = Hn\OTn.
There are other half-spaces homeomorphic to ЕГ = K™n>0. Ιη fact, for a
fixed с G R and for any nonzero linear function λ G (Mn)*, we define
RJ>c = {a e Rn : λ(α) > с},
which includes W£k>c and I^fc<c. We will also use the notations Мд<с, Мд=с,
etc., whose meanings are obvious.
The space Hn is not a manifold because points on the boundary do not
have open neighborhoods homeomorphic to any open set in a Euclidean
space. However, Hn will be our model for the following generalization. A
topological manifold with boundary (of dimension n) is a paracompact
Hausdorff topological space Μ such that each point ρ e Μ is contained in
some open set Up that is homeomorphic to an open subset2 in Hn. Clearly,
we could also use charts with images in other half-spaces as mentioned above,
but we initially stick with Hn for purposes of the theoretical development.
A point ρ e Μ that is mapped to дШп under some homeomorphism φ :
Up —> V^(p) is called a boundary point and the set of all boundary points
of Μ is called the boundary of Μ and is denoted dM. The manifold's
interior consists of those points of Μ that are mapped to points of int(Hn).
It is a corollary to Brouwer's theorem that these concepts are well-defined
independently of the homeomorphism used.
In developing manifolds with boundary, a preference is often given to
Hn = Μ£η>ο· However, this is just a convenience that reduces notation. In
fact, from the point of view of "manifold orientation" developed in a later
chapter, the spaces of the form IK^i<0 have distinct and often unnoticed
advantages which we explain in due course.
2The reader should recall carefully the meaning of an open set in Hn; these certainly need
not be open in the containing Rn.
10
1. Differentiable Manifolds
Exercise 1.19. Show that int(M) DdM = 0.
Exercise 1.20. The boundary dM of an η-dimensional topological
manifold with boundary is an (n — l)-dimensional topological manifold (without
boundary).
Prom now on, by "manifold" we shall mean a manifold without boundary
unless otherwise indicated or implied by context.
Example 1.21. Let [a,b] С Μ be a closed interval. If TV is a topological
manifold of dimension η — 1, then TV χ [α, b] is an η-dimensional topological
manifold with boundary д (Ν χ [α,6]) = (Ν χ {α}) U (Ν χ {6}).
Example 1.22. If TV is a topological manifold with boundary cW, then
Ν χ R is a topological manifold with boundary and d(N xR) = dN χ Ш.
Example 1.23. The closed cube C* = {x : max^x1] ,..., |xn|} < 1} with
its subspace topology inherited from Rn is a topological manifold with
boundary. The boundary is (homeomorphic to) an (n — l)-dimensional
sphere.
Recall that the (topological) boundary of a set S in a topological space
X is defined as the set of all points ρ e X with the property that every
open set containing ρ also contains elements of both S and X\S. This is
not quite the same idea as boundary in the sense of manifold with boundary
defined above. For example, the subset of R2 defined by
S:={(x,y) gR2 : 1 <x2 + y2 < 2 or 2 < x2 + y2 < 3}
is a manifold with boundary
dS = {(ж, у) e R2 : x2 + y2 = 1 or x2 + y2 = 3}.
On the other hand, the topological boundary of S also contains the circle
given by x2 + y2 = 2.
Topological manifolds, both with or without boundary, are paracompact,
Hausdorff and hence also normal. This means that given any pair of disjoint
closed sets Fi, F2 С Μ, there are open sets U\ and Ό2 containing Fi and F2
respectively such that U\ and /У2 are also disjoint. Recall that a topological
space X (or its topology) is called metrizable if there exists a metric on
the space which induces the given topology of X. Since they are normal,
manifolds are, a fortiori, regular. According to the Urysohn metrization
theorem, every second countable regular Hausdorff space is metrizable. Now
if every connected component of a space is metrizable, then the whole space
is also metrizable by a standard trick which modifies the metric so as to
make the distance between points in a given component always less than
one while a pair of points from distinct components has distance one. Since
1.3. Charts, Atlases and Smooth Structures
11
every paracompact Hausdorff manifold has second countable components,
we see that all manifolds, as defined here, are metrizable. Thus, the reader
may as well think of manifolds as special kinds of metric spaces. For more
on manifold topology, consult [Matsu].
1.3. Charts, Atlases and Smooth Structures
In this section we introduce the notion of charts or coordinate systems. The
existence of such charts is what allows for a well-defined notion of what it
means for a function on a manifold to be differentiable and also what it
means for a map from one manifold to another to be differentiable.
Definition 1.24. Let Μ be a set. A chart on Μ is a bijection of a subset
U С М onto an open subset of some Euclidean space Rn.
We say that the chart takes values in Rn or simply that the chart is
Unvalued. A chart χ : U -> x(U) С Rn is traditionally indicated by the
pair ({7, x). If pr^ : Rn -> R is the projection onto the г-th factor given
by ргДа1,... ,αη) = аг, then хг is the function defined by хг = pr^ ο χ
and is called the г-th coordinate function for the chart ({7, x). We often
write χ = (x1,... ,xn). If ρ G Μ and (/7, x) is a chart with ρ G U and
x(p) = 0 G Mn, then we say that the chart is centered at p.
Definition 1.25. Let А = {(С/а, ха)}аед be a collection of Rn-valued charts
on a set M. We call A an Rn-valued atlas of class Cr if the following
conditions are satisfied:
(i) UaeA U* = M.
(ii) The sets of the form χα(^α Π ί/^) for α, β G A are all open in Rn.
(iii) Whenever Ua Π ΙΙβ is not empty, the map
χβ ο χ'1 : ха (С/а П ϋβ) -> Χβ (С/а Π υβ)
is a Cr diffeomorphism.
Remark 1.26. Here x^ ο χ"1 is really a shorthand for
Х/з|с/аПС//3ОХа1|ха(с/апС//3)'
but this notation is far too pedantic and cluttered for most people's tastes.
The maps x^ox"1 in the definition are called overlap maps or change
of coordinate maps. It is exactly the way we have required the overlap
maps to be diffeomorphisms that will allow us to have a well-defined and
useful notion of what it means for a function on Μ to be differentiable of
class Cr. An atlas of class Cr is also called a Cr atlas.
12
1. Differentiable Manifolds
There exist various simplifying but occasionally ambiguous notational
conventions regarding coordinates. Consider an arbitrary pair of charts
(/7, x) and (V, y) and the overlap map у о х-1 : x(J7 Π V) -> j(U DV). If we
write χ = (x1,..., xn) and у = (у1,..., yn), then we have
(1.1) y*(p) = yiox-1(x1(p),...,xn(p))
for any ρ G U Π V, which makes sense, but in the literature we also see
(1.2) уг = у\х\...,хп).
In considering this last expression, one might wonder if the x1 are functions
or numbers. But this ambiguity is sort of purposeful. For if (1.1) is true for
all ρ e U Π V\ then (1.2) is true for all (ж1,..., xn) <E x(U Π V), and so we
are unlikely to be led into error.
Definition 1.27. Two Cr atlases A\ and Λ2 on Μ are said to be
equivalent provided that A\ U Λ2 is also a Cr atlas for M. A Cr differentiable
structure on Μ is an equivalence class of Cr atlases. A C°° differentiable
structure will also be called a smooth structure.
Exercise 1.28. Show that the notion of equivalence of atlases given above
defines an equivalence relation.
The union of all the Cr atlases in an equivalence class, is itself an atlas
that is in the equivalence class. Such an atlas is called a maximal Cr
atlas since it is not properly contained in any larger atlas. Atlases obtained
in this way are precisely those that are maximal with respect to partial
ordering of atlases by inclusion. Thus every Cr atlas is contained in a
unique maximal Cr atlas which is the union of all the atlases equivalent to
it. We thereby obtain a 1-1 correspondence of the set of equivalence classes
of Cr differentiable structures and the set of maximal Cr atlases. Thus an
x(i/nv)
yoX"
y(UC)V)
Figure 1.1. Chart overlaps
1.3. Charts, Atlases and Smooth Structures
13
alternative way to define a Cr differentiable structure is as a maximal Cr
atlas. We shall use this alternative quite often without comment. As soon
as we have any Cr atlas, we have a determined Cr differentiable structure.
Indeed, we just take the equivalence class of this atlas. Alternatively, we
take the maximal atlas that contains the given atlas.
A pair of charts, say (/7, x) and (V, y), are said to be Cr-related if either
U Π V = 0 or both x(U Π V) and y({7 Π V) are open and χ о у-1 and у о χ-1
are Cr maps. Thus an atlas is just a family of mutually Cr-related charts
whose domains form a cover of the manifold. We say that a chart (/7, x)
on Μ is compatible with a Cr atlas A if A U {(/7, x)} is also a Cr atlas.
This just means that (/7, x) is Cr-related to every chart in A. The maximal
Cr atlas determined by A is exactly composed of all charts compatible with
A. We also say that a chart from the maximal atlas that gives the smooth
structure is admissible. Charts will be assumed admissible unless otherwise
indicated.
Example 1.29. The space W1 itself has an atlas consisting of the single
chart (id, Mn), where id : W1 —> Rn is just the identity map. This atlas
determines a differentiable structure.
Lemma 1.30. Let Μ be α set with α Cr structure given by an atlas A =
{(/7α,χα)}α€Α· If {U, x) and (V, у) are charts compatible with A such that
U DV φ 0, then the charts (UnV,x\UnV) and {U r\V,y\UnV) are also
compatible with A and hence are in the maximal atlas generated by A.
Furthermore, if О is an open subset ofx(U) for some compatible chart (/7, x),
then taking V = x_1 (O) we have that (V, x\v) is also a compatible chart.
Proof. The assertions of the lemma are almost obvious: If χ ο χ"1, χαοχ_1,
у ο χ"1, χα ο у-1 are all Cr diffeomorphisms, then certainly the restrictions
x\unV ° ΧαΧ^ χα ° xlt/nV У\ипУ ° X^ and xa ° V\UnV are als0' 0ne miSht
just check that the natural domains of these maps are indeed open in Rn.
For example, the domain of x|c/ny ° x^ is xa (uanunv) = xa{uanu)n
xa{Ua Π V) and both χα(ί7α Π U) and χα(ί7α Π V) are open because of what
it means for (y, U) to be compatible. The last assertion of the lemma is
equally easy to prove. D
It follows that the family of sets which are the domains of charts from
a maximal atlas provide a basis for a topology on M, which we call the
topology induced by the Cr structure on Μ or simply the manifold
topology if the Cr structure is understood. Thus the open sets are exactly
the empty set plus arbitrary unions of chart domains from the maximal
atlas. Since a Cr atlas determines a Cr structure, we will also call this the
topology induced by the atlas. This topology can be characterized as
follows: A subset V С Μ is open if and only if xQ(/7a Π V) is an open subset
14
1. Differentiable Manifolds
of Euclidean space for all charts (ί/α,χα) in any atlas {(/7α,χα)}α€Λ giving
the Cr structure.
Exercise 1.31. Show that if A\ is a subatlas of Л2? then they both induce
the same topology.
Proposition 1.32. Let Μ be α set with α Cr structure given by an atlas A.
We have the following:
(i) If for every two distinct points p,q e M, we have that either ρ
and q are respectively in disjoint chart domains Ua and Ι/β from
the atlas, or they are both in a common chart domain, then the
topology induced by the atlas is Hausdorff.
(ii) If A is countable, or has a countable subatlas, then the topology
induced by the atlas is second countable.
(iii) If the collection of chart domains {Ua}aeA from the atlas A is
such that for every fixed cto G A the set {a G A : Ua Π Uao φ
0} is at most countable, then the topology induced by the atlas is
paracompact. Thus, if this condition holds and if Μ is connected,
then the topology induced by the atlas is second countable.
Proof. We leave the proofs of (i) and (ii) as a problem, or the reader may
consult the online supplement [Lee, Jeff].
We prove (iii). Give Μ the topology induced by the atlas. It is enough
to show that each connected component has a countable basis. Thus we may
as well assume that Μ is connected. By (ii) it suffices to show that A is
countable. Let Uai be a particular chart domain from the atlas. We proceed
inductively to define a sequence of sets starting with X\ = Uai. Now given
Xn-i let Xn be the union of those chart domains Ua which intersect Xn-\-
It follows (inductively) that each Xn is a countable union of chart domains
and hence the same is true of the union X = [jnXn. By construction, if
some chart domain Ua meets X, then it is actually contained in X since to
meet Xn-i is to be contained in Xn. All that is left is to show that Μ = X.
We have reduced to the case that Μ is connected, and since X is open, it
will suffice to show that M\X is also open. If M\X = 0 we are done. If
ρ G M\X, then it is in some /7a, and as we said, Ua cannot meet X without
being contained in X. Thus it must be the case that Ua Π Χ = 0 and so
Ua С M\X. We see that M\X is open as is X. Since Μ is connected, we
conclude that Μ = X (and M\X = 0 after all). D
This leads us to a principal definition:
Definition 1.33. A differentiable manifold of class Cr is a set Μ
together with a specified Cr structure on Μ such that the topology induced
1.3. Charts, Atlases and Smooth Structures
15
by the Cr structure is Hausdorff and paracompact. If the charts are
Unvalued, then we say the manifold has dimension n. We write dim(M) for
the dimension of M.
In other words, an η-dimensional differentiable manifold of class Cr is
a pair (M, Л), where Л is a maximal (Rn-valued) Cr atlas and such that
the topology induced by the atlas makes Μ a topological manifold. A
differentiable manifold of class Cr is also referred to as a Cr manifold. A
differentiable manifold of class C°° is also called a smooth manifold and
a C°° atlas is also called a smooth atlas.3 Notice that if 0 < r\ < Г2,
then any CV2 atlas is also a CTl atlas and so any CV2 manifold is also a
Cri manifold (0 < η < Г2). In fact, it is a result of Hassler Whitney that
for r > 0, every maximal Cr atlas contains a C°° atlas. For this and other
reasons, we will be mostly concerned with the C°° case. For every integer
η > 0, the Euclidean space W1 is a smooth manifold where, as noted above,
there is an atlas whose only member is the chart (Mn,id) and this atlas
determines a maximal atlas providing the usual smooth structure for Rn.
If V is an η-dimensional real vector space, then it is a smooth manifold of
dimension η in a natural way. Indeed, for each choice of basis (ei,..., en) we
obtain a chart whose coordinate functions хг are defined so that xl(v) = аг
when ν = Σ a%ei- The overlap maps between any two such charts are linear
and hence smooth. Thus we have a smooth atlas which defines a smooth
structure.
It is important to notice that if r > 0, then a Cr manifold is much more
than merely a topological manifold. Note also that we have defined
manifolds in such a way that they are necessarily paracompact and Hausdorff.
For many purposes, neither assumption is necessary. We could have just
defined a Cr manifold to be a set with a Cr structure and with the topology
induced by the Cr structure. Proposition 1.32 tells us how to determine,
from knowledge about a given atlas, whether the topology is indeed
Hausdorff and/or paracompact. In Problem 3, we ask the reader to check that
these topological conditions hold for the examples of smooth manifolds that
we give in this chapter.
Notation 1.34. As defined, a Cr manifold is a pair (Μ, Λ). However,
we follow the tradition of using the single symbol Μ itself to denote the
differentiable manifold if the atlas is understood.
Now we come to an important point. Suppose that Μ already has some
natural or previously given topology. For example, perhaps it is already
known that Μ is a topological manifold. If Μ is given a Cr structure, then
it is important to know whether this topology is the same as the topology
3In some contexts we will just say "atlas" when we mean "smooth atlas".
16
1. Differentiable Manifolds
induced by the Cr structure. For this consideration we have the following
lemma which we ask the reader to prove in Problem 8:
Lemma 1.35. Let (M,T) be α topological space which also has a Cr atlas.
If each chart of the atlas has open domain and is a homeomorphism with
respect to this topology, then Τ will be the same as the topology induced by
the Cr structure.
A good portion of the examples of Cr manifolds that we provide will be
of the type described by this lemma. In fact, one often finds differentiable
manifolds defined as topological manifolds that have a Cr atlas consisting
of charts that are homeomorphisms. In expositions that use this alternative
definition, the fact that one can start out with a set, provide charts, and
then end up with an appropriate topology is presented as a separate lemma
(see for example [Lee, John] or [ONI]).
Exercise 1.36. Let Μ be a smooth manifold of dimension η and let ρ G M.
Show that for any r > 0 there is a chart (/7, x) with ρ G U and such that
x({7) = B(0,r) := {x G W1 : \x\ < r}. Show that for any p G U we may
further arrange that x(p) = 0.
Remark 1.37. Prom now on all manifolds in this book will be assumed
to be smooth manifolds unless otherwise indicated. Also, let us refer to an
η-dimensional smooth manifold as an "n-manifold".
As mentioned above, it is certainly possible for there to be two different
differentiable structures on the same topological manifold. For example, if
φ : Μ1 —> Μ1 is given by φ(χ) = χ3, then {(М1,^)} is a smooth atlas for
Ш1, but the resulting smooth structure is different from the usual structure
provided by the atlas {(M1, id)}. The problem is that the inverse of χ \-> x3
is not differentiable (in the usual sense) at the origin. Now we have two
differentiable structures on the line R1. Actually, although the two atlases
do give distinct differentiable structures, they are equivalent in another sense
(Definition 1.62 below).
If U is some open subset of a smooth manifold Μ with atlas Am , then
U is itself a differentiable manifold with an atlas of charts being given by all
the restrictions (Ua Π /7, Ха|[/аП[/ ) where (f7Q,xQ) G Am. We shall refer to
such an open subset U С М with this differentiable structure as an open
submanifold of M. Open subsets of Шп might seem to be very uninteresting
manifolds, but in fact they can be quite complex. For example, much can
be learned about a knot К С Ш3 by studying its complement M3\/i and the
latter is an open subset of Ш3.
We now give several examples of smooth manifolds. All of the examples
are easily seen to be Hausdorff and paracompact.
1.3. Charts, Atlases and Smooth Structures
17
Figure 1.2. Stereographic projection
Example 1.38. Consider the sphere S2 CK3. We have the usual spherical
coordinates (0,0), where φ is the polar angle measured from the north pole
(z = 1). We want the domain of this chart to be open so we restrict to
the set where 0 < φ < π and 0 < θ < 2π. We can also use projection
onto the coordinate planes as charts. For instance, let U* be the set of
all (я, у, ζ) Ε S2 such that ζ > 0. Then (ж,у, 2) ь-> (ж, у) provides a chart
/7+ —> Ш2. The various overlap maps of all of these charts are smooth (some
can be computed explicitly without much trouble). It is easy to show that
the topology induced by the atlas is the usual topology.
Example 1.39. We can also use stereographic projection to give charts
on S2. More generally, we can provide the η-sphere Sn С Mn+1 with a
smooth structure using two charts (Us^s) and (Un^n)· Here,
Us = {x = (zi,..., xn+i) € Sn : χη+ι φ 1},
UN = {x = (xi,...,xn+i) <E Sn : χη+ι φ -1}
and ips '· Us —> Mn (resp. ψ ν : /7дг —> Rn) is stereographic projection from
the north pole рдг = (0,0,..., 0,1) (resp. south pole ps = (0,0,..., 0, -1)).
Note that фз maps from the southern open set containing ps. Explicitly we
have
^s{x) = jz Ахи · · ·, *n) e Мд,
(1 -Zn+l)
<Μχ) = 7ΓΊ r(xi,...,xn) eRn.
(1 + xn+i)
Exercise 1.40. Show that ^s{Un Π Us) = ^n{Un Π Us) = Mn\ {0} and
that φδ ° ψ χ1 (у) = у I |М|2 = Ψν° ^sl{y) for all у е Шп\ {0}. Thus we
have an atlas {{Us^s)AUn^n)}· Verify that the topology induced by
this atlas is the same as the usual topology on Sn (as a subspace of Mn+1)
and that all the maps involved are smooth.
18
1. Differentiable Manifolds
If we identify R2 with C, then the overlap maps for charts on S2 from
the last example become
(1.3) φδ ο ψχ\ζ) = z~l = φΝ ο φ-\ζ)
for all ζ G C\{0}. This observation will come in handy later.
Example 1.41 (Projective spaces). The set of all lines through the origin
in R3 is denoted RP2 and is called the real projective plane. Let Uz be
the set of all lines £ G RP2 not contained in the я, у plane. Every line £ G Uz
intersects the plane ζ = 1 at exactly one point of the form (x(£),y(£),l).
We can define a bijection ψζ : Uz —> R2 by letting £ ь-> (x(£),y(£)). This is a
chart for MP2, and there are obviously two other analogous charts (Ux,(px)
and (Uy,ipy). These charts cover RP2 and form a smooth atlas since they
have smooth overlap maps.
More generally, the set RPn of all lines through the origin in Rn+1 is
called real projective η-space. We have the surjective map π : Mn+1\{0}—>
RPn given by letting π (ж) be the line through χ and the origin. We give
RPn the quotient topology where U С RPn is open if and only if π-1 (U)
is open. Also RPn is given an atlas consisting of charts of the form (СД, φι),
where
Ui = {£ G RPn : £ is not contained in the hyperplane xl = 0},
and ψί{£) is the unique (г/1,..., un) such that (г/1,... гх1"1,1, гхг,..., wn) G ^.
Once again it can be checked that the overlap maps are smooth so that
we have a smooth atlas. The topology induced by the atlas is exactly the
quotient topology, and we leave it as an exercise to show that it is both
paracompact and Hausdorff.
It is often useful to view RPn as a quotient of the sphere Sn. Consider
the map Sn —> RPn given by χ ь-> £х, where £x is the unique line through
the origin in Mn+1 which contains x. Notice that if £x = £y for x,y G 5n,
then χ = ±y. It is not hard to show that RPn is homeomorphic to 5n/~,
where χ ~ у if and only if χ = ±y. We can use this homeomorphism to
transfer the differentiable structure to 5n/~ (see Exercise 1.64). We often
identify 5n/~ with RPn.
Exercise 1.42. Show that the overlap maps for RPn are indeed smooth.
Example 1.43. In this example we consider a more general way of getting
charts for the projective space RPn. Let а : Жп —> Rn+1 be an affine map
whose image does not contain the origin. Thus α has the form a(x) =
Lx + 6, where L : Rn -» Rn+1 is linear and b G Rn+1 is nonzero. Let
π : Mn+1\{0} —> RPn be the projection defined above. The composition
π ο α can be easily shown to be a homeomorphism onto its image, and we
1.3. Charts, Atlases and Smooth Structures
19
call this type of map an affine parametrization. The inverses of these maps
form charts for a smooth atlas. The charts described in the last example are
essentially special cases of these charts and give the same smooth structure.
Notation 1.44 (Homogeneous coordinates). For (xi,... , xn+i) £ Kn+1,
let [xi,... , xn+i] denote the unique / G ШРп such that the line / contains
the point (xi,... ,xn+i). The numbers (xi,...,xn+i) are said to provide
homogeneous coordinates for / because [λχι,..., λχη+ι] = [χι,..., xn+i]
for any nonzero λ G Ш. In terms of homogeneous coordinates, the chart map
ψι : Ui —> Шп is given by
(/?;([xi,...,xn+i]) = (xix~ , ...,l,...,xn+ix~ ),
where the caret symbol л means we have omitted the 1 in the г-th slot to
get an element of Шп.
Example 1.45. By analogy with the real case we can construct the complex
projective n-space CPn. As a set, CPn is the family of all 1-dimensional
complex subspaces of Cn+1 (each of these has real dimension 2). In tight
analogy with the real case, CPn can be given an atlas consisting of charts
of the form (C/i, φι), where
Ui = {£ G CPn : I is not contained in the complex hyperplane zl = 0}
and ψι{ί) is the unique (z1,... ,zn) such that (21,... zl~l, 1, zl..., zn) G L
Here ψι : Ui —> Cn = R2n and so CPn is a manifold of (real) dimension
2n. Homogeneous coordinate notation [21,..., zn+i] is defined as in the real
case, but now the multiplier λ is complex.
Exercise 1.46. In reference to the last example, compute the overlap maps
4>i ο φ'1 : Ui Π Uj -» Cn. For CP1, show that Ui Π U2 = C\{0}, and
that ψ2 ο ψϊι(ζ) = z~l = (/?! ο ψ21(ζ) for ζ G C\{0}. Show also that if
we define ψι : U\ —> С by φι(£) = φι{£), where the bar denotes complex
conjugation, then {(f/i, <£i), (^2,^2)} is an atlas for CP1 giving the same
smooth structure as before. Show that
ψ2 ° ψ\1{ζ) = Z~l =ψ\° Ψ21(Ζ) f°r Z ^ C\ {0} .
Notice that with the atlas {(/7i,v?i), (f/2,^2) } f°r CP1 from the last
example, we have the same overlap maps as for S2 (see (1.3)). This suggests
that CP1 is diffeomorphic to S2. Let us construct a diffeomorphism S2 —>
CP1. For each χ = (х1,х2,хз) G S2 with x3 ф -1, let z(x) := jf^ + xf^i,
and for each χ = (хьХ2,^з) G S2 with X3 ф 1 let w(x) := j^ ϊ^Γ^·
Define / by
f(„\ - / [*(*)> 4 if ^3^-1,
/W"\ [1,Цх)] ifx3^l.
20
1. Differentiable Manifolds
Fixed
Figure 1.3. Double pendulum
Using the fact that l — x\ = x\ + x%, one finds that for χ = (χι, £2> #з) £ S2
with -1 < £3 < 1, we have w(x) = z(x)~l. It follows that for such x, we
have [z(x), 1] = [1,ги(ж)]. This means that / is well-defined.
Exercise 1.47. Show that the map /
diffeomorphism.
CP1 defined above is a
Exercise 1.48. Show that RP1 is diffeomorphic to S1.
Example 1.49. The set of all m χ η real matrices Mmxn(IR) is an mn-
manifold. We only need one chart since it is clear that Mmxn(IR) is in
one-to-one correspondence with M.mn by the map [aij] ь-> (ац, αΐ2,..., amn).
Also, the set of all nonsingular matrices GL(n, Ж) is an open submanifold of
Mnxn^Rn\
If we have two manifolds M\ and M2 of dimensions n\ and 712
respectively, we can form the topological Cartesian product M\ χ Μ2. We may
give M\ χ Μ2 a differentiable structure in the following way: Let Лмх and
Am2 be atlases for M\ and M2. Take as charts on M\ χ Μ2 the maps of the
form
xxy:Uax F7->Mni x»n2,
where (/7,x) is a chart from Лмг and (V,y) a chart from Лм2- This gives
Μι χ Μ2 an atlas called the product atlas, which induces a maximal atlas and
hence a differentiable structure. With this product differentiable structure,
Μι χ Μ2 is called a product manifold. The product of several manifolds is
also possible by an obvious iteration. The topology induced by the product
atlas is the product topology, and so the underlying topological manifold is
the product topological manifold discussed earlier.
Example 1.50. The circle S1 is a 1-manifold, and hence so is the product
T2 = Sl χ 51, which is a torus. The set of all configurations of a double
pendulum constrained to a plane and where the arms are free to swing past
each other can be taken to be modeled by T2 = S1 χ S1. See Figure 1.3.
1.3. Charts, Atlases and Smooth Structures
21
Example 1.51. For any smooth manifold Μ we can construct the
"cylinder" Μ χ 7, where I = (a, b) is some open interval in R.
We now discuss an interesting class of examples. Let G (n, k) denote
the set of /c-dimensional subspaces of W1. We will exhibit a natural dif-
ferentiable structure on this set. The idea is the following: An alternative
way of defining the points of projective space is as equivalence classes of
η-tuples (v\...,vn) GRn\{0}, where (v1,... ,vn) ~ (Аг;1,... ,Аг;п) for any
nonzero λ. This is clearly just a way of specifying a line through the origin.
Generalizing, we shall represent a /.-plane as an η χ A: matrix whose column
vectors span the /c-plane. Thus we are putting an equivalence relation on
the set of η χ к matrices where A ~ Ag for any nonsingular к х к matrix
д. Let М^хк be the set of η χ к matrices with rank к < η (full rank).
Two matrices from M^k are equivalent exactly if their columns span the
same /c-dimensional subspace. Thus the set G(/c,n) := M^k/^ of
equivalence classes is in one-to-one correspondence with the set of k-dimensional
subspaces of Rn.
Let U be the set of all [A] G G(/c, n) such that any representative A has
its first к rows linearly independent. This property is independent of the
representative A of the equivalence class [A], and so U is a well-defined set.
Now every element [A] G U С G(k,n) is an equivalence class that has a
unique member Ao of the form
Г hxk 1
L z \
which is obtained by Gaussian column reduction. Thus we have a map on
U defined by Φ : [А] ь-> Ζ G M(n_/c)x/c ^ Rk(n~k). We wish to cover G(k,n)
with sets similar to U and define similar maps. Consider the set Uilt„ik
of all [A] G G(fc,n) such that any representative A has the property that
the к rows indexed by z*i,..., %k are linearly independent. The permutation
that puts the к rows indexed by ii,..., %k into the positions 1,..., к without
changing the relative order of the remaining rows induces an obvious bijec-
tion aib..ifcfrom f/ib..ifc onto U = /7ι.../~. We now have maps Ф^...^ : ^u...ifc —^
№(n-k)xk — Шк(п~к>} given by composition Φ^...^ := Φοσ^...^. These maps
form an atlas {(/7^...^, ^h...ik)} f°r G(k,n) that gives it the structure of a
smooth manifold called the Grassmann manifold of real /c-planes in Rn.
The topology induced by the atlas is the same as the quotient topology, and
one can check that this topology is Hausdorff and paracompact.
We have defined Cr manifold for 0 < r < oo for r an integer or oo.
We can also define Cu manifolds (analytic manifolds) by requiring that the
charts are related by analytic maps. This means that the overlaps maps have
component functions that may be expressed as convergent powers series in
22
1. Differentiable Manifolds
a neighborhood of any point in their domains. For convenience, we agree to
take oo < ω.
1.4. Smooth Maps and Diffeomorphisms
Definition 1.52. Let Μ and TV be smooth manifolds with corresponding
maximal atlases Am and An- We say that a map / : Μ —> N is of class Cr
(or r-times continuously differentiable) at ρ G Μ if there exists a chart
(V, y) from An with f(p) G V, and a chart (C7, x) from Am with ρ G /7, such
that / (U) С V and such that у о / ο χ-1 is of class Cr. If / is of class Cr
at every point ρ G M, then we say that / is of class Cr (or that / is a Cr
map). Maps of class C°° are called smooth maps.
Exercise 1.53. Show that a Cr map is continuous. [Hint: Consider
compositions у-1 о (у о / ο χ-1) о χ.]
Exercise 1.54. Show that a composition of Cr maps is a Cr map.
The family of Cr manifolds together with the family of smooth maps
determines a category called the Cr category (see Appendix A). The C°°
category is called the smooth category. Smooth structures are often tailor
made so that certain maps are smooth. For example, given smooth manifolds
M\ and M2, the smooth structure on a product manifold M\ χ Μ<2 is designed
to make the projection maps onto M\ and M2 smooth.
Even when dealing with smooth manifolds, we may still be interested in
maps which are only of class Cr for some r < 00. This is especially so when
one wants to do analysis on smooth manifolds. In fact, one could define
what it means for a map to be Lebesgue measurable in a similar way. It is
obvious from the way we have formulated the definition that the property
of being of class Cr is a local property.
Let / : Μ —> N be a map and suppose that (C7, x) and (V, y) are
admissible charts for Μ and TV respectively. If f~l(V) Π U is not empty, then we
have a composition
yofox-1:x(f-1(V)nU)-+y(V).
Maps of this form are called the local representative maps for /. Notice
that if / is continuous, then f~l(V) Π U is open. Definition 1.52 does
not start out with the assumption that / is continuous, but is constructed
carefully so as to imply that a function that is of class Cr (at a point)
according to the definition is automatically continuous (at the point). But
if / is known to be continuous, then we may check Cr differentiability using
representative maps with respect to atlases that are not necessarily maximal:
Proposition 1.55. Let {(Ua,xa)}aeA and {(V^,y^)}^€^ be (not
necessarily maximal) Cr atlases for Μ and N respectively. A continuous map
1.4. Smooth Maps and Diffeomorphisms
23
f : Μ -^ Ν is of class Cr if for each a and β, the representative map
у β ο f ο χ"1 is Cr on its domain xQ (/_1(V#) Π Ua) ·
Proof. Suppose that a continuous / is given and that all the representative
maps у β о f о χ"1 are Cr. Let ρ G Μ and choose (C/Q, xQ) and (V/?, у β) with
ρ G Ua and f(p) G Vfi. Letting U := /_1(^)nf/a, we have a chart (C7, ха\ц)
with ρ G U and f(U) С V£ such that у о / ο χ\~λ is Cr. Thus / is Cr at ρ
by definition. Since ρ was arbitrary, we see that / is of class Cr. D
If / is continuous, then the condition that / be Cr at ρ G Μ for r > 0
can be seen to be equivalent to the condition that for some (and hence every)
choice of charts (/7, x) from Am and (V,y) from An such that ρ G U and
f(p) G V, the map
yo/ox-^xtf-^tOntO-^yOO
is Cr. Note the use of the phrase "and hence every" above. The point is
that if we choose another pair of charts (x;, U') and (y;, V') with ρ G U' and
f(p) G V1', then /ο/οχ/_1 must be Cr on some open neighborhood of x;(p)
if and only if у о / ο χ-1 is Cr on some neighborhood4 of x(p). This is true
because the overlap maps x'ox"1 and у' о у-1 are Cr diffeomorphisms (the
chain rule is at work here of course). Without worrying about domains, the
point is that
У о/ox'"1
= у' ° (у-1 ° у) ° / ° (χ_1 °χ) °χ/_1
= (у/оУ_1) ° (У0/0Х_1) ° (x'ox-1)" .
Now the reader should be able to see quite clearly why we required overlap
maps to be diffeomorphisms.
A representative map / = у о / ο χ-1 is defined on an open subset of Rn
where η = dim(M). If dim(TV) = A:, then / = (/*,..., fk) and each fl is a
function of η variables. If we denote generic points in W1 as (u1,..., un), and
those inIRk as (г;1,... ,ι^), then we may write vl = fl(ul,... ,wn), 1 < г < к.
It is also common and sometimes psychologically helpful to simply write
yl — fl(x1,... ,xn). The bars over the /'s are also sometimes dropped.
Another common way to indicate у о / οχ-1 is with the notation fvu which
is very suggestive and tempting, but it has a slight logical defect since there
may be many charts with domain U and many charts with domain V.
Exercise 1.56. Consider the map π : S2 —> RP2 given by taking the point
(#,y, z) to the line through this point. Using an atlas on each of these
manifolds such as the atlases introduced previously, show that π is smooth.
Recall that our convention is that a neighborhood is assumed to be open unless indicated.
24
1. Differentiable Manifolds
(At least check one of the representative maps with respect to a chart on S2
and a chart on MP2.)
As a special case of the above, we note that a function / : Μ —> R (resp.
C) is Cr differentiable at ρ G Μ if and only if it is continuous and
/ ο χ"1 : X(J7) -» R (resp. C)
is Cr-differentiable for some admissible chart (C/, x) with ρ G U. And, / is
of class Cr if it is of class Cr at every p. The set of all Cr maps Μ —> N is
denoted Cr(M,N) and Cr(M,R) is abbreviated to Cr(M). Both Cr(M,R)
and Cr(M, C) are rings and also algebras over the respective fields R and С
(Definition in Appendix D). The addition, scaling, and multiplication are
defined pointwise so that (/ + g)(p) := f(p) + g(p), etc.
Definition 1.57. Let ({7, x) be a chart on an η-manifold Μ with ρ G U.
We write χ = (χ1,... ,xn) as usual. For / G (7Χ(Μ), define a function -^
on U by
Ά(ρ) := lim
дхг v J h-+o
f о χ-V, ...,a4V..,aV/ox"V,...,an)
h
where x(p) = (a1,..., an). In other words,
^(p):=ft(/ox-l)(x(P)) = ^^)(x(P)),
where (ti1,... ,wn) denotes the standard coordinates on Mn.
Recall that D% is the notation for i-th partial derivative with respect to
the decomposition W1 = R χ · · · χ R . Thus if g : U С R -»R is differentiable
at α G IR, then Dig(a) : IR —»IR is a linear map but is often identified with
the single entry dig (a) of the lxl matrix that represents it with respect
to the standard basis on R. Thus one sometimes sees the definition above
written as
giP^Atfox-1)^)).
If / is a Cr function, then df/дх1 is clearly Cr~l. Notice also that /
really only needs to be defined in a neighborhood of ρ and differentiable at
ρ for the expression q^(p) to make sense. This definition makes precise the
notation that is often encountered in calculus courses. For example, if Τ
is the "temperature" on a sphere 52, then Τ takes as arguments points ρ
on S2. On the other hand, using spherical coordinates, we often consider
дТ/дф and дТ/дв as being defined on S2 rather than on some open set in
a "0, #-space".
1.4. Smooth Maps and Diffeomorphisms
25
Finally, notice that if / and g are C1 and defined at least on the domain
of the chart (/7, x), then we easily obtain that on U
d(af + bg) df , dg c , ^
v i . } = a^- + b-^- for any a, b e R,
ox1 ox1 ox1
and
Let (/7, x) and (V,y) charts on an η-manifold with pG U (~)V. Then it is
easy to check using the usual chain rule and the definitions above that for
any smooth function / defined at least on a neighborhood of ρ we have the
following version of the chain rule:
df , ч v-^ df / ч dxi , ч
A map / which is defined only on some proper open subset of a manifold
is said to be Cr if it is Cr as a map of the corresponding open submanifold,
but this is again just to say that it is Cr at each point in the open set. We
shall often need to consider maps that are defined on subsets S С М that
are not necessarily open.
Definition 1.58. Let S be an arbitrary subset of a smooth manifold M.
Let / : S —> N be a continuous map where TV is a smooth manifold. The
map / is said to be Cr if for every s G S there is an open set О С М
containing s and a map / that is Cr on О and such that /
= /.
sno
In a later exercise we ask the reader to show that a function / with
domain S is smooth if and only if it has a smooth extension to some open
set containing all of S. In particular, a curve defined on a closed interval
[a, b] is smooth if it has a smooth extension to an open interval containing
[a,b].
We already have the notion of a diffeomorphism between open sets of
some Euclidean space M.n. We are now in a position to extend this notion
to the realm of smooth manifolds.
Definition 1.59. Let Μ and iV be smooth (or Cr) manifolds. A homeo-
morphism f : Μ —> N such that / and f~l are Cr differentiable with r > 1
is called a Cr diffeomorphism. In the case r — oo, we shorten C°°
diffeomorphism to just diffeomorphism. The set of all Cr diffeomorphisms of a
manifold Μ onto itself is a group under the operation of composition. This
group is denoted Diffr(M). In the case r = oo, we simply write Diff(M)
and refer to it as the diffeomorphism group of M.
26
1. Differentiable Manifolds
We will use the convention that Diff (M) denotes the group of homeo-
morphisms of Μ onto itself. Also, it should be pointed out that if we refer
to a map between open subsets of manifolds as being a Cr diffeomorphism,
we mean that the map is a Cr diffeomorphism of the corresponding open
submanifolds.
Example 1.60. The map re : S2 —> S2 given by
r$(x, y, z) = (xcos# — у sin 0, χ sin 0 + ycosO, z)
for x2 + y2 + z2 = 1 is a diffeomorphism.
Exercise 1.61. Let 0 < θ < 2π. Consider the map / : S2 -» S2 given by
Λ(χ,ϊ/, ζ) = (xcos((l - ζ2)θ) - j/sin((l - *2)0),xsin((l - ζ2)θ) + j/cos((l -
ζ2)θ),ζ). Is this map a diffeomorphism? Try to picture this map.
Definition 1.62. Cr manifolds Μ and TV will be called (Cr) diffeomor-
phic and then said to be in the same diffeomorphism class if and only if
there is a Cr diffeomorphism f : Μ -* N.
Exercise 1.63. Let Μ and TV be smooth manifolds with respective maximal
atlases Am and An- Show that a bijection / : Μ —> N is a diffeomorphism
if and only if the following condition holds:
(i/,y) € An if and only if (ГЧ^.У ° /) € AM.
Exercise 1.64. Show that if Μ is a Cr manifold and φ : Μ —> X is
any bijection, then there is a unique Cr structure on X such that φ is a
diffeomorphism. This process is called a transfer of structure. If X is a
topological space and φ is a homeomorphism, then the topology induced by
the transferred structure is the original topology.
For another example, consider the famous Cantor set С С [0,1] С Ш.
Consider R as a coordinate axis subspace of R2 set Mc := M2\C It can
be shown that Mc is diffeomorphic to a surface suggested by Figure 1.4.
Once again we see that open sets in a Euclidean space can have interesting
differential topology.
In the definition of diffeomorphism, we have suppressed explicit reference
to the maximal atlases, but note that whether or not a map is differentiable
(Cr or smooth) essentially involves the choice of differentiable structures on
the manifolds. Recall that we can put more than one differentiable structure
on R by using the function χ as a chart. This generalizes in the obvious
way: The map ε : (χ1, χ2,..., χη) ь-> ((я1)3, χ2,..., χη) is a chart for Rn,
but is not C°°-related with the standard (identity) chart. It is globally
defined and so provides an atlas that induces the usual topology again, but
the resulting maximal atlas is different! Thus we seem to have two smooth
manifolds (Mn, A\) and (Mn, A2) both with the same underlying topological
1.4. Smooth Maps and Diffeomorphisms
27
Figure 1.4. Interesting surface
space. Indeed, this is true. Technically, they are different. But they are
equivalent and therefore the same in another sense. Namely, they are diffeo-
morphic via the map ε. So it may be that the same underlying topological
space Μ carries two different differentiate structures, and so we really have
two differentiable manifolds with the same underlying set. It remains to ask
whether they are nevertheless diffeomorphic. It is an interesting question
whether a given topological manifold can carry differentiable structures that
are not diffeomorphic. It has been shown that there are 28 pairwise non-
diffeomorphic smooth structures on the topological space S7 and more than
16 million on 531. Each Rk for к ф 4 has only one diffeomorphism class
compatible with the usual topology. On the other hand, it is a deep result
that there exist infinitely many truly different (nondiffeomorphic)
differentiable structures on Ш4. The existence of exotic differentiable structures on
R4 follows from the results of [Donaldson] and [Freedman]. The reader
ought to be wondering what is so special about dimension four. Note that
when we mention Μ4, 57, 531, etc. as smooth manifolds, we shall normally
assume the usual smooth structures unless otherwise indicated.
Definition 1.65. Let TV and Μ be smooth manifolds of the same dimension.
A map / : Μ —> N is called a local diffeomorphism if and only if every point
ρ G Μ is contained in an open subset U С Μ such that /1^ : U —> f(U) is
a diffeomorphism onto an open subset of N. For Cr manifolds, a Cr local
diffeomorphism is defined similarly.
Example 1.66. The map π : 52 —>· RP2 given by taking the point (x, y, z)
to the line through this point and the origin is a local diffeomorphism, but
is not a diffeomorphism since it is 2-1 rather than 1-1.
Example 1.67. The map (x,j/) ь-> (x/z(x,y),y/z(x,y)), where
28
1. Differentiable Manifolds
is a diffeomorphism from the open disk Б(0,1) = {(я, у) : χ2 + у2 < 1} onto
the whole plane. Thus Б(0,1) and R2 are diffeomorphic and in this sense
are the "same" differentiable manifold.
Sometimes it is only important how maps behave near a certain point
Let Μ and TV be smooth manifolds and consider the set £(p, Μ, Ν) of all
smooth maps into TV which are defined on some open neighborhood of the
fixed point ρ G M. Thus,
S(p,M,N):= (J С°°(/У,Л0,
where λίρ denotes the set of all open neighborhoods of ρ G M. On this set
we define the equivalence relation where / and g are equivalent at ρ if and
only if they agree on a neighborhood of p. The equivalence class of / is
denoted [/], or by [f]p if the point in question needs to be made clear. The
set of equivalence classes S(p, M, N)/~ is denoted C™(M, N).
Definition 1.68. Elements of C™(M, N) are called germs, and if / and g
are in the same equivalence class, we write f ~p g and we say that / and g
have the same germ at p.
The value of a germ at ρ is well-defined by [f](p) = f(p)· Taking N = R
we see that C£°(M, R) is a commutative R-algebra if we make the definitions
a[f]+b\g]:=[af + bg] for a, b G M,
The C-algebra of complex-valued germs C£°(M, C) is defined similarly.
1.5. Cut-off Functions and Partitions of Unity
There is a special and extremely useful kind of function called a bump
function or cut-off function, which we now take the opportunity to introduce.
Recall that given a topological space X, the support, supp(/), of a function
/ : X —> Ш is the closure of the subset on which / takes nonzero values.
The same definition applies for vector space-valued functions / : X —> V. It
is a standard fact that there exist smooth functions defined on Ш that have
compact support. For example, we have the smooth function Φ : R —> Ш
defined by
m = I e~m-x2) for 1*1 < i.
\ 0 otherwise.
Lemma 1.69 (Existence of cut-off functions). Let Μ be a smooth manifold.
Let К be a compact subset of Μ and О an open set containing K. There
exists a smooth function β on Μ that is identically equal to 1 on K, takes
values in the interval [0,1], and has compact support in O.
1.5. Cut-off Functions and Partitions of Unity
29
Proof. Special case 1: Assume that Μ = Rn and that О = Б(0, R) and
К = Б(0, r) for 0 < г < R. In this case we may take
.M f\*9(t)dt
where
(i) = / e-^)"^^-^"1 if r < ί < Д,
\ 0 otherwise.
It is an exercise in calculus to show that g is a smooth function and thus
that φ is smooth. Clearly, a composition with a translation gives the result
for a ball centered at an arbitrary point.
Special case 2: Assume again that Μ = Rn. Let К С О be as in
the hypotheses. For each point ρ G К let Up be an open ball centered at
ρ and contained in O. Let Kp be the closed ball centered at ρ of half the
radius of Up. The interiors of the Kps form an open cover for К and so
by compactness we can reduce to a finite subcover. Thus we have a finite
family {Κι} of closed balls of various radii such that К С U^> and w^h
corresponding concentric open balls С/г- С О. For each Щ, let φι be the
corresponding function provided in the proof of Special case 1 so that φι
has support in U{ and is identically 1 on^. Examination of the following
function will convince the reader that it is well-defined and provides the
needed cut-off function:
/3(х) = 1-Ц(1-&(х)).
г
General case: From the second special case above it is clear that we
have the result if К is contained in the domain U of a chart (/7, x). If
К is not contained in such a chart, then we may take a finite number of
charts (J7i, xi),..., (£4, x/J and compact sets K\,..., K^ with К С Ui=i^b
Ki С C/i, and [jUi С О. Now let φι be identically 1 on K{ and identically 0
on Щ = M\Ui. Then the function β we are looking for is given by
к
/3 = l-J](l-fc). D
г=1
Let [/] e C£°(M,IR) (or e C™(MX)) and let / be a representative of
the equivalence class [/]. We can find an open set U containing ρ such that
U is compact and contained in the domain of /. If β is a cut-off function that
is identically equal to 1 on C/, and has support inside the domain of /, then
β/ is smooth and it can be extended to a globally defined smooth function
that is zero outside of the domain of /. Denote this extended function by
(/3/)ext. Then (/?/)ext G [/] (usually, the extended function is just written
30
1. Differentiable Manifolds
as /3/). Thus every element of C™(M, R) has a representative in C°°{M, R).
In short, each germ has a global representative. (The word "global" means
defined on, or referring to, the whole manifold.)
A partition of unity is a technical tool that can help one piece together
locally defined smooth objects with some desirable properties to obtain a
globally defined object that also has the desired properties. For example,
we will use this tool to show that on any (paracompact) smooth manifold
there exists a Riemannian metric tensor. As we shall see, the metric tensor
is the basic object whose existence allows the introduction of notions such
as length and volume.
Definition 1.70. A partition of unity on a smooth manifold Μ is a
collection {φα}αβλ of smooth functions on Μ such that
(i) 0 < ψα < 1 for all a.
(ii) The collection of supports {supp((^a)}aG^ is locally finite; that
is, each point ρ οι Μ has a neighborhood Wp such that Wp Π
supp((/?a) = 0 for all but a finite number of a G A.
(iii) ΣαβΑΨ<*(ρ) = 1 f°r all ρ G Μ (this sum has only finitely many
nonzero terms by (ii)).
If О = {Оа}аел is an open cover of Μ and supp((/?a) С Оа for each
qGA, then we say that {φα}αβΑ is a partition of unity subordinate to
О = {Оа}аел-
Remark 1.71. Let U = {Ua}aeA be a cover of Μ and suppose that W =
{νΫβ}β£Β is a refinement of U. If {φβ}β^Β is a partition of unity subordinate
to W, then we may obtain a partition of unity {^a} subordinate to U.
Indeed, if / : В —> A is such that W$ С Uf^ for every /3gB, then we may
let ψα :=Σ/3€/-ΐ(α)#·
Our definition of a smooth manifold Μ includes the requirement that
Μ be paracompact (and Hausdorff). Paracompact Hausdorff spaces are
normal spaces, but the following theorem would be true for a normal
locally Euclidean space with smooth structure even without the assumption
of paracompactness. The reason is that we explicitly assume the local finite-
ness of the cover. For this reason we put the word "normal" in parentheses
as a pedagogical device.
Theorem 1.72. Let Μ be a (normal) smooth manifold and {Ua}aeA be a
locally finite cover of M. If each Ua has compact closure, then there is a
partition of unity {ψα}αβΑ subordinate to {Ua}aeA-
Proof. We shall use a well-known result about normal spaces sometimes
called the "shrinking lemma". Namely, if {Ua}a^A is a locally finite (and
1.6. Coverings and Discrete Groups
31
hence "point finite") cover of a normal space M, then there exists another
cover {V^Jae^ of Μ such that Va С Ua. This is Theorem B.4 proved in
Appendix B.
We do this to our cover and then notice that since each Ua has compact
closure, each Va is compact. We apply Lemma 1.69 to obtain nonnegative
smooth functions ψα such that supp^a С Ua and ψα\γ = 1· Let φ :=
ΣαβΑ^α and notice that for each ρ G M, the sum ^2ае^Фа{р) is a finite
sum and ψ(ρ) > 0. Let ψα := ψα/Ψ· It is now easy to check that {φα}αβλ
is the desired partition of unity. D
If we use the paracompactness assumption, then we can show that there
exists a partition of unity that is subordinate to any given cover.
Theorem 1.73. Let Μ be a (paracompact) smooth manifold and {Ua}a^A
a cover of M. Then there is a partition of unity {φα}αβΑ subordinate to
{UafaeA-
Proof. By Remark 1.71 and the fact that Μ is locally compact we may
assume without loss of generality that each Ua has compact closure. Then
since Μ is paracompact, we may find a locally finite refinement of {Ua}a^A
which we denote by {VJ}^/· Now use the previous theorem to get a partition
of unity subordinate to {Κ}^/· Finally use remark 1.71 one more time to
get a partition of unity subordinate to {Ua}aeA- Π
Exercise 1.74. Show that if a function is smooth on an arbitrary set S С М
as defined earlier, then it has a smooth extension to an open set that contains
S.
Now that we have established the existence of partitions of unity we
may show that the analogue of Lemma 1.69 works with К closed but not
necessarily compact:
Exercise 1.75. Let Μ be a smooth manifold. Let К be a closed subset
of Μ and О an open set containing K. Show that there exists a smooth
function β on Μ that is identically equal to 1 on Κ, takes values in the
interval [0,1], and has compact support in O.
1.6. Coverings and Discrete Groups
1.6.1. Covering spaces and the fundamental group. In this section,
and later when we study fiber bundles, many of the results are interesting
and true in either the purely topological category or in the smooth category.
Let us agree that a C° manifold is simply a topological manifold. Thus all
32
1. Differentiable Manifolds
relevant maps in this section are to be Cr, where if r = 0 we just mean
continuous and then only require that the spaces be sufficiently nice topological
spaces. Also, "C° diffeomorphism" just means homeomorphism.
In the definition of path connectedness and path component given before,
we used continuous paths, but it is not hard to show that if two points on a
smooth manifold can be connected by a continuous path, then they can be
connected by a smooth path. Thus the notion of path component remains
unchanged by the use of smooth paths.
Definition 1.76. Let /0 : X -» Υ and /i : X -» Υ be Cr maps. A Cr
homotopy from /o to /i is a Cr map Я:1х[0,1]-)7 such that
#(x, 0) = /o(x) and
tf(x,l) = /i(x)
for all x. If there exists such a Cr homotopy, we then say that /o is Cr
QT
homotopic to f\ and write /q ~ /i. If А С X is a closed subset and if
H(a,s) = /ο(α) = /ι(α) for all α G A and all s G [0,1], then we say that
cr
/o is Cr homotopic to /i relative to A and we write /ο ~ /ι (rel A). The
map Η is called a Cr homotopy.
In the above definition, the condition that Η : Χ χ [0,1] —> Υ be Cr
for r > 0 can be understood by considering Χ χ [0,1] as a subset of Χ χ R.
Homotopy is obviously an equivalence relation.
Exercise 1.77. Show that /o : X —> Υ and /i : X —> У are Cr homotopic
(rel A) if and only if there exists a Cr map Η : Χ χ Ш -+ Υ such that
#(x, s) = fo(x) for all χ and 5 < 0, #(x, s) = fi{x) for all я and s > 1 and
#(a, s) = /o(a) = /i(a) for all α G A and all 5.
At first it may seem that there could be a big difference between C°°
and C° homotopies, but if all the spaces involved are smooth manifolds,
then the difference is not big at all. In fact, we have the following theorems
which we merely state. Proofs may be found in [Lee, John].
Theorem 1.78. If f : Μ -ϊ Ν is α continuous map on smooth manifolds,
then f is homotopic to a smooth map /0 : Μ —> N. If the continuous map
f : Μ -+ N is smooth on a closed subset A, then it can be arranged that
/ ~ fo(relA).
Theorem 1.79. If fo : Μ -ϊ Ν and f\ : Μ —> N are homotopic smooth
maps, then they are smoothly homotopic. If /0 is homotopic to f\ relative
to a closed subset A, then /0 is smoothly homotopic to f\ relative to A.
Because of these last two theorems, we will usually simply write / — /0
cr
instead of / ~ /0, the value of r being of little significance in this setting.
1.6. Coverings and Discrete Groups
33
Figure 1.5. Coverings of a circle
Definition 1.80. Let Μ and Μ be Cr manifolds. A surjective Cr map
ρ : Μ —> Μ is called a Cr covering map if every point ρ G Μ has an
open connected neighborhood U such that each connected component Ui of
p~1(U) is Cr diffeomorphic to U via the restrictions p\fj : C/j —> £/. In
this case, we say that /7 is evenly covered by ρ (or by the sets Ui). The
triple (M, p, M) is called a covering space. We also refer to the space Μ
(somewhat informally) as a covering space for M.
Example 1.81. The map R —> S1 given by t ь-> elt is a covering. The set of
points {elt : θ — π < t < θ + π} is an open set evenly covered by the intervals
In in the real line given by In := (Θ — π + 2πη, θ + π + 2πη) for η G Ζ.
Exercise 1.82. Explain why the map (—2π,2π) —>· 51 given by t ь-> ezi is
noi a covering map.
Definition 1.83. A continuous map / is said to be proper if f~l(K) is
compact whenever К is compact.
Exercise 1.84. Show that a Cr proper map between connected smooth
manifolds is a smooth covering map if and only if it is a local Cr diffeomor-
phism.
The set of all Cr covering spaces is the set of objects of a category. A
morphism between Cr covering spaces, say (Mi, pi, Mi) and (M2, p2, M2),
is a pair of Cr maps (/, /) such that the following diagram commutes:
Mi
/
Pi
M2
P2
f
M2
Mi-
This means that / ο ρ1 = p2 ° f ■ Similarly, the coverings of a fixed space Μ
are the objects of a category where the morphisms are maps Φ : Mi —> M2
34
1. Differentiable Manifolds
that make the following diagram commute:
~ Φ ~
Mi >■ M2
meaning that pi = P20 Φ· Let (Μ, ρ, Μ) be a Cr covering space. The Cr
diffeomorphisms Φ that are automorphisms in the above category, that is,
diffeomorphisms for which ρ = ρ ο Φ, are called deck transformations
or covering transformations. The set of deck transformations is a group
of Cr diffeomorphisms of Μ called the deck transformation group, which
we denote by Deck(p) or sometimes by Deck(M). A deck transformation
permutes the elements of each fiber p~l (p). In fact, it is not hard to see that
if U С Μ is evenly covered, then Φ permutes the connected components of
Proposition 1.85. // ρ : Μ —> Μ is α Cr covering тар with Μ path
connected, then the cardinality of p~l(p) is independent of p. In the latter
case, the cardinality of p~l(p) is called the multiplicity of the covering.
Proof. Fix a cardinal number k. Let B^ be the set of all points such that
p~l(p) has cardinality k. For any fixed ρ G M, there is a connected
neighborhood Uv that is evenly covered, and it is easy to see that the cardinality
of p~l(q) is the same for all points q G Up. Using this, it is easy to show
that Β^ is both open and closed and so, since Μ is connected, B^ is either
empty or all of M. D
Since we are mainly interested in the smooth case, the following theorem
is quite useful:
Theorem 1.86. Let Μ be a Cr manifold with r > 0 and suppose that
ρ : Μ —> Μ is a C° covering map with Μ paracompact. Then there exists
a (unique) Cr structure on Μ making ρ a Cr covering map.
Proof. Choose an atlas {(/7α,χα)}α€Α such that each domain Ua is small
enough to be evenly covered by p. Thus we have that p~l(Ua) is a disjoint
union of open sets /71 with each restriction ρ\Ίμ a homeomorphism. We
now construct charts on Μ using the maps xa о р\ц% defined on the sets /7^
1.6. Coverings and Discrete Groups
35
(which cover M). The overlap maps are smooth since if Ula Π Ui Φ 0 then
(ха°И^)°(х/?°рЦ)
= Xa°plt/. °(р|^) °x/
= xa °x^1·
We leave it to the reader to show that Μ is Hausdorff if Μ is Hausdorff. D
The following is a special case of Definition 1.76.
Definition 1.87. Let a : [0,1] -» Μ and /3 : [0,1] -» Μ be two Cr maps
(paths) both starting at ρ G Μ and ending at g. A Cr fixed endpoint ho-
motopy from a to /3 is a family of Cr maps Hs : [0,1] —> Μ parameterized
by s e [0,1] such that
1) Η : [0,1] χ [0,1] -» Μ defined by #(£, 5) := #e(i) is Cr;
2) H0 = a and Я1 = /3;
3) #5(0) = ρ and H8(l) = q for all s e [0,1].
Definition 1.88. If there is a Cr homotopy from a to /3, then we say that
a is Cr homotopic to β and write a ~ β (Cr). If r = 0, we speak of paths
being continuously homotopic.
Remark 1.89. By Theorems 1.79 and 1.78 above we know that in the case
of smooth manifolds, if a and β are smooth paths, then we have that a ~ β
(C°) if and only if a ~ β (Cr) for r > 0. Thus we can just say that a is
homotopic to β and write a ~ β. In case a and β are only continuous, they
may be replaced by smooth paths a' and β' with a' ~ a and β' ~ /3.
It is easily checked that homotopy is an equivalence relation. Let P(p, q)
denote the set of all continuous (or smooth) paths from ρ to q defined on
[0,1]. Every a G P(p,q) has a unique inverse (or reverse) path a^~ defined
by
a*~{t) :=a(l-t).
If Pi,P2 and рз are three points in M, then for а € Ρ(ρι,ρ2) and /3 €
Р(р2,Рз) we can "multiply" the paths to get a path α*β € Р(рг,рз) defined
by
Г a(2t) for 0 < t < 1/2,
α*/3(ί) := <
[ /3(2t — 1) for 1/2 <t< 1.
Notice that a*/3 is a path that follows along a and then β in that order.5 An
important observation is that if αϊ ~ ot<i and β\ ~ /З2, then αϊ */3χ ~ «2 *^2·
In some settings, it is convenient to reverse this convention.
36
1. Differentiable Manifolds
The homotopy between αι*/3ι and a2*/?2 is given in terms of the homotopies
Ηα : αϊ ~ 0L2 and Ηβ : β\ ~ β2 by
Г Ha(2t,s) forO<£< 1/2,
#(M) := <{ andO < s < 1.
[ Hfi(2t-l,s) for 1/2 <t < 1,
Similarly, if αϊ ~ α2, then a^~ ~ a^-. Using this information, we can define
a group structure on the set of homotopy equivalence classes of loops, that
is, of paths in P(p,p) for some fixed ρ G M. First of all, we can always form
α * β for any α, β G P(p,p) since we always start and stop at the same point
p. Secondly, we have the following result.
Proposition 1.90. Letni(M,p) denote the set of fixed endpoint homotopy
classes of paths from Ρ (p, p). For [α], [β] G πι(Μ,ρ), define [οϊ\·[β] := [a*/3].
This is a well-defined multiplication, and with this multiplication πι(Μ,ρ)
is a group. The identity element of the group is the homotopy class 1 of the
constant map lp : t ь-> р, the inverse of a class [a] is [a*~].
Proof. We have already shown that [α] · [β] := [a * β] is well-defined. One
must also show that
1) For any a, the paths α ο a^~ and a^~ ο α are both homotopic to the
constant map lp.
2) For any α G P(p,p), we have lp * α ~ α and α * lp ~ a.
3) For any a,/3,7 G P(p,p), we have (a * β) * η ~ a * (β * 7).
Proof of 1): lp is homotopic to α ο of~ via
( a(2t) for 0 < 2i < 5,
H(t,s) = l a(s) for s<2£<2-s,
[ α<"(2ί - 1) for 2 - 5 < 2i < 2,
where 0 < s < 1. Interchanging the roles of α and a^ we also get that lp is
homotopic to a^~ ο a.
Proof of 2): Use the homotopy
H(t,s) =
a(^-st) for 0 < t < 1/2 + 5/2,
Ρ
for l/2 + s/2< t < 1.
Proof of 3): Use the homotopy
H(t,s)= {
β(Φ-ψ))
for 0 < ί < i±«,
for^<i<^
for 2±« < ί < 1.
D
The group 7Γι(Μ,ρ) is called the fundamental group of Μ at p. If
desired, one can take the equivalence classes in π\(Μ,ρ) to be represented
1.6. Coverings and Discrete Groups
37
by smooth maps. If 7 : [0,1] —> Μ is a path from ρ to g, then we have a
group isomorphism πι(Μ, q) —> πι(Μ,ρ) given by
[a] h-> [7 * a * 74-].
It is easy to show that this prescription is a well-defined group
isomorphism. Thus, for any two points p, q in the same path component of M,
the groups πι(Μ,ρ) and πι(Μ, q) are isomorphic. In particular, if Μ is
connected, then the fundamental groups based at different points are all
isomorphic. Because of this, if Μ is connected, we may simply refer to the
fundamental group of M, which we write as πι(Μ).
Definition 1.91. A path connected topological space is called simply
connected if 7Γι(Μ) = {1}.
The fundamental group is actually the result of applying a functor (see
Appendix A). Consider the category whose objects are pairs (M,p), where
Μ is a Cr manifold and ρ is a distinguished point (base point), and whose
morphisms / : (M,p) —> (iV, q) are Cr maps f : Μ -* N such that f(p) =
q. The pairs are called pointed Cr spaces and the morphisms are called
pointed Cr maps (or base point preserving maps). To every pointed space
(M,p), we assign the fundamental group πχ(Μ,ρ), and to every pointed
Cr map / : (M,p) —> (N,f(p)) we may assign a group homomorphism
7Γι(/) : πι (Μ, ρ) -» m(NJ(p)) by
πι(/)([α]) = [/οα].
It is easy to check that this is a covariant functor, and so for pointed maps /
and д that can be composed, (M, x) —> (iV, у) Л (Ρ, ζ), we have 71*1(0 ° /) =
πι(ίΟ°πι(/).
Notation 1.92. To avoid notational clutter, we will often denote πι(/) by
/#·
Definition 1.93. Let ρ : Αί_^ Μ be a Cr covering and let / : Ρ -> Μ be
a Cr map. A map / : Ρ —> Μ is said to be a lift of the map / if ρ ο f = /.
Theorem 1.94. Lei ρ : Μ -^ Μ be a Cr covering, let 7 : [a, 6] —> Μ be a
Cr curve and pick a point у in ρ~ι(η(α)). Then there exists a unique Cr
lift 7 : [a, b] —> Μ of 7 such that 7(a) = y. Thus the following diagram
commutes:
Μ
38
1. Differentiable Manifolds
Figure 1.6. Lifting a path to a cover
// two paths a and β with a(a) = β (a) are fixed endpoint homotopic via a
homotopy h, then for a given point у in ρ~ι(η(α)), we have the corresponding
lifts a and β starting at y. In this case, the homotopy h lifts to a fixed
endpoint homotopy h between 5 and β. In short, homotopic paths lift to
homotopic paths.
Proof. We just give the basic idea and refer the reader to the extensive
literature for details (see [Gre-Hrp]). Figure 1.6 shows the way. Decompose
the curve 7 into segments that lie in evenly covered open sets by using the
Lebesgue number lemma. Lift inductively starting by using the inverse of ρ
in the first evenly covered open set. It is clear that in order to connect up
continuously, each step is forced and so the lifted curve is unique. A similar
argument shows how to lift the homotopy h. A little thought reveals that if
ρ is a Cr covering, then the lifts of Cr maps are Cr. D
There are several important corollaries to this result. One is simply
that if a : [0,1] —> Μ is a path starting at a base point ρ e M, then since
there is one and only one lift 5 starting at a given pi in the fiber p~l(p),
the endpoint 5(1) is completely determined by the path a and by the point
p' from which we want the lifted path to start. In fact, the endpoint only
depends on the homotopy class of a (and the choice of starting point p').
To see this, note that if α, β : [0,1] —> Μ are fixed endpoint homotopic
paths in Μ beginning at p, and if 5 and β are the corresponding lifts with
5(0) = /3(0) = p', then by the second part of the theorem, any homotopy
ht : a ~ β lifts to a unique fixed endpoint homotopy fit : δ ~ β. This then
1.6. Coverings and Discrete Groups
39
implies that 5(1) = /3(1). Applying these ideas to loops based at ρ G M, we
will next see that the fundamental group πι (Μ, ρ) acts on the fiber p~l (p) as
a group of permutations. (This is a right action as we will see.) In case the
covering space Μ is simply connected, we will also obtain an isomorphism
of the group πι(Μ,ρ) with the deck transformation group (which acts from
the left on M). Before we delve into these matters, we state, without proof,
two more standard results (see [Gre-Hrp]):
Theorem 1.95. Let ρ : Μ —> Μ be α Cr covering. Fix α point q G Q and
a point ρ G M. Let φ : Q —> Μ be a Cr map with <fi(q) = p(p). If Q is
connected, then there is at most one lift φ : Q —> Μ of φ such that ф(р) = р.
If 0#(7Ti(Q,g)) С ρ#(ττι(Μ,ρ)), then φ has such a lift. In particular, if Q
is simply connected, then the lift exists.
Theorem 1.96. Every connected topological manifold Μ has a C° simply
connected covering space which is unique up to isomorphism of coverings.
This is called the universal cover. Furthermore, if Η is any subgroup of
πι(Μ,ρ), then there is a connected covering ρ : Μ —ϊ Μ and a point ρ G Μ
such that p#(7i*i(M,p)) = Η.
If follows from this and Theorem 1.86 that if Μ is a Cr manifold, then
there is a unique Cr structure on the universal covering space Μ so that
ρ : Μ —> Μ is a Cr covering.
Since a deck transformation is a lift, we have the following corollary.
Corollary 1.97. Let ρ : Μ —ϊ Μ be a Cr covering map and choose a base
point ρ G M. If Μ is connected, there is at most one deck transformation
φ that maps a given p\ G p~l(p) to a given p2 G p~l(p). If Μ is simply
connected, then such a deck transformation exists and is unique.
Theorem 1.98. If Μ is the universal cover of Μ and ρ : Μ —ϊ Μ is the
corresponding universal covering map, then for any base point po G M, there
is an isomorphism πι(Μ,ρο) — Deck(p).
Proof. Fix a point ρ G p_1(po)· Let α G πι(Μ,ρο) and let α be a loop
representing a. Lift to a path a starting at p. As we have seen, the point
a(l) depends only on the choice of ρ and a = [a]. Let φα be the unique deck
transformation such that φα(ρ) = δ(1). The assignment α \-+ φα gives a
map πι (Μ,ρο) ~> Deck(p). For α = [a] and b = [β] chosen from πι(Μ,ρο),
we have the lifts 5 and /3, and we see that φα ο β is a path from φα(ρ) to
φα(β(1)) = Фа{Фь{р))- Thus the path 7 := 5 * (φα ο β) is defined. Since
40
1. Differentiable Manifolds
Ρ ° Φα = Ρ, we have
ρ ο η = ρ ο 5 * (φα ο β)\
= (ρ Ο 5)* [ρ Ο (φα°β))
= (ρ ο α) * [{ρ ο φα) о Щ
= (ρ ο α) * (ρ ο β) = α* β.
Since α * β represents the element аЪ G πι(Μ,ρο)? we have фаь(р) — 7(1) —
0α(06 (ρ))· Since Μ is connected, Corollary 1.97 gives 0α& = фа ° Фь- Thus
the map is a group homomorphism.
It is easy to see that the map α ь-> 0α is onto. Indeed, given / G Deck(p),
we simply take a curve 7 from ρ to /Q5), and then we have f = фд, where
9 = [p°l\ £ πι(Μ,ρο).
Finally, if 0a = id, then we conclude that any loop α G [a] = a lifts to a
loop δ based at p. But Μ is simply connected, and so 5 is homotopic to a
constant map to p, and its projection a is therefore homotopic to a constant
map to p. Thus α = [a] = 0 and so the homomorphism is 1-1. D
1.6.2. Discrete group actions. Groups actions are ubiquitous in
differential geometry and in mathematics generally. Felix Klein emphasized the
role of group actions in the classical geometries (see [Klein]). More on
classical geometries can be found in the online supplement [Lee, Jeff]. In this
section, we discuss discrete group actions on smooth manifolds and show
how they give rise to covering spaces.
Definition 1.99. Let G be a group and Μ a set. A left group action on
Μ is a map / : G χ Μ ->> Μ such that
1) K92,l{gux)) = Kg29ux) for all gug2 G G and all χ <Ε Μ;
2) /(e,x) = χ for all χ G M, where e is the identity element of G.
We often write g · χ or just gx in place of the more pedantic notation
l(g,x). Using this notation, we have #2(#i#) = (#2#i)# and ex = x.
Similarly, we define a right group action as a map r : Μ χ G —ϊ Μ with
r(r(x,#1),#2) = r(x,9i92) for all #ι,#2 £ G and all χ G Μ and r(x,e) = χ
for all χ G M. In the case of right actions, we write r(x, #) as χ · g or xg.
If / : G χ Μ —> Μ is a left action, then for every g G G we have a map
lg : Μ -^ Μ defined by lg(x) = l(g,x), and similarly a right action gives for
every #, a map rg : Μ —ϊ Μ. For every result about left actions, there is an
analogous result for right actions. However, mathematical conventions are
such that while g \-+ lg is a group homomorphism from G to the group of
1.6. Coverings and Discrete Groups
41
permutations of M, the map д н-> гд is a group anti-homomorphism which
means that rQl о rQ2 = rg29l for all #1, #2 £ G (notice the order reversal).
Given a left action, the sets of the form Gx = {gx : g G G} are called
orbits. The set Gx is called the orbit of x. Two points χ and у are in the
same orbit if and only if there is a group element g such that gx = y. The
orbits are equivalence classes and so they partition M. Let G\M be the set
of orbits and let ρ : Μ —> G\M be the projection taking each χ to Gx. We
give G\M the quotient topology. By definition, U is open in G\M if and
only if p~l(U) is open. This makes ρ continuous, but in this case, it is also
an open map. To see this, let U be open. Then p~l (p(U)) is the union
UaeGdU which is open and so p(U) is open by the definition of quotient
topology.
Definition 1.100. Suppose G acts on a set Μ by / : G χ Μ ->> Μ. We say
that G acts transitively if for any x, у G Μ there is a g such that gx = y.
Equivalently, the action is transitive if the action has only one orbit. We
say that the action is effective provided that lg = \ам implies that g = e.
If the action has the property that gx = χ for some χ G Μ only when g = e,
then we say that G acts freely (or that the action is free). In other words,
an action is free provided that the only element of G that fixes any element
of Μ is the identity element.
Similar statements and definitions apply for right actions except that
the orbits have the form xG. The quotient space (space of orbits) will then
be denoted by M/G.
Warning: The notational distinction between G\M and M/G is not
universal.
Example 1.101. Let ρ : Μ —> Μ be a covering map. Fix a base point
po G Μ and a base point po G p_1(po)· If α G πι(Μ,ρο), then for each
χ G ρ~λ(ρο), we define ra(x) '-= xa := 5(1), where a is the lift of any loop
a representing a. The reader may check that ra is a right action on the set
p~4po)·
Example 1.102. Recall that if ρ : Μ —> Μ is a universal Cr covering
map (so that Μ is simply connected), we have an isomorphism πχ(Μ,ρο) ->
Deck(p), which we denote by о и фа. This means that /(а,ж) = фа(х)
defines a left action of πχ(Μ,ρο) οη Μ.
Let G be a group and endow G with the discrete topology so that, in
particular, every point is an open set. In this case, we call G a discrete
group. If Μ is a topological space, then we endow G χ Μ with the product
topology. What does it mean for a map a : G χ Μ —> Μ to be continuous?
The topology of G χ Μ is clearly generated by sets of the form S χ /7, where
42
1. Differentiable Manifolds
S is an arbitrary subset of G and U is open in M. The map а : G χ Μ —> Μ
will be continuous if for any point (go,xo) G GxM and any open set /7 С М
containing a(po?^o) we can find an open set 5 χ V containing (50? #0) such
that a(S x V) С U. Since the topology of G is discrete, it is necessary and
sufficient that there is an open V such that a(go χ V) С U. It is easy to
see that a necessary and sufficient condition for a to be continuous on all of
GxMis that the partial maps ag := a(g, ·) are continuous for every g G G.
Definition 1.103. Let G be a discrete group and Μ a manifold. A left
discrete group action is a group action / : G χ Μ —ϊ Μ such that for
every g G G the partial map lg(·) := l(g, ·) is continuous. A right discrete
group action is defined similarly.
It follows that if / : G χ Μ —> Μ is a discrete action, then each partial
map lg is a homeomorphism with l~l = lg-i.
Definition 1.104. A discrete group action is Cr if Μ is a Cr manifold and
each lg (resp. rg ) is a Cr map.
Example 1.105. Let φ : Μ —> Μ be a diffeomorphism and let Ζ act on Μ
by η · χ := φη(χ) where
φ := idM ,
φη := φ ο ... о φ for n > О,
φ-" := (0-1)" for η > 0.
This gives a discrete action of Ζ on Μ.
Definition 1.106. A discrete group action / : G χ Μ —> Μ is said to be
proper if for every two points x,y e Μ there are open neighborhoods Ux
and Uу respectively such that the set {g G G : gUx Π Uy φ 0} is finite.
There is a more general notion of proper action which we shall meet
later. For free and proper discrete actions, we have the following useful
characterization.
Proposition 1.107. A discrete group action I : G χ Μ —ϊ Μ is proper
and free if and only if the following two conditions hold:
(i) Each χ G Μ has an open neighborhood U such that gU Π U = 0
for all g except the identity e. We shall call such open sets self-
avoiding.
(ii) If x,y Ε Μ are not in the same orbit, then they have self-avoiding
neighborhoods Ux and Uy such that gllx nUy = Φ for all g G G.
Proof. Suppose that the action / is proper and free. Let χ be given. We
then know that there is an open V containing χ such that gVOV = 0 except
1.6. Coverings and Discrete Groups
43
for a finite number of g, say, #1, ·.., #ь which are distinct. One of these, say
51, must be e. Since the action is free, we know that for each fixed г > 1
we have gix G M\{x}. By using continuity and then the fact that Μ is a
regular topological space, we can replace V by a smaller open set (called V
again) such that giV С M\{x} for all г = 2,..., к or, in other words, that
χ <£ g2V U · · · U gkV. Let U = V\(g2V U · · · U gkV). Notice that U is open
and we have arranged that U contains x. We show that U Π gU is empty
unless g = e. So suppose g φ e = g\. Since U Π gU Cl^fl gV', we know that
this is empty for sure in all cases except maybe where g = gi for г = 2,..., к.
If χ G U Π giU for such an i, then χ e U and so χ ^ g^V by the definition of
U. But we also have χ G ^^ С #г^Л which is a contradiction. We conclude
that (i) holds.
Now suppose that x,y G Μ are not in the same orbit. We know that
there exist open sets Ux and Uy with χ G Ux, у G Uy and such that gllx Π C/y
is empty except possibly for some finite set of elements which we denote by
51,..., gk- Since the action is free, g\x,..., g^x are distinct. We also know
that у is not equal to any of g\x,... ,^x, and so since Μ is a Hausdorff
space, there exist pairwise disjoint open sets Oi,..., Оь Oy with gix G Oi
and у Ε Oy. By continuity, we may shrink Ux so that <7ii/x С Οι for all
г = 1,..., /с, and then we also replace Uy with Oy Π Uy (renaming this Uy
again). As a result we now see that gUx DUy = 0 for g = #1,..., </*. and
hence for all g. By shrinking the sets Ux and Uy further we may make them
self-avoiding.
Next we suppose that (i) and (ii) always hold for a given discrete action
/. First we show that / is free. Suppose that χ — gx. Then for every open
neighborhood U of χ the set gU Π U is nonempty, which by (i) means that
g = e. Thus the action is free. Next pick x, у G M. If x, у are not in the same
orbit, then by (ii) we may pick Ux and Uy so that {g G G : gllx C\Uy ^ 0}
is empty and so certainly a finite set. If x,y are in the same orbit, then
у = дох for a unique #0 since we now know that the action is free. Choose
an open neighborhood U of χ so that gU Π U = 0 for g φ e. Let Ux = U and
Uy = goU. Then, gUxDUy = gUDg0U. IfgUDgoU φ 0, then g'1 gUDU φ 0
and so g$lg = e and g = #0· Thus the only way that gUxDUy is nonempty
is if g — go and so the set {g G G : gllx Π Uy φ 0} has cardinality one. In
either case, we may choose Ux and Uy so that the set is finite, which is what
we wanted to show. D
It is easy to see that if U С М is self-avoiding, then any open subset
V С U is also self-avoiding. Thus, if the discrete group G acts freely and
properly on M, then the open sets of (i) and (ii) in the above proposition
can be taken to be connected chart domains.
44
1. Differentiable Manifolds
Proposition 1.108. Let Μ be an η-manifold and let I : G χ Μ —>· Μ be
a smooth discrete action which is free and proper. Then the quotient space
G\M has a natural smooth structure such that the quotient map is a smooth
covering map.
Proof. Giving G\M the quotient topology makes ρ : Μ —> G\M
continuous. Using (ii) of Proposition 1.107, it is easy to show that the quotient
topology on G\M is Hausdorff. By Proposition 1.107, we may cover Μ
by charts whose domains are self-avoiding and connected. Let (/7, x) be
such a chart and consider the restriction p\v. This restricted map is open
since, as remarked above, ρ is an open map. If x,y G U and p(x) = p(y),
then χ and у are in the same orbit and so у = дх for some g. Therefore
у G gU Π /7, which means that gU Π U is not empty and so g = e since U is
self-avoiding. Thus χ = у and we conclude that p\v is injective. Since p\v
is also surjective, we see that it is a bijection and hence a homeomorphism.
I.e., since p\v is also open, it has a continuous inverse and so it is a
homeomorphism. Since U is connected, the connected components of p~l (p(U))
are exactly the sets gU for g G G. Since ρ ο lg = ρ for all G, it is easy to see
that ρ restricts to a homeomorphism on each connected component gU of
p~l (p (U)). Thus, p(U) is evenly covered by ρ and so ρ is a covering map.
For every such chart (/7, x), we have a map
хоМс/Г1:^)-^),
which is a chart on G\M. This map is clearly a homeomorphism. Given any
other map constructed in this way, say у о (р|у)~~ , the domains p(U) and
p(V) only meet if there is a g G G such that gU meets V and lg maps an
open subset of U diffeomorphically onto a subset of V. In fact, by Exercise
1.109 below, the map (HvO~~ ° p\U \s defined on an open set each point of
which has a neighborhood on which this map is a restriction of lg for some
g. Thus(HvO"loH
ц is smooth and for the overlap map we have
У ° (НуГ1 ° (x °(p\u)j = У ° (HvO_1 ° P\u ° x_1>
which is smooth. Thus we have an atlas on G\M and the topology induced
by the atlas is the same as the quotient topology since we have already
established that the charts are homeomorphisms. D
Exercise 1.109. In the context of the proof above, show that (ρ\ν)~ ο ρ\ν
is defined on an open set Ο = {ρ\ν)~ (p(U) Π p{V)). Show that each
χ G О has a neighborhood on which the map (p|vO~ ° p\u c°incides with a
restriction of a map χ \-> gx for some fixed g. Conclude that (p|vO~ ° p\u ls
a Cr map. [Outline of solution: For χ G O, we must have (ρ\ν)~λο ρ\ν (x) =
gx for some g. Now O' = UC\g~1V is an open set that contains x. But also,
1.6. Coverings and Discrete Groups
45
p{9~lV) = p(V) so О = U П g~lV С (p\ Uyl (p{U) П p(V)). Let x1 € O.
Then since p(x') € p(U) Π p(V), it follows that
My)"10 Hi/00 = *".
where ж" is the unique point in V such that p(x") = p(xf). But gxf G V
and ρ (до') = ж" so gxf = x" ]
Example 1.110. We have seen the torus previously presented as T2 =
51 χ S1. Another presentation that uses a group action is given as follows:
Let the group Ζ χ Ζ = Ζ2 act on R2 by
(ra,n) · (x,y) := (x + m,y + n).
It is easy to check that Proposition 1.108 applies to give a manifold R2/Z2.
This is actually the torus in another guise, and we have the diffeomorphism
φ : Ш2/I? -> S1 x S1 =T2 given by [(x,y)] *-> (ei2nx,ei2ny). The following
diagram commutes:
R2 >- S1 χ S1
M2/Z2
Exercise 1.111. Let ρ : Μ —> G\M be the covering arising from a free
and proper discrete action of G on Μ and suppose that Μ is connected. Let
Tq '·= {lg £ Diff(M) : J G G}\ then G is isomorphic to Γ^ by the obvious
map g *-> lg and furthermore Г<з = Deck(p).
Covering spaces ρ : Μ —>· Μ that arise from a proper and free discrete
group action are special in that if Μ is connected, then the covering is a
normal covering, which means that the group Deck(p) acts transitively on
each fiber p~l(p) (why?).
Example 1.112. Recall the multiplicative abelian group Z2 = {1,-1} of
two elements. Let Z2 act on the sphere Sn С Μη+1 by (±1) · χ := ±x.
Thus the action is generated by letting — 1 send a point on the sphere to its
antipode. This action is also easily seen to be free and proper. The quotient
space is the real projective space RPn (See Example 1.41),
Rpn = 3ηβ^
If Μ is simply connected and we have a free and proper action by G as
above, then we can define a map φ : G —> 7Ti(G\M, bo) as follows: Fix a
base point xo G Μ with p(xo) = bo· Given g G G, let 7 : [0,1] —> Μ be a
path with 7(0) = xo and 7(1) = gxo. Then
Φ{9) := [Ρ°Ί] G7Ti(G\M,b0).
46
1. Differentiable Manifolds
This is well-defined because Μ is simply connected. In fact, we already know
from Theorem 1.98 that there is an isomorphism 7Ti(G\M,bo) = Deck(p).
But by Exercise 1.111, we know that G = Deck(p) by the map g ь-> lg.
Composing, we obtain an isomorphism φ : πχ(0\Μ, bo) -^ G. Recalling the
definition of the isomorphism constructed in the proof of Theorem 1.98, we
see that the map φ : G —> πι((7\Μ,6ο) defined above is just the inverse
of the isomorphism φ : πι(£τ\Μ, bo) —> G. Thus we obtain the following
theorem, which is essentially a variation of Theorem 1.98.
Theorem 1.113. If Μ is simply connected, then the map
0:G->^(G\M,bo)
defined above is a group isomorphism.
The reader may wish to try to prove directly that φ is a group
isomorphism.
Corollary 1.114. m(RPn) ** Z2.
1.7. Regular Submanifolds
A subset S of a smooth η-manifold Μ is called a regular submanifold of
dimension к if every point ρ G S is in the domain of a chart (/7, x) that has
the following regular submanifold property with respect to S:
x{U Π S) = x{U) Π (Rk χ {c}) for some с е Mn_/c.
Usually с is chosen to be 0, which can always be accomplished by
composition with a translation of W1. The terminology here does not seem to be
quite standardized. If a subset S С Μ is covered by charts of Μ of the above
type, then S itself is said to have the (regular) submanifold property.
We will refer to such charts as being single-slice charts (adapted to S).
For every such single-slice chart (/7, x), we obtain a chart (U Π 5,xs) on 5,
where xs := pr о xl^^ and pr : Rk χ Rn~k —> Rk is projection onto the first
к coordinates. In other words, if ж1,..., xn are the coordinate functions of a
single-slice chart, then the restrictions of ж1,..., xk to U Π S are coordinate
functions on S. These charts provide an atlas for S (called a submanifold
atlas) making it a smooth manifold in its own right. Indeed, one checks that
the overlap maps for such charts are smooth.
Exercise 1.115. Prove this last statement.
We will see more general types of submanifolds in the sequel. An
important aspect of regular submanifolds is that the topology induced by the
smooth structure is the same as the relative topology. The integer η — к
is called the codimension of S (in M), and we say that S is a regular
submanifold of codimension η — к.
1.7. Regular Submanifolds
47
Figure 1.7. Projection chart
Example 1.116. The unit sphere Sn С Rn+1 is a regular submanifold of
Rn+1. To see this, let W* := {(a1,... ,an+1) € Rn+l : ±o* > 0}. Then
define i\)f : Wf -» ф±{Шп+1) by
^(a1,...,a"+1) = (a1,...,ai-1,a<+1,...,an+1,||a||-l).
These can easily be checked to give charts on Rn+l smoothly related to
the standard chart. If ρ G S2 then ρ is in the domain of one of the
charts ipf. On the other hand, identifying Rn+1 with Rn χ Μ, we have
^HWz±nSn) = ^HU)n(Rn x {°}) and so these charts have the submanifold
property with respect to Sn. Let pr : Rn+1 = W1'1 xR-4 W1'1. The
resulting charts on Sn given by pr ο ^|μ^±ηοη have the form (a1,... ,an+1) t-^
г
(a1,..., a*"1, al+1,..., an+1). These are projections from Sn onto
coordinate hyperplanes in Rn+1 and are easily checked to give the same smooth
structure as the stereographic charts given earlier.
Exercise 1.117. Show that a continuous map f \ Ν -ϊ Μ that has its
image contained in a regular submanifold S is differentiable with respect to
the submanifold atlas if and only if it is differentiable as a map into M.
Exercise 1.118. Show that the graph of any smooth map Rn —> Rm is a
regular submanifold of Rn χ Rm.
Exercise 1.119. Show that if Μ is a fc-dimensional regular submanifold of
En, then for every ρ G M, there exists at least one fc-dimensional coordinate
plane Ρ such that the orthogonal projection Rn —> Ρ = Rk restricts to a
coordinate chart for Μ defined on some neighborhood of p. [Hint: If (/7, x)
is a single-slice chart for Μ so that ж1,..., xk restrict to coordinates on M,
then xk+\..., xn together give a map f : U С Rn = Rk x Rn~k -» Rn~k
48
1. Differentiable Manifolds
such that MDlI = f г(0). Argue that if г/1,..., un are standard coordinates
on Mn, then after a suitable renumbering we must have
д(ик+\...,ип) Ψ '
Now use the implicit mapping theorem to show that Μ is locally the graph
of a smooth function.]
1.8. Manifolds with Boundary
For the general Stokes theorem, where the notion of flux has its
natural setting, we will need to have the concept of a smooth manifold with
boundary. We have already introduced the notion of a topological
manifold with boundary, but now we want to see how to handle the issue of the
smooth structures. Some basic two-dimensional examples to keep in mind
are the upper half-plane Ky>0 '·= {(x,y) £ IK2 : у > 0}, the closed unit disk
D = {(x, y) G Ш2 : x2 + y2 < 1}, and the closed hemisphere which is the set
of all (x,y,z) G S2 with ζ > 0. Recall that in Section 1.2 we defined the
closed η-dimensional Euclidean half-spaces Щ>с := {α G W1 : λ(α) > с}.
Of course, Щ<с = M^A>_C so we are including "both M^<0 and H^fc>0. We
also write Щ^=с = {а еШп : λ(α) = с}. Give Мд>с the relative topology as a
subset of W1. Since №%>c С Mn, we already have a notion of differentiability
for a map U —> Mm, where U is a relatively open subset of Мд>с. We just
invoke Definition 1.58. We can extend definitions a bit more:
Definition 1.120. Let U С Щ1>С1 and / : U -» M^2>C2. We say that / is
Cr if it is Cr as a map into Rm/if both / : U -> /(/7)~and f~l : f(U) -> U
are homeomorphisms of relatively open sets and Cr in this sense, then / is
called a Cr diffeomorphism.
For convenience, let us introduce for an open set U С Щ^>с (relatively
open) the following notations: Let dU denote Кд=с П U and int({7) denote
U \ dU. In particular, дЩ>с = Щ=с. Notice that dU is clearly an (n - 1)-
manifold.
We have the following three facts:
(1) First, let / : U С Шп -> Rk be Cr differentiable (with r > 1) and
g another such map with the same domain. If / = g on Щ^>с П /7,
then Df(x) = Dg(x) for all χ G Щ>с DU.
(2) If / : U С W1 -» M^>c is Cr differentiable (with r > 1) and
/(ж) G M^=c = дЩ>с for all χ G f/, then Df(x) must have its
image in Кд=0.
1.8. Manifolds with Boundary
49
Figure 1.8. Manifold with boundary
(3) Let / : U\ С Щ1>С1 ->U2 CK^2>C2 be a diffeomorphism (in our new
extended sense). Assume that dU\ and дЩ are not empty. Then
/ induces diffeomorphisms dU\ —> 3U2 and int(J7i) —> int(C/2).
These three claims are not exactly obvious, but they are very intuitive. On
the other hand, none of them is difficult to prove (see Problem 19).
We can now form a definition of smooth manifold with boundary in a
fashion completely analogous to the definition of a smooth manifold without
boundary. A half-space chart χ for a set Μ is a bijection of some subset
U of Μ onto an open subset of some half-space Щ>с. A Cr half-space
atlas is a collection (C/Q,xQ) of half-space charts such that for any two, say
(f7Q, xQ) and (ί/β, Χβ), the map XqOxI1 is a Cr diffeomorphism on its natural
domain. Notice carefully that we allow the half-space to vary from chart
to chart, but we will keep η fixed for a given Μ and refer to the charts as
η-dimensional half-space charts.
Definition 1.121. An η-dimensional Cr manifold with boundary is
a pair (M, A) consisting of a set Μ together with a maximal atlas of n-
dimensional half-space charts A. The manifold topology is that generated
by the domains of the charts in the maximal atlas. The boundary of Μ
is denoted by dM and is the set of points whose image under any chart is
contained in the boundary of the associated half-space.
The three facts listed above show that the notion of a boundary is a well-
defined concept and is a natural notion in the context of smooth manifolds;
it is a "differentiable invariant".
50
1. Differentiable Manifolds
Colloquially, one usually just refers to Μ as a manifold with
boundary and forgoes the explicit reference to the atlas. Also we refer to an
η-dimensional C°° manifold with boundary as an η-manifold with
boundary. The interior of a manifold with boundary is M\dM. It is a manifold
without boundary and is denoted int(M) or M.
Exercise 1.122. Show that dM is a closed set in M.
If no component of a manifold without boundary is compact, it is called
an open manifold. For example, the interior int(M) of a connected
manifold Μ with nonempty boundary is never compact and is an open manifold
in the above sense if every component of Μ contains part of the boundary.
Remark 1.123. We avoid the phrase "closed manifold", which is sometimes
taken to refer to a compact manifold without boundary.
Let Μ be an η-dimensional Cr manifold with boundary and ρ G dM.
Then by definition there is a chart (/7, x) with x(p) G Жд>с. The image of
the restriction х|с/п^д/ is contained in Жд>с for some λ and с depending on
the chart. By composing this restriction with any fixed linear isomorphism
Жд>с —> Mn_1, we obtain a bijection, say χ#μ> of U Π dM onto an open
subset of Mn_1 which provides a chart (Ua Π дМ, хам) for dM. The family
of charts obtained in this way is an atlas for dM. The overlaps are smooth
and so we have the following:
Proposition 1.124. If Μ is an η-manifold with boundary, then dM is an
(n — 1)-manifold.
Exercise 1.125. Show that the overlap maps for the atlas just constructed
for dM are smooth.
Exercise 1.126. The closed unit ball B(p, 1) in Rn is a smooth manifold
with boundary dB(p,l) = 5η_1. Also, the closed hemisphere S+ = {x G
Sn : xn+1 > 0} is a smooth manifold with boundary.
Exercise 1.127. Is the Cartesian product of two smooth manifolds with
boundary necessarily a smooth manifold with boundary?
Exercise 1.128. Show that the concept of smooth partition of unity makes
sense for manifolds with boundary. Show that such exist.
Problems
51
Problems
(1) Prove Proposition 1.32. The online supplement [Lee, Jeff] outlines the
proof.
(2) Prove Lemma 1.11.
(3) Check that the manifolds given as examples are indeed paracompact and
Hausdorff.
(4) Let Μι, Μ2 and M3 be smooth manifolds.
(a) Show that (Μι χΜ2) χ M3 is diffeomorphic to Μλ χ(Μ2 χ Μ3) in
a natural way.
(b) Show that / : Μ -> Μχ χ Μ2 is С°° if and only if the composite
maps prx о / : Μ -)► Mi and pr2 о / : Μ -> M2 are both C°°.
(5) Show that a Cr manifold Μ is connected as a topological space if and
only it is Cr path connected in the sense that for any two points p\, p2 G
Μ there is a Cr map с : [0,1] —>· Μ such that c(0) = pi and c(l) = p2.
(6) A fc-frame in Mn is a linearly independent ordered set of vectors
(i>i,... ,Vfc). Show that the set of all fc-frames in Rn can be given the
structure of a smooth manifold. This kind of manifold is called a Stiefel
manifold.
(7) For a product manifold Μ χ TV, we have the two projection maps prx :
Μ χ Ν -> Μ and pr2 : Μ χ Ν -> N defined by (x, у) ι—> χ and
(χ, у) ι—> у respectively. Show that if we have smooth maps fi'.P—ϊΜ
and /2 : Ρ -> TV, then the map (f,g) : Ρ -> Μ χ Ν given by (/, g) (p) =
(f(p),g(p)) is the unique smooth map such that prx о (/,g) = f and
Pr2°(/,i0 =g.
(8) Prove (i) and (ii) of Lemma 1.35.
(9) Show that the atlas obtained for a regular submanifold induces the
relative topology inherited from the ambient manifold.
(10) The topology induced by a smooth structure is not necessarily Hausdorff:
Let S be the subset of R2 given by the union (R χ 0) U {(0,1)}. Let U
be Μ χ 0 and let V be the set obtained from U by replacing the point
(0,0) by (0,1). Define a chart map χ on U by x(x, 0) = χ and a chart у
on V by
, ν Γ χ if χ φ 0,
y^0) = l0 ifx = 0.
Show that these two charts provide a C°° atlas on 5, but that the
topology induced by the atlas is not Hausdorff.
52
1. Differentiable Manifolds
(11) As we have defined them, manifolds are not required to be second
countable and so may have an uncountable number of connected components.
Consider the set M2 without its usual topology. For each aGK, define
a bijection фа : R χ {α} —> R by фа{х, о) = х- Show that the family of
sets of the form U χ {α} for U open in R and aGK provide a basis for
a paracompact topology on R2. Show that the maps фа are charts and
together provide an atlas for R2 with this unusual topology. Show that
the resulting smooth manifold has an uncountable number of connected
components (and so is not second countable).
(12) Show that every connected manifold has a countable atlas consisting of
charts whose domains have compact closure and are simply connected.
Hint: We are assuming that our manifolds are paracompact, so each
connected component is second countable.
(13) Show that every second countable manifold has a countable fundamental
group (a solution can be found in [Lee, John] on page 10).
(14) If С χ С is identified with R4 in the obvious way, then S3 is exactly the
subset of Cx С given by {(zi,z2) : |^i|2 + |^2|2 = 1}· Let p, q be coprime
integers and ρ > q > 0. Let ω be a primitive p-th root of unity so that
Zp = {Ι,ω,. ..,ωρ~1}. For (ζι,ζ2) G 53, let ω\ζ\, z2) := (uzi,ujqz2) and
extend this to an action of Zp on S3 so that uk · (z\, z2) = (uukz\,uuqkz2).
Show that this action is free and proper. The quotient space ZP\S3 ie
called a lens space and is denoted by L(p; q).
(15) Let Sl be realized as the set of complex numbers of modulus one. Defim
a map θ : S1xS1 -> S1 χ S1 by 0(z, w) = (-z, w) and note that θοθ = id
Let G be the group {id, Θ}. Show that Μ := (S1 x S1) /G is a smootl
2-manifold.
(16) Show that if S is a regular /c-dimensional submanifold of an n-manifol<
M, then we may cover S by special single-slice charts from the atlas с
Μ which are of the form χ : U -> Vi χ V2 С Шк х Шп~к = Шп with
x(UHS) = Vix {0}
for some open sets V\ С Жк, V2 С Мп_/\ Show that we may arrange fc
V\ and V2 to both be Euclidean balls or cubes. (This problem should b
easy. Experienced readers will likely see it as merely an observation.)
(17) Show that π\(Μ χ TV, (p,g)) is isomorphic to πι(Μ,ρ) χ ni(N,q).
(18) Suppose that Μ = U U V, where U and V are simply connected ar
open. Show that if UC\V is path connected, then Μ is simply connecte
(19) Prove the three properties about maps involving the model half-spac
<ЖЪП listed in Section 1.8.
Problems
53
ρ χ (-ι,ΐ)
Ш
Figure 1.9. Smoothly connecting manifolds
(20) Let Μ and N be smooth η-manifolds with boundaries dM and dN. Let
Ρ be a smooth manifold diffeomorphic to both dM and dN via maps
a and /3. Suppose that there are open neighborhoods U and V of <9M
and dN respectively and diffeomorphisms
0i:l7-*Px[O,l),
02: V->Px(-l,0]
such that 0i(ρ) = (α(ρ),0) whenever p G dM Π /7, and similarly
02(p) = (/3 (p), 0) whenever pedNnU. Then 0.J1 o0x : {7 -» V is a dif-
feomorphism. Let f := φ^1 ο φλ\ dM so that f = β~ι ο a \ dM -» 97V.
Let Μ U/ TV be the topological space defined by identifying χ G dM
with /(x) G dN (see the online supplement [Lee, Jeff] for a discussion
of identification spaces). Show that there is a unique smooth structure
on ΜU/TV such that the inclusion maps Μ c-> MUfN and N c-> MUfN
are smooth and such that the induced map U U/ V —> Ρ x (—1,1) is a
diffeomorphism. Here U\JfV is the image of U\JV under the projection
map MUN -+M\JfN.
(21) Define
Rl = {иеЖп:и* >0fori = 1,2,...,n},
R£ = {иеШп :ul >0for i = l,2,...,n}.
A boundary point of R+ is a point such that at least one of its
coordinates ul is 0. A corner point of R+ is a point such that at least two of
its coordinates ul,v? are 0. We consider a set M. An IR^-valued chart
on Μ is a pair (/7, x), where U С Μ and χ : U —> χ (U) is a bijection
onto an open subset χ (U) of M+, where the latter has the relative
topology as a subset of Rn. A smooth atlas for Μ is a family {(Ua,xa)}aeA
of M^-valued charts whose domains cover Μ and such that whenever
54
1. Differentiable Manifolds
Ua Π Uβ is nonempty, the composite map
χβ ο χ"1 : χα({7α Π ΙΙβ) -» χβ(υα Π ϋβ)
is smooth. A maximal atlas of M+-valued charts of this type gives Μ
the structure of a smooth manifold with corners (of dimension n).
We also use the terminology η-manifold with corners.
(a) Suppose that ρ G Μ and xa(p) is a corner point in R+. Show
that if ρ is in the domain of another chart (ΙΙβ,Χβ) in the atlas (as
above), then Χβ(ρ) is also a corner point. Use this to define "corner
points" on M. Do the same for "boundary points". Thus the
boundary contains the set of corner points. Explain why a manifold
with corners whose set of corner points is empty is a manifold with
(possibly empty) boundary.
(b) Define the notion of smooth functions on a manifold with corners
and the notion of smooth maps between manifolds with corners.
(c) Show that the boundary of a manifold with corners is not
necessarily a manifold with corners.
(22) Prove Theorem 1.113 and its corollary.
Chapter 2
The Tangent Structure
In this chapter we introduce the notions of tangent space and cotangent
space of a smooth manifold. The union of the tangent spaces of a given
manifold will be given a smooth structure making this union a manifold
in its own right, called the tangent bundle. Similarly we introduce the
cotangent bundle of a smooth manifold. We then discuss vector fields and
their integral curves together with the associated dynamic notions of Lie
derivative and Lie bracket. Finally, we define and discuss the notion of a
1-form (or covector field), which is the notion dual to the notion of a vector
field. One can integrate 1-forms along curves. Such an integration is called a
line integral. We explore the concept of exact 1-forms and nonexact 1-forms
and their relation to the question of path independence of line integrals.
2.1. The Tangent Space
If с : (—б, б) —» RN is a smooth curve, then it is common to visualize the
"velocity vector" c(0) as being based at the point ρ = c(0). It is often
desirable to explicitly form a separate TV-dimensional vector space for each
point p, whose elements are to be thought of as being based at p. One
way to do this is to use {ρ} χ RN so that a tangent vector based at ρ is
taken to be a pair (p, v) where ν G RN. The set {ρ} χ RN inherits a vector
space structure from RN in the obvious way. In this context, we provisionally
denote {ρ} χ RN by TPRN and refer to it as the tangent space at p. If we write
c(t) = (x1^),..., xN(t)), then the velocity vector of a curve с at time1 t = 0
is (P> ljjr(0)> * * *' lf(0))» which is based at ρ = c(0). Ambiguously, both
1 It is common to refer to the parameter t for a curve as "time", although it may have nothing
to do with physical time in a given situation.
55
56
2. The Tangent Structure
(P. #(°)> · · ·' ТГ(°)) and (¥(°). · · · - ТГ(°)) are often denoted ЬУ έ(°) or
c'(0). A bit more generally, if V is a finite-dimensional vector space, then V
is a smooth manifold and the tangent space at ρ G V can be provisionally
taken to be the set {ρ} χ V. We use the notation vp := (p, v). If vp := (p, г;)
is a tangent vector at p, then υ is called the principal part of vp.
We have a natural isomorphism between RN and TpRN given by υ ι—>
(ρ, г>), for any p. Of course we also have a natural isomorphism TpRN =
TqRN for any pair of points given by (p,v) t-> (q,v). This is sometimes
referred to as distant parallelism. Here we see the reason that in the
context of calculus on RN, the explicit construction of vectors based at a
point is often deemed unnecessary. However, from the point of view of
manifold theory, the tangent space at a point is a fundamental construction.
We will define the notion of a tangent space at a point of a differentiable
manifold, and it will be seen that there is, in general, no canonical way to
identify tangent spaces at different points.
Actually, we shall give several (ultimately equivalent) definitions of the
tangent space. Let us start with the special case of a submanifold of RN. A
tangent vector at ρ can be variously thought of as the velocity of a curve,
as a direction for a directional derivative, and also as a geometric object
which has components that depend in a special way on the coordinates
used. Let us explore these aspects in the case of a submanifold of RN. If
Μ is an η-dimensional regular submanifold of RN, then a smooth curve
с : (—6, б) —> Μ is also a smooth curve into RN and c(0) is normally thought
of as a vector based at the point ρ = c(0). This vector is tangent to M. The
set of all vectors obtained in this way from curves into Μ is an n-dimensional
subspace of the tangent space of RN at ρ (described above). In this special
case, this subspace could play the role of the tangent space of Μ at p. Let
us tentatively accept this definition of the tangent space at ρ and denote it
by TpM.
Let vp := (p,v) G TpM. There are three things we should notice about
vp. First, there are many different curves с : (—6, e) —> Μ with c(0) = ρ
which all give the same tangent vector i>p, and there is an obvious equivalence
relation among these curves: two curves passing through ρ at t = 0 are
equivalent if they have the same velocity vector. Already one can see that
perhaps this could be turned around so that we can think of a tangent vector
as an equivalence class of curves. Curves would be equivalent if they agree
infinitesimally in some appropriate sense.
The second thing that we wish to bring out is that a tangent vector can
be used to construct a directional derivative operator. If vp = (p,v) is a
tangent vector in TpRN, then we have a directional derivative operator at ρ
which is a map C°°(RN) —> R given by / \-+ Df(p)v. Now if vp is tangent
2.1. The Tangent Space
57
to M, we would like a similar map C°°(M) —> R. If / is only defined on M,
then we do not have Df to work with but we can just take our directional
derivative to be the map given by
A*:/->(/oc)'(0),
where с : 7 —> Μ is any curve whose velocity at t = 0 is vp. Later we use
the abstract properties of such a directional derivative to actually define the
notion of a tangent vector.
Finally, notice how vp relates to charts for the submanifold. If (/7, y) is
a chart on Μ with ρ G /7, then by inverting we obtain a map y_1 : V —> M,
which we may then think of as a map into the ambient space RN. The map
y"1 parameterizes a portion of M. For convenience, let us suppose that
y"1 (0) = p. Then we have the "coordinate curves" уг ь-> у-1 (0,..., уг,..., 0)
for i = 1,..., п. The resulting tangent vectors Ei at ρ have principal parts
given by the partial derivatives so that
It can be shown that (E\,..., En) is a basis for TPM. For another coordinate
system у with y_1(0) = p, we similarly define a basis (£Ί,..., Ёп). If vp =
ΣΓ=ι aiEi = ΣΓ=ι δ<^ί» then lettinS a = (α1 τ --,аП) and δ = (α1,... ,αη),
the chain rule can be used to show that
a= D(yoy-l)\y(p)a,
which is classically written as
i=i y
Both (a1,..., an) and (a1,..., an) represent the tangent vector vp, but with
respect to different charts. This is a simple example of a transformation
law.
The various definitions for the notion of a tangent vector given below in
the general setting will be based in turn on the following three ideas: (1)
Equivalence classes of curves through a point. (2) Transformation laws for
the components of a tangent vector with respect to various charts. (3) The
idea of a "derivation" which is a kind of abstract directional derivative. Of
course we will also have to show how to relate these various definitions to
see that they are really equivalent.
2.1.1. Tangent space via curves. Let ρ be a point in a smooth n-
manifold M. Suppose that we have smooth curves c\ and C2 mapping into M,
each with open interval domains containing 0 G Μ and with ci(0) = сг(0) =
p. We say that c\ is tangent to c<i at ρ if for all smooth real-valued functions
58
2. The Tangent Structure
f defined on an open neighborhood of p, we have (/ о ci)' (0) = (/ о c^)' (0).
This is an equivalence relation on the set of all such curves. The reader
should check that this really is an equivalence relation and also do so when
we introduce other simple equivalence relations later. Define a tangent
vector at ρ to be an equivalence class under this relation.
Notation 2.1. The equivalence class of с will be denoted by [c], but we also
denote tangent vectors by notation such as vp or Xp, etc. Eventually we will
often denote tangent vectors simply as v,w, etc., but for the discussion to
follow we reserve these letters without the subscript for elements of Rn for
some n.
If vv = [c] then we will also write c(0) = vp. The tangent space TpM is
defined to be the set of all tangent vectors at ρ G M. A simple cut-off
function argument shows that c\ is equivalent to c2 if and only if (/ о с\) (0) =
(/ ° С2У (0) for all globally defined smooth functions / : Μ —> Ш.
Lemma 2.2. c\ is tangent to c2 at ρ if and only if (f о c\) (0) = (/ о c2)' (0)
for all Rk -valued functions f defined on an open neighborhood of p.
Proof. If / = (A ..., /"), then (/ о Cl)' (0) = (/ о с2)' (0) if and only if
(f о Cl)' (0) = (f о c2)' (0) for i = 1,... Л Thus (/ о Cl)' (0) = (/ о с2)' (0)
if ci is tangent to C2 at p. Conversely, let g be a smooth real-valued function
defined on an open neighborhood of ρ and consider the map / = (<?, 0,..., 0).
Then the equality (/ о ci)' (0) = (/oc2);(0) implies that (g о a)'(0) =
(goc2)'(0). □
The definition of tangent space just given is very geometric, but it has
one disadvantage. Namely, it is not immediately obvious that TpM is a
vector space in a natural way. The following principle is used to obtain a
vector space structure:
Proposition 2.3 (Consistent transfer of linear structure). Suppose that S
is a set and {Va}aeA is a family of n-dimensional vector spaces. Suppose
that for each a we have a bisection ba : Va —> S. If for every α, β G A the
map b^oba : Va -^ Vp is a linear isomorphism, then there is a unique vector
space structure on the set S such that each ba is a linear isomorphism.
Proof. Define addition in S by 51 + 52 := ba(b~1(s\) + b~1(s2))- This
definition is independent of the choice of a. Indeed,
ba(b-Hs,) + b-1(s2)) = baib-1 о Ъ$ о b^(8i) + б"1 о Ь0 о b^(s2)}
= bao b~l о bfi [^(si) + ^(«2)]
= bfi(bp1(si) + bp1(s2)).
2.1. The Tangent Space
59
The definition of scalar multiplication is а · s '.— ba(flba^(s)), and this is
shown to be independent of α in a similar way. The axioms of a vector
space are satisfied precisely because they are satisfied by each Va. D
We will use the above proposition to show that there is a natural vector
space structure on TpM. For every chart (xa,Ua) with ρ G /7, we have a
map ba : Rn —> TPM given by ν \-+ [7^], where ην : t \-+ x~1(xa(p) + tv) for t
in a sufficiently small but otherwise irrelevant interval containing 0.
Lemma 2.4. For each chart (C/Q, xQ), the map ba : Kn —> TpM is a bijection
and b^1 о ba = D (χβ ο χ"1) (xa(p))·
Proof. We have
(Χα°7υ)'(0) = —
- A
~ ~dt
xa ο χαλ(xa(p) + tv)
t=0
(χα(ρ) +ίν) = v.
t=0
Suppose that [7^] = [jw] for v,w G Rn. Then by Lemma 2.2 we have
υ = (xQ о 7V); (0) = (xQ о 7^); (0) = w.
This means that ba is injective.
Next we show that ba is surjective. Let [c] G TpM be represented by
с : (-6, б) -» Μ. Let υ := (xQ о с/ (0) <Е Mn. Then we have ba(v) = Ы,
where ην : t \-> x~1(xa(p)+tv). But [7^] = [c] since for any smooth / defined
near ρ we have
(/°7,)'(0)=!
t=o
/ о х-^ХоЫ + tv) = D(fo χ-1) (χα(ρ)) ■ ν
= D{fo χ"1) (χα(ρ)) · (χα о с)' (0) = (/ о с)' (0).
Thus ba is surjective. Prom Lemma 2.2 we see that the map [c] \-> (xQ о с)' (0)
is well-defined, and from the above we see that this map is exactly b~l. Thus
bp1oba(v)= — xpox~1(xa(p) + tv)) = D(xpox-1)(xa(p))v. D
at\t=o
The above lemma and proposition combine to provide a vector space
structure on the set of tangent vectors. Let us temporarily call the
tangent space defined above, the kinematic tangent space and denote it by
{TpM)kin. Thus, if Cp is the set of smooth curves с defined on some open
interval containing 0 such that c(0) = p, then
(rpM)kin = с,/-,
where the equivalence is as described above.
60
2. The Tangent Structure
Exercise 2.5. Let c\ and c2 be smooth curves mapping into a smooth
manifold M, each with open interval domains containing OgR and with
Cl(0) = c2(0) = p. Show that
(/oci),(0) = (/oc2),(0)
for all smooth / if and only if the curves χ ο c\ and χ о c2 have the same
velocity vector in Шп for some and hence any chart ({7, x).
2.1.2. Tangent space via charts. Let Л be the maximal atlas for an n-
manifold M. For fixed ρ e M, consider the set Γρ of all triples (p, i>, (/7, x)) G
{ρ} χ Rn χ Л such that ρ G /7. Define an equivalence relation on Γρ by
requiring that (p, i>, (/7, x)) ~ (p, it;, (V, y)) if and only if
(2-1) w=D(yox-l)\x{p).v.
In other words, the derivative at x(p) of the coordinate change у о х-1
"identifies" ν with w. The set Γρ/~ of equivalence classes can be given a
vector space structure as follows: For each chart (/7, x) containing p, we have
a map 6(t/)X) : Mn —> Гр/~ given by υ \-> [p, i>, (C/, x)], where [p, г;, (С/, х)]
denotes the equivalence class of (p, г;, (С/, χ)). To see that this map is a
bijection, notice that if [p, v, (/7,x)] = [p, ги, ({7, χ)], then
υ= 0(хох~г)\х(р) -v = w
by definition. By Proposition 2.3 we obtain a vector space structure on
Гр/~ whose elements are tangent vectors. This is another version of the
tangent space at p, and we shall (temporarily) denote this by (TPM) h .
The subscript "phys" refers to the fact that this version of the tangent space
is based on a "transformation law" and corresponds to a way of looking
at things that has traditionally been popular among physicists. If vp =
[ρ, ν, (Ϊ7, x)] G (TPM) h , then we say that υ G Mn represents vp with respect
to the chart ({7, x).
This viewpoint takes on a more familiar appearance if we use a more
classical notation. Let ({7, x) and (V, y) be two charts containing ρ in their
domains. If an η-tuple (г;1,..., vn) represents a tangent vector at ρ from the
point of view of ({7, x), and if the η-tuple (ги1,..., wn) represents the same
vector from the point of view of (V,y), then (2.1) is expressed in the form
x(p)
VJ,
.7 = 1
where we write the change of coordinates as уг = уг(х1,... ,xn) with 1 <
г < п.
Notation 2.6. It is sometimes convenient to index the maximal atlas: Л =
{(/7α,χα)}α£Λ· Then we would consider triples of the form (ρ,υ,α) and let
2.1. The Tangent Space
61
the defining equivalence relation for (TPM) , be (ρ, υ, α) ~ (p, w, β) if and
only if
D^ox-1^
Xa(p)
V = W.
2.1.3. Tangent space via derivations. We abstract the notion of
directional derivative for our next approach to the tangent space. There are
actually at least two common versions of this approach, and we explain both.
Let Μ be a smooth manifold of dimension n. A tangent vector vp at ρ is
a linear map vp : C°°(Ai) -» R with the property that for f,g e C°°(M),
Vp(fg) = g(p)vP (/) + f(p)vp (g).
This is the Leibniz law. We may say that a tangent vector at ρ is a
derivation of the algebra C°°(M) with respect to the evaluation map evp
at ρ defined by evp(/) := f(p). Alternatively, we say that vp is a
derivation at p. The set of such derivations at ρ is easily seen to be a vector
space which is called the tangent space at ρ and is denoted by TpM. We
temporarily distinguish this version of the tangent space from (TpM)k[n and
(TPM) h defined previously by denoting it (TpM), and referring to it as
the algebraic tangent space. We could also consider the vector space of
derivations of Cr(M) at a point for r < oo, but this would not give a finite-
dimensional vector space and so is not a good candidate for the definition
of the tangent space (see Problem 18). Recall that if ({7, x) is a chart on an
η-manifold M, we have defined df /дхг by
g(p):=A(/ox-l)(x(p))
Definition 2.7. Given ({7, x) and ρ as above, define the operator g^| :
(see Definition 1.57).
Definition 2.7.
C°°(M) -» R by
_d_
dxl
/:=Ι?ω·
It is often helpful to use the easily verified fact that if & : (—6, e) —> Μ
is the curve defined for sufficiently small e by
Ci(t) :=x~l(x(p) + ei),
where e^ is the г-th member of the standard basis of Mn, then
д
дхг
/ = liB/W*))-/W
/ι->ο h
ρ
Prom the usual product rule it follows that -^ | is a derivation at ρ and so
is an element of (TpM)aig. We will show that (gfr| , · · ·, g^L) *s a basis
for the vector space (TpM)aig.
62
2. The Tangent Structure
Lemma 2.8. Let vp G (TpM)aig. Then
(i) if f,9 £ C°°(M) are equal on some neighborhood ofp, then vp (/) =
(ii) ifhe C°°(M) is constant on some neighborhood ofp, then vp (h) =
0.
Proof, (i) Since vp is a linear map, it suffices to show that if / = 0 on a
neighborhood U of p, then vp (/) = 0. Of course vp(0) = 0. Let β be a
cut-off function with support in U and β(ρ) = 1. Then we have that β} is
identically zero and so
0 = νΡ(βί) = ί(ρ)νΡ(β) + β(ρ)νΡ(ί)
= vp (/) (since β(ρ) = 1 and f(p) = 0).
(ii) Prom what we have just shown, it suffices to assume that h is equal
to a constant с globally on M. In the special case с = 1, we have
vp (1) = vp(l · 1) = 1 · vp (1) + 1 · vp (1) = 2vp (1),
so that vp (1) = 0. Finally we have vp (c) = vp (lc) = с (vp(l)) = 0. Π
Notation 2.9. We shall often write vpf or vp · / in place of vp (/).
We must now deal with a technical issue. We anticipate that the action
of a derivation is really a differentiation and so it seems that a derivation at
ρ should be able to act on a function defined only in some neighborhood U of
p. It is pretty easy to see how this would work for -J^ | . But the domain of
a derivation as defined is the ring C°°(M) and not C°°(U). There is nothing
in the definition that immediately allows an element of (TpM)aig to act on
C°°(U) unless U = M. It turns out that we can in fact identify (7pf7)al
with (ΤρΜ)ά1, and the following discussion shows how this is done. Once
we reach a fuller understanding of the tangent space, this identification will
be natural and automatic. So, let ρ G U С Μ with U open. We construct
a rather obvious map Φ : (TpE/)al —>■ (ТрМ)л1 by using the restriction
map C°°(M) -» C°°(U). For each wp e TPU, we define uTp : C°°(M) -> R
by Щ)(Л '·= wp(f\u)· -^ ls easy t° show that ufp is a derivation of the
appropriate type and so vfp G (TpM)ul . Thus we get a linear map Φ :
(TpZ7)al —> (7pM)al . We want to show that this map is an isomorphism,
but notice that we have not yet established the finite-dimensionality of either
(Tpi7)al or (ΤρΜ)ά1. First we show that Φ : wp \-> vfp has trivial kernel. So
suppose that 2^ = 0, i.e. uTp(f) = 0 for all / G C°°(M). Let h G C°°(i7).
Pick a cut-off function β with support in U so that β\ι extends by zero to
a smooth function / on all of Μ that agrees with feona neighborhood of
2.1. The Tangent Space
63
p. Then by the above lemma, wp(h) = wp(f\u) = wp(f) = 0. Thus, since h
was arbitrary, we see that wp = 0 and so Φ has trivial kernel.
Next we show that Φ is onto. Let vp G {ΤΡΜ)&1 . We wish to define wp G
(TpZ7)al by wp(h) := νρ(βΚ), where β is as above and βΚ is extended by zero
to a function in C°°(M). If β\ is another similar choice of cut-off function,
then βΚ and β\Η (both extended to all of M) agree on a neighborhood of p,
and so by Lemma 2.8, νρ(βΚ) = νρ(β\Κ). Thus wp is well-defined. Thinking
of β{ί\ν) as defined on Μ, we have Wp(f) -^w^f^) = νρ(β/\υ) =vp(f)
since β/\υ and / agree on a neighborhood of p. Thus Φ : (TpU)al —>
(ΤΡΜ), is an isomorphism.
Because of this isomorphism, we tend to identify {TpU)al with (TpM),
and in particular, if ({7, x) is a chart, we think of the derivations g~| ,
1 < г < η as being simultaneously elements of both {TPU)&1 and (TpM)..
In either case the formula is the same: ^| / = ^ J*t ^(x(p)).
Notice that agreeing on a neighborhood of a point is an important
relation here and this provides motivation for employing the notion of a germ
of a function (Definition 1.68). First we establish the basis theorem:
Theorem 2.10. Let Μ be an η-manifold and (/7, x) a chart with ρ G U.
Then the η-tuple of vectors (derivations) ί ^γ| ,..., g^| j is a basis for
(ΤρΜ)ά1 . Furthermore, for each vp G (TpM), we have
Proof. Prom our discussion above we may assume that x(U) is a convex
set such as a ball of radius ε in Rn. By composing with a translation we
assume that x(p) = 0. This makes no difference for what we wish to prove
since vp applied to a constant is 0. For any smooth function д defined on
the convex set χ(U) let
• Ы := / ^-(tu) dt for all и G x(U).
Jo dulX
The fundamental theorem of calculus can be used to show that g = g(0) +
Y,giu\ We see that #(0) = jM . For a function / e C°°({7), we let
g := /ox-1. Using the above, we arrive at the expression / = f(p) + J2fiXl1
and applying ^\ we get fi(p) = §^\ . Now apply the derivation vp to
64
2. The Tangent Structure
ρ
f = f(p) + Σ ίίχ1 to obtain
This shows that vp = Σνρ(χι) -Jj^\ and thus we have a spanning set.
To see that (g^r| , · · ·, g^rL) ls a linearly independent set, let us assume
that Σα% ]j& | = 0 (the zero derivation). Applying this to xj gives 0 =
Σ)α* j}&\ = Σα^ = °? ·> an(i since J was arbitrary, we get the result. D
Remark 2.11. On the manifold W\ we have the identity map id : Rn-> Rn
which gives the standard chart. As is often the case, the simplest situations
have the most confusing notation because of the various identifications that
may exist. On R, there is one coordinate function, which we often denote
by either и or i. This single function is just id^. The basis vector at ίο G Ш
associated to this coordinate is -^\t (or щ\ ). If we think of the tangent
space at ίο G IR as being {ίο} χ Μ, then ^ \t is just (ίο, 1). It is also common
to denote -^\t by "1" regardless of the point ίο·
Above, we used the notion of a derivation as one way to define a tangent
vector. There is a slight variation of this approach that allows us to worry
a bit less about the relation between (Гр/7)а1 and (ΤΡΜ)^ . Let Tv —
α^(Μ,Μ) be the algebra of germs of functions defined near p. Recall that
if / is a representative for the equivalence class [/] G Tv, then we can
unambiguously define the value of [/] at ρ by [f](p) = f(p)- Thus we have
an evaluation map evp : Tv —> Ш.
Definition 2.12. A derivation (with respect to the evaluation map evp)
of the algebra Tv is a map Vv : Tv —> Ш such that 2?p([/][<;]) = f(p)T>p[g] +
5(p)Pp[/]forall[/],[5]eJp.
The set of all these derivations on Tv is easily seen to be a real vector
space and is sometimes denoted by Der(Jrp).
Remark 2.13. The notational distinction between a function and its germ
at a point is not always maintained; Vpf is taken to mean Vp[f].
Let Μ be a smooth manifold of dimension n. Consider the set of all
germs of C°° functions Tv at ρ G M. The vector space Der(Fp) of derivations
of Tp with respect to the evaluation map evp could also be taken as the
definition of the tangent space at p. This would be a slight variation of
what we have called the algebraic tangent space.
2.2. Interpretations
65
2.2. Interpretations
We will now show how to move from one definition of tangent vector to the
next. Let Μ be a (smooth) η-manifold. Consider a tangent vector vp as an
equivalence class of curves represented by с : I —> Μ with c(0) = p. We
obtain a derivation by defining
/ d
foe.
t=0
This gives a map (ТрМ)ып —^ (TpM)aig which can be shown to be an
isomorphism. We also have a natural isomorphism (ТрМ)^[П —> (TpM)phys.
Given [c] G (TpM)kin, we obtain an element vp G (TpM)phys by letting vp be
the equivalence class of the triple (p,v, (/7, x)), where vl := ^.\t=Qxl о с for
a chart (/7, x) with ρ e U.
If vp is a derivation at ρ and ({7, x) an admissible chart with domain
containing p, then u<pi as a tangent vector in the sense of Definition 2.1.2, is
represented by the triple (p, v, (C/, x)), where ν = (г;1,..., vn) is given by
vl — vpxl (vp is acting as a derivation).
This gives us an isomorphism (TpM)aig —> (TpM)p\lys.
Next we exhibit the inverse isomorphism (TpM)phys —> (7pM)aig.
Suppose that [(p,v, (C/,x))] G (TpM)phys where г> G Mn. We obtain a derivation
by defining
vpf=D(fox-1)\
x(p)
г;.
In other words,
n я
дхг
f
for v = (г;1,... ,vn). It is an easy exercise that vp defined in this way is
independent of the representative triple (p, г;, (/7, χ)).
We now adopt the explicitly flexible attitude of interpreting
a tangent vector in any of the ways we have described above
depending on the situation. Thus we effectively identify the spaces
(TpM)kim (rpM)phys and (TpM)aig. Henceforth we use the notation TPM
for the tangent space of a manifold Μ at a point p.
Definition 2.14. The dual space to a tangent space TpM is called the
cotangent space and is denoted by Τ* Μ. An element of T*M is referred
to as a covector.
The basis for T*M that is dual to the coordinate basis (gfr | , · · ·, gjpr | )
described above is denoted (dxl I ..... dxn\S). By definition dxlI ( -A-1 ) =
b%y (Note that δ%3■ = 1 if г = j and δ1- = 0 if г Ф j. The symbols δ^ and
66
2. The Tangent Structure
fh'dt
διΐ are defined similarly.) The reason for the differential notation dx1 will
be explained below. Sometimes one abbreviates g|j| and dxl\ to g|j and
dx1 respectively, but there is some risk of confusion since later g|j and dx1
will more properly denote not elements of the vector spaces TpM and T*M,
but rather fields defined over a chart domain. More on this shortly.
2.2.1. Tangent space of a vector space. Our provisional definition of
the tangent space at point ρ in a vector space V was the set {ρ} χ V, but this
set does not immediately fit any of the definitions of tangent space just given.
This is remedied by finding a natural isomorphism {ρ} χ V = TpV. One may
pick a version of the tangent space and then exhibit a natural isomorphism
directly, but we take a slightly different approach. Namely, we first define
a natural map jp : V —> TpV. We think in terms of equivalence classes of
curves. For each υ G V, let cp^v : R —>V be the curve cp,v(t) := ρ + tv. Then
Jp(v) := [Cp^v] e TpV.
As a derivation, jp(v) acts according to
f(p + tv).
10
On the other hand, we have the obvious projection pr2 : {ρ} χ V —> V. Then
our natural isomorphism {p} xV = TpV is just jp о pr2. The isomorphism
between the vector spaces {ρ} χ V and TpV is so natural that they are often
identified. Of course, since TpV itself has various manifestations ((7pV)al ,
(TpV)phys, and (TPV)
kin' we now have a multitude of spaces which are
potentially being identified in the case of a vector space.
Note: Because of the identification of {ρ} χ V with TpV , we shall often
denote by pr2 the map TpV —> V. Furthermore, in certain contexts, TpV
is identified with V itself. The potential identifications introduced here are
often referred to as "canonical" or "natural" and jp : V —> TpV is often
called the canonical or natural isomorphism. The inverse map TpV —> V
is also referred to as the canonical or natural isomorphism. Context will
keep things straight.
2.3. The Tangent Map
The first definition given below of the tangent map at ρ G Μ of a smooth
map f : Μ -* N will be considered our main definition, but the others are
actually equivalent. Given / and ρ as above, we wish to define a linear map
Tpf : TpM —> Tf(p)N. Since we have several definitions of tangent space, we
expect to see several equivalent definitions of the tangent map. For the first
definition we think of TpM as (TpM)kin.
2.3. The Tangent Map
67
Figure 2.1. Tangent map
Definition 2.15 (Tangent map I). If we have a smooth function between
manifolds
and we consider a point ρ G Μ and its image q = f(p) G iV, then we define
the tangent map at p,
in the following way: Suppose that i>p G TPM and we pick a curve с with
c(0) = ρ so that г;р = [c]; then by definition
TPf -Vp = [foc]e TqN,
where [/ о с] G T^iV is the vector represented by the curve foe. (Recall
Notation 1.3.)
Another popular way to denote the tangent map Tpf is /p*, but a further
abbreviation to /* is dangerous since it conflicts with a related meaning for
/* introduced later.
Exercise 2.16. Let / : Μ -» TV be a smooth map. Show that Tpf : TpM -»
TqN is a linear map and that if / is a diffeomorphism, then Tpf is a linear
isomorphism. [Hint: Think about how the linear structure on (TpM)k[n was
defined.]
We have the following version of the chain rule for tangent maps:
Theorem 2.17. Let f : Μ -* N and д : Ν -^ Ρ be smooth maps. For each
peM we have Тр(д о f) = (Tf{p)g) о Tpf.
68
2. The Tangent Structure
Proof. Let ν G TPM be represented by the curve с so that ν = [с]. Then
foe represents Tpf(v) and we have
TP(gof)(v) = [(gof)oc] = \go(foc)]
= (Tf{p)g) (Tpf(v)) = ((Tf{p)g) оTpf) (v). D
For the next alternative definition of tangent map, we consider TVM as
№>M)phys.
Definition 2.18 (Tangent map II). Let / : Μ —> TV be a smooth map and
consider a point ρ e Μ with image q = f(p) G N. Choose any chart (/7, x)
containing ρ and a chart (V, y) containing q = f(p) so that for vp G TpM
we have the representative (p, г>, (/У, x)). Then the tangent map Tp/ :
TPM —> Tf(p)N is defined by letting the representative of Tp/ · i>p in the
chart (V,y) be given by (q,w, (V,y)), where
w = D(y о f ox-1) · v.
This uniquely determines Tpf · i>, and the chain rule guarantees that this is
well-defined (independent of the choice of charts).
Another alternative definition of tangent map is given in terms of
derivations:
Definition 2.19 (Tangent map III). Let Μ be a smooth η-manifold.
Continuing our set up above, we define Tpf · vp as a derivation by
(Tpf · vp)g = vP(g о f)
for each smooth function g. It is easy to check that this defines a derivation
and so a tangent vector in TqM. This map is yet another version of the
tangent map Tpf.
In the above definition, one could take g to be the germ of a smooth
function defined on a neighborhood of f(p) and then Tpf · vp would act as
a derivation of such germs.
One can check the chain rule for tangent maps using the above definition
in terms of derivations as follows: If / : Μ —> N and g : Ν -^ Ρ are smooth
maps and vp G TpM, then for h G C°°(M) we have
(TP(g о f) · vp)h = vp(h о (go /)) = Vp((h о g) о f)
= (Tpf . Vp) (hog) = Tf{p)g . (Tpf . vp) h,
so since h and vp were arbitrary, we conclude again that Tp(gof) = (Tf^g) о
Tpf. We have just proved the same thing using different interpretations of
tangent space and tangent map, but the isomorphisms between the versions
are so natural that we may use any version convenient for a given purpose
and then draw conclusions about all versions. This could be formalized
2.3. The Tangent Map
69
using category theory arguments but we shall forgo the endeavor. Just as
with the idea of the tangent space, we will think of the above versions of the
tangent map as a single abstract thing with more than one interpretation
depending on the interpretation of tangent space in play.
Now we introduce the differential of a function.
Definition 2.20. Let Μ be a smooth manifold and let ρ G M. For / G
C°°(M), we define the differential of / at ρ as the linear map df(p) :
TPM -» R given by
df(p) · vP = vPf
for all vp G TPM. Thus df(p) G T*M.
The notation dfp or df\ is also used in place of df(p). One may view
dfp(vp) as an "infinitesimal" aspect of the composition /и/07, where
7;(0) = vp. It is easy to see that df(p) is just Tpf followed by the natural
map 7/(p)IR —> Ш. In a way, df(p) is just a version of the tangent map that
takes advantage of the identification of 7/(p)R with R (recall Remark 2.11).
Let (/7, x) be a chart with χ = (χ1,..., xn) and let ρ G U. We previously
denoted the basis dual to (gfr(p), · · ·, g|n (p)) by (dxl\ ,..., dxn\p), and
now this notation is justified since we can check directly that we really do
have dxl\ (Al ) = δ].
\p V οχ3 \ρ) 3
Definition 2.21. Let I = (a, 6) be an interval in R. If с : I —> Μ is a smooth
map (a curve), then the velocity at to G / is the vector c(io) G TC^M
defined by
c(io):=TfoC·- ,
ι to
where J^| is the coordinate basis vector at ίο G Tt0I = Tt0R associated to
the standard coordinate function on Ш (denoted here by u).
Note: We may also occasionally write c' for c.
Thus if / is a smooth function defined in a neighborhood of c(io), then
έ(ίο) acts as a derivation as follows:
m-s-i
/°c;
t=tQ
c(io) is also denoted by ^ | с In this notation, с = -^с is a map that assigns
to every ί G / the tangent vector c(i) := Ttc · ^ \t, and this is referred to as
the velocity field along the curve с Also note that we may view c(io) as the
equivalence class of the curve ί \-> c(t + ίο).
70
2. The Tangent Structure
The differential can be generalized:
Definition 2.22. Let V be a vector space. For a smooth / : Μ —> V with
ρ G Μ as above, the differential df(p) : TpM —> V is the composition of the
tangent map Tpf and the canonical map TyY —> V where у = /(ρ),
df(p) : TPM Ц TyV -* V.
The notational distinction between Tpf and dfp is not universal, and dfp
is itself often used to denote Tpf.
Exercise 2.23. Let / : Μ —> N be a smooth map. Show that if Tpf = 0
for all ρ G M, then / is locally constant (constant on connected components
of M).
We now consider the inclusion map ι : U ^-> Μ where U is open. For
ρ G /7, we get the tangent map Tpl : TpZ7 —> TPM. Let us look at this map
from several points of view corresponding to the various ways one can define
the tangent space. First, consider tangent spaces from the derivation point
of view. From this point of view the map Tpl is defined for vp G TpU as
acting on C°°(M) as follows: Tpt(vp)f = υρ (f ο t) = vp(f\U). We have
seen this map before where we called it Φ : (ΤρΙΙ)ά1 —> (ΤρΜ)ά1 , and it was
observed to be an isomorphism and we decided to identify (TpU), with
(TpM),. From the point of view of equivalence classes of curves, the map
Tpi sends [7] to [i о 7], But while 7 is a curve into /7, the map ι ο η is
simply the same curve, but thought of as mapping into M. We leave it to
the reader to verify the expected fact that Tpl is a linear isomorphism. Thus
it makes sense to identify [7] with [l о 7] and so again to identify TPU with
TpM via this isomorphism. Next consider vp G TpU to be represented by
a triple (p, i>, (f/Q,xQ)) where (C/Q,xQ) is a chart on the open manifold U.
Since (C/Q,xQ) is also a chart on M, the triple also represents an element of
TpM which is none other than Tpl · vp. The map Tpl looks more natural and
trivial than ever, and we once again see the motivation for identifying TpU
and TpM.
More generally, when S is a regular submanifold of M, then the tangent
space TpS at ρ G S С Μ is intuitively a subspace of TPM. Again, this is
true as long as one is not bent on distinguishing a curve in S through ρ from
the "same" curve thought of as a map into M. If one wants to be pedantic,
then we have the inclusion map l : S ^-> M, and if с : I —> S is a curve into
5, then loc : I -+ Μ is a, map into M. At the tangent level this means that
c(0) G TPS while (ι о с);(0) G TpM. Thus Tpl : TPS -» Tpl(TpS) С ТрМ.
When convenient, we just identify TpS with Tpl(TpS) and so think of TpS
as a subspace of TpM and take Tpl to be an inclusion.
2.3. The Tangent Map
71
Recall that a smooth pointed map / : (M,p) —> (N,q) is a smooth map
f : Μ -* N such that f(p) = q. Taking the set of all pairs (M,p) as objects
and pointed maps as morphisms we have an obvious category.
Definition 2.24. The "pointed" version of the tangent functor Τ takes
(M,p) to TpM and a map / : (M,p) -» (N,q) to the linear map Tpf :
TPM -» TqM.
The following theorem is the inverse mapping theorem for
manifolds and will be used repeatedly.
Theorem 2.25. If f : Μ -± N is α smooth map such that Tpf : TpM -»
TqN is an isomorphism, then there exists an open neighborhood О of ρ such
that f(0) is open and f\Q : О —> f(0) is a diffeomorphism. If Tpf is an
isomorphism for all ρ G M, then f : Μ -+ N is a local diffeomorphism.
Proof. The proof is a simple application of the inverse mapping theorem,
Theorem C.l found in Appendix C. Let (/7, x) be a chart centered at ρ
and let (V,y) be a chart centered at q = f(p) with f(U) С V. Prom the
fact that Tpf is an isomorphism we easily deduce that Μ and TV have the
same dimension, say n, and then D(y ο / ο χ-1)(χ(ρ)) : Шп —> Шп is an
isomorphism. Prom Theorem C.l if follows that у о / ο χ-1 restricts to a
diffeomorphism on some neighborhood O' of 0 G W1. Prom this we obtain
that f\Q : О —> f(0) is a diffeomorphism where О = х-1 (0;). The second
part follows from the first. D
2.3.1. Tangent spaces on manifolds with boundary. Recall that a
manifold with boundary is modeled on the half-spaces Мд>с := {α G Шп :
λ (α) > с}. If M is a manifold with boundary, then the tangent space TpM
is defined as before. For instance, even if ρ G <9M, the fiber TVM may still be
thought of as consisting of equivalence classes where (ρ, ν, α) ~ (ρ,νο,β) if
and only if ΰ(χβ ο χ~χ) |χ , yv = w. Notice that for a given chart (C/Q, xQ),
the vectors υ in (ρ,υ,α) still run through all of Rn and so TpM still has
dimension η even if ρ G dM. On the other hand, if ρ G <9M, then for any
half-space chart χ : U —> Кд>с with ρ in its domain, Гх-1(Гх(р)Мд=с) is a
subspace of TpM. This is the subspace of vectors tangent to the boundary
and is identified with TpdM, the tangent space to dM (also a manifold).
Exercise 2.26. Show that this subspace does not depend on the choice of
chart.
If one traces back through the definitions, it becomes clear that because
of the way charts and differentiability are defined for manifolds with
boundary, any smooth function defined on a neighborhood of a boundary point
can be thought of as being the restriction of a smooth function defined
72
2. The Tangent Structure
dM
ITPM
Tp(dM)
Figure 2.2. Tangents at a boundary point
slightly "outside" M. More precisely, the representative function always
has a smooth extension from a (relatively open) neighborhood in Мд>с to
a neighborhood in Rn. The derivatives of the extended function at points
of Жд>с = Мд=с are independent of the extension. These considerations
can be used to show that tangent vectors at boundary points of a smooth
η-manifold with boundary can still be considered as derivations of germs
of smooth functions. A closed interval [a, b] is a one-dimensional manifold
with boundary, and with only minor modifications in our definitions we can
also consider equivalence classes of curves to define the full tangent space at
a boundary point. It also follows that if с is a smooth curve with domain
[a, b], then we can make sense of the velocities c(d) and c(b), and this is true
even if c{d) or c(b) is a boundary point. The major portion of the theory of
manifolds extends in a natural way to manifolds with boundary.
2.4. Tangents of Products
Suppose that / : M\ χ Μ<ι —> N is a smooth map. For fixed ρ G Μχ and fixed
q G M2, consider the "insertion" maps iv : у \-ь (ρ, у) and tq : χ \-> (x,g).
Then fotq and fotp are the maps sometimes denoted by /(·, q) and /(p, ·).
Definition 2.27. Let / : Μχ χ Μ<ι —> TV be as above. Define the partial
tangent maps d\f and $2/ by
(dif) (p, q) := Tv (/ о fi) : TpMx -+ Tf{m)N,
(d2f) (p, q) := Tq (/ о Lp) : TqM2 -> Tf{m)N.
Next we introduce another natural identification. It is obvious that a
curve с : / —> M\ χ Μ<ι is equivalent to a pair of curves
c\\I -* Mi,
c2 : /-» M2.
2.4. Tangents of Products
73
The infinitesimal version of this fact gives rise to a natural identification on
the tangent level. If c(t) = (ci(t),c2(t)) and c(0) = (p, g), then the map
Т{т)рт1 x T(Pj(?)pr2 : T(M)(Mi x M2) -+ ΓρΜχ χ TqM2
is given by [с] н-> ([ci], [сг]), which is quite natural. This map is an
isomorphism. Indeed, consider the insertion maps iv : q ь-> (ρ, g) and iq : ρ ь-> (p, g),
P*l Pr2
(Μι,ρ) ^ (A^i x Af2, (p,?)) Z^ (M2,P) ·
We have linear monomorphisms Tiq(p) : TVM\ -> T^V^{M\ χ Μ2) and
Пр{д) : TgM2 -> η,,,,) (Μι χ M2),
Τ(ρ,9)ΡΓ1 Γ(Ρ,9)ΡΓ2
ТрМг ^ZZ T(p,?)Mi x M2 ^ TPM2 .
J ρ L 1 q Lp
Then, we have the map
Ttq + Ttp : TpMi χ TqM2 -» Γ(Μ)(Μι χ M2),
which sends (v,w) G ΓρΜχ χ TqM2 to T^(v) + Tlp(w). It can be checked
that this map is the inverse of [с] ь-> ([ci],[c2])· Thus we may identify
Γ(ρ^)(Μι χ M2) with ΓρΜχ χ TqM2. Let us say a bit about the naturalness
of this identification. In the smooth category, there is a direct product
operation. The essential point is that for any two manifolds Μχ and M2,
the manifold Μχ χ Μ2 together with the two projection maps serves as the
direct product in the technical sense that for any smooth maps / : TV —> Μχ
and д : N —> M2 we always have the unique map f χ д : N —> Μχ χ Μ2
which makes the following diagram commute:
Μχ ^ Μχ χ M2 ^ M2
For a point χ e N, write ρ = f(x) and q = g(x). On the tangent level we
have
TXN
τρΜι J^!^ тш(мг χ м2) _!k^!u TqM2
which is a diagram in the vector space category. In the category of vector
spaces, the product of TPM\ and TPM2 is ΓρΜχ χ TPM2 together with the
74
2. The Tangent Structure
projections onto the two factors. Corresponding to the maps Tpf and Tqg
we have the map Tpf χ Tqg. But
(Γ(ρ.ς)ΡΓ1 X T(p,c?)Pr2) °Tx(f X g)
= (Тшртг oTx(fx g)) χ (Г(м)рг2 оГЛ/х р))
= Tx (ΡΓι О (/ Χ ρ)) Χ Γ, (РГ2 Ο (/ Χ g))
= Txf χ Txg.
Thus under the identification introduced above the map Tx(f x g)
corresponds to Txf χ Txg. Now if υ G ΓρΜχ and ги G ΤςΜ2, then (г;, ги) represents
an element of Τ(Μ)(Μχ χ M2), and so it should act as a derivation. In fact,
we can discover how this works by writing (г;, w) = (г>, 0) + (0, w). If c\ is a
curve that represents ν and c<i is a curve that represents w, then (г>,0) and
(0, w) are represented by t \-> (ci(i), q) and t \-> (p, сг(*)) respectively. Then
for any smooth function / on M\ χ M2 we have
(t;,ti;)/ = (i;,0)/ + (0,ti;)/ = |
/(ci(t),i)+^
/(P,c2(t))
0
= г;[/ог9] +гу[/огр].
Lemma 2.28 (Partials lemma). For α map f : M\ χ M2 —> N, we have
where we have used the aforementioned identification T(pq}(Mi χ Μ2) =
ΓρΜιχΓςΜ2.
Proving this last lemma is much easier and more instructive than reading
the proof so we leave it to the reader in good conscience.
2.5. Critical Points and Values
Definition 2.29. Let / : Μ -» TV be a Cr-map and ρ G M. We say that
ρ is a regular point for the map / if Tpf is a surjection. Otherwise, ρ
is called a critical point or singular point. A point q in TV is called a
regular value of / if every point in the inverse image f~1{q} is a regular
point for /. This includes the case where f~1{q} is empty. A point of TV
that is not a regular value is called a critical value.
Most values of a smooth map are regular values. In order to make this
precise, we will introduce the notion of measure zero on a second countable
smooth manifold. It is actually no problem to define a Lebesgue measure
on such a manifold, but for now the notion of measure zero is all we need.
Definition 2.30. A subset A of Rn is said to be of measure zero if for
any б > 0 there is a sequence of cubes {W{} such that А С \JW{ and
Σ vo\(Wi) < 6. Here, vo\(Wi) denotes the volume of the cube.
2.5. Critical Points and Values
75
In the definition above, if the W{ are taken to be balls, then we arrive at
the very same notion of measure zero. It is easy to show that a countable
union of sets of measure zero is still of measure zero. Our definition is
consistent with the usual definition of Lebesgue measure zero as defined in
standard measure theory courses.
Lemma 2.31. Let U С W1 be open and f : U -» Rn a C1 map. If А С U
has measure zero, then f(A) has measure zero.
Proof. Since A is certainly contained in the countable union of compact
balls (all of which are translates of a ball at the origin), we may as well
assume that U = B(0,r) and that A is contained in a slightly smaller ball
B(0, r — δ) С β(0, r). By the mean value theorem (see Appendix C), there is
a constant с depending only on / and its domain such that for x, у G B(0, r)
we have ||/(y) — f(x)\\ < с ||x — y||. Let e > 0 be given. Since A has measure
zero, there is a sequence of balls B(xi, ei) such that А С \JB(x{, ei) and
Х>1(В(х4,£0)<2^.
Thus f(B(Xi,€i)) С В(/(х<),2с€<), and while f(A) С \JB(f(xi),2cei), we
also have
vol(UB№i),2cC<)) < 2vol(B(/(xi),2c6i))
< ^vol(B!)(2c60n < 2ncnY^vo\(B(xuei)) < 6,
where B\ = Б(0,1) is the ball of radius one centered at the origin. Since e
was arbitrary, it follows that A has measure zero. D
The previous lemma allows us to make the following definition:
Definition 2.32. Let Μ be an η-manifold that is second countable. A
subset А С М is said to be of measure zero if for every admissible chart
(17, x) the set x(A Π U) has measure zero in W1.
In order for this to be a reasonable definition, the manifold must be
second countable so that every atlas has a countable subatlas. This way we
may be assured that every set that we have defined to be measure zero is
the countable union of sets that are measure zero as viewed in some chart.
It is not hard to see that in this more general setting it is still true that a
countable union of sets of measure zero has measure zero. Also, we still have
that the image of a set of measure zero under a smooth map, has measure
zero.
Proposition 2.33. Let Μ be second countable as above and A = {(ί7α, χα)}
a fixed atlas for M. If χα(Α Π Ua) has measure zero for all a, then A has
measure zero.
76
2. The Tangent Structure
Proof. The atlas A has a countable subatlas, so we may as well assume from
the start that A is countable. We need to show that given any admissible
chart (/7, x) the set x(A Π U) has measure zero. We have
x(A Π U) = [J x(A Π U Π Uа) (а countable union).
α
Since χα(ΑΠ/7Π{7α) С χα(ΑΠί7α), we see that xa(Ar\UDUa) has measure
zero for all a. But x(A Π U Π Ua) = χ ο χ"1 о ха(А Π U Π ί7α), and so by
the lemma above, x(A f)U DUa) also has measure zero. Thus x(A Π /7) has
measure zero since it is a countable union of sets of measure zero. D
We now state the famous and useful theorem of Arthur Sard.
Theorem 2.34 (Sard). Let N be an η-manifold and Μ an m-manifold,
both assumed second countable. For a smooth map / : iV —> M, the set of
critical values has measure zero.
The somewhat technical proof may be found in the online supplement
[Lee, Jeff] or in [Bro-Jan].
Corollary 2.35. If Μ and N are second countable manifolds, then the set
of regular values of a smooth map f : Μ -^ Ν is dense in N.
2.5.1. Morse lemma. If we consider a smooth function / : Μ —> R, and
assume that Μ is a compact manifold (without boundary), then / must
achieve both a maximum at one or more points of Μ and a minimum at one
or more points of M. Let pe be one of these points. The usual argument
shows that df\ = 0. (Recall that under the usual identification of Ш
with any of its tangent spaces we have df\ = TPef'.) Now let ρ be some
point for which df\ = 0, i.e. ρ is a critical point for /. Does / achieve
either a maximum or a minimum at pi How does the function behave in a
neighborhood of ρΊ As the reader may well be aware, these questions are
easier to answer in case the second derivative of / at ρ is nondegenerate.
But what is the second derivative in this case?
Definition 2.36. The Hessian matrix of / at one of its critical points ρ
and with respect to coordinates χ = (χ1,... ,xn), is the matrix of second
partials:
[Hf*]T = ; ·:
<92/ox l / \ <92/ox * / \
L dxndxl \X°) '" дхпдхп\Х°) J
where xq = x{p). The critical point ρ is called nondegenerate if Η is
nonsingular.
2.5. Critical Points and Values
77
Any such matrix Η is symmetric, and by Sylvester's law of inertia, it
is congruent to a diagonal matrix whose diagonal entries are either 0 or 1
or —1. The number of — l's occurring in this diagonal matrix is called the
index of the critical point. According to Problem 13 we may define the
Hessian Hf# : TPM χ TPM —> R, which is a symmetric bilinear form at
each critical point ρ of /, by letting Hf^p(v,w) = Xp(Y f) = Yp(Xf) for
any vector fields X and Υ which respectively take the values ν and w at
p. Thus, we may give a coordinate free definition of a nondegenerate point
for /. Namely, ρ is a nondegenerate point for / if and only if Hf# is a
nondegenerate bilinear form. The form Ядр is nondegenerate if for each
fixed nonzero υ G TpM the map Яу?р(г;, ·) : TpM —> Ш is a nonzero element
of the dual space T*M.
Exercise 2.37. Show that the nondegeneracy is well-defined by either of
the two definitions given above and that the definitions agree.
Exercise 2.38. Show that nondegenerate critical points are isolated. Show
by example that this need not be true for general critical points.
The structure of a function near one of its nondegenerate critical points
is given by the following famous theorem of M. Morse:
Theorem 2.39 (Morse lemma). Let f : Μ —> Ш be α smooth function and
let xo be a nondegenerate critical point for f of index v. Then there is a
local coordinate system (/7, x) containing xo such that the local representative
fu'-=f° x_1 for f has the form
fu(x\ ...,xn) = /(x0) + Σ Κίΐχϊχ3
hj
and it may be arranged that the matrix h = (hij) is a diagonal matrix of the
form diag(—1,..., —1,1,..., 1) for some number (perhaps zero) of ones and
minus ones. The number of minus ones is exactly the index v.
Proof. This is clearly a local problem and so it suffices to assume that
/ : U —> Μ for some open U сШп and also that /(0) = 0. Our task is to show
that there exists a diffeomorphism φ : Rn —> Rn such that / ο φ(χ) = xlhx
for a matrix of the form described. The first step is to observe that if
g : U С Ш.п —> Ш is any function defined on a convex open set U and
5(0) = 0, then
g{ui,...,un)= / —g(tui,...,tun)dt
ι η
У^ Uidig(tui,..., tun) dt.
i=l
-L
78
2. The Tangent Structure
Thus д is of the form д = Σ™=1 щд{ for certain smooth functions gi, 1 < г < η
with the property that dig(0) = gi(0). Now we apply this procedure first
to / to get / = Y^l=iUifi where dif(0) = fa(0) = 0 and then apply the
procedure to each fa and substitute back. The result is that
η
(2.3) /(ггь...,ггп) = Y^UiUjhlJ(ui,...,un)
for some functions W with the property that /г1·7' is nonsingular at, and
therefore near 0. Next we symmetrize the matrix h = (W) by replacing W
with \{W + №г) if necessary. This leaves (2.3) untouched. The index of
the matrix (/ιυ(0)) is ι/, and this remains true in a neighborhood of 0. The
trick is to find a matrix C(x) for each χ in the neighborhood that effects the
diagonalization guaranteed by Sylvester's theorem: D = C(x)h(x)C(x)~l.
The remaining details, including the fact that the matrix C(x) may be
chosen to depend smoothly on x, are left to the reader. D
2.6. Rank and Level Set
Definition 2.40. The rank of a smooth map / at ρ is defined to be the
rank of Tpf.
If / : Μ —> N is a smooth map that has the same rank at each point,
then we say it has constant rank. Similarly, if / has the same rank for
each ρ in a open subset /7, then we say that / has constant rank on U.
Theorem 2.41 (Level submanifold theorem). Let f : Μ -^ N be α smooth
map and consider the level set f~l(qo) for go £ iV. // / has constant rank
к on an open neighborhood of each ρ G /_1(#o)> then f~1(qo) is a closed
regular submanifold of codimension k.
Proof. Clearly /_1Ы is a closed subset of M. Let po G / 1(qo) and
consider a chart (/7, φ) centered at po and a chart (V,^) centered at go
with f(U) С V. We may choose U small enough that / has rank к on U.
By Theorem C.5, we may compose with diffeomorphisms to replace (/7, φ)
by a new chart (U',x) also centered at po and replace (V,-0) by a chart
(V', y) centered at go such that /:=yo/o x_1 is given by (a1,..., an) ь->
(a1,..., ak, 0,..., 0), where η = dim(M). We show that
U' П Г1Ы = {p6l/':i1(p) = - = хЧр) = 0}·
lipeU'D Γ^ο). then у о f(p) = 0 and у о / ο χ"1 (χ1 (ρ),... ,χη(ρ)) = 0
or
X1(p) = ...=Xk(p)=0.
On the other hand, suppose that ρ € £/' and xl(p) = ■ ■ · = xk(p) = 0. Then
we can reverse the logic to obtain that у ° /(p) = 0 and hence f(p) = qo.
2.6. Rank and Level Set
79
Since po was arbitrary, we have verified the existence of a cover of / l(qo)
by single-slice charts (see Section 1.7). D
Proposition 2.42. Let Μ and N be smooth manifolds of dimension m
and η respectively with η > m. Consider any smooth map f : Μ —> N.
Then if q G N is a regular value, the inverse image set f~1(q) is a regular
submanifold.
Proof. It is clear that since / must have maximal rank in a neighborhood
of /_1(g), it also has constant rank there. We may now apply Theorem
2.41. D
Example 2.43 (The unit sphere). The set Sn~l = {x e Rn : £ (x1)2 =
1} is a codimension 1 submanifold of Rn. For this we apply the above
proposition with the map / : Rn —> R given by χ ^ ^ (xl) and with the
choice q = 1 G R.
Example 2.44. The set of all square matrices Mnxn is a manifold by virtue
of the obvious isomorphism Mnxn = Rn . The set sym(n, R) of all
symmetric matrices is a smooth n(n + l)/2-dimensional manifold by virtue of the
obvious 1-1 correspondence sym(n,R) = Rn(n+1)/2 given by using n(n +1)/2
entries in the upper triangle of the matrix as coordinates. It can be shown
that the map / : Mnxn —> sym(n, R) given by А и A1 A has full rank
on 0(n,R) = f~1(I) and so we can apply Proposition 2.42. Thus the set
0(n,R) of all η χ η orthogonal matrices is a submanifold of Mnxn. We
leave the details to the reader, but note that we shall prove a more general
theorem later (Theorem 5.107).
The following proposition shows an example of the simultaneous use of
Sard's theorem and Proposition 2.42.
Proposition 2.45. Let S be a connected submanifold o/Rn and let L be a
codimension one linear subspace ofW1. Then there exist χ G Rn such that
(x + L)f) S is a submanifold of S.
Proof. Start with a line / through the origin that is normal to L. Let
pr : Rn —> S be orthogonal projection onto / . The restriction π := pr|5 —> I
is easily seen to be smooth. If π(5) were just a single point x, then π_1(χ) =
(x + L)f)S would be all of 5, so let us assume that π (S) contains more than
one point. Now, π(5) is a connected subset of / = R, so it must contain an
open interval. This implies that π(5) has positive measure. Thus by Sard's
theorem there must be a point χ G tt(S) С / that is a regular value of π.
Then Theorem 2.42 implies that π_1(χ) is a submanifold of S. But this is
the conclusion since π_1(χ) = (χ + L) Π S. D
We can generalize Theorem 2.42 using the concept of transversality.
80
2. The Tangent Structure
Definition 2.46. Let / : Μ —> N be a smooth map and ScJVa subman-
ifold of N. We say that / is transverse to S if for every ρ G f~l(S) we
have
Tf(P)N = Tf{p)S + Tpf(TpM).
If / is transverse to £, we write / iti S.
Theorem 2.47. Let f : Μ -^ N be α smooth map and S С N a submanifold
of N of codimension к and suppose that f iti S and f~l(S) Φ 0. Then
f~1{S) is a submanifold of Μ with codimension k. Furthermore we have
Tpif-^S)) = Tf-l(Tf{p)S) for all ρ e /^(S).
Proof. Let q = f(p) G S and choose a single-slice chart (V, x) centered at
q G V so that x(S П V) = x(V) П (Rn~k χ 0). Let U := f~l(V) so that
ρ G U. If π : Rn~k χ Rk -» Rk is the second factor projection, then the
transversality condition on U implies that 0 is a regular value of π ο χ ο /Ι^.
Thus (π ο χ ο /Ι^)-1 (0) = f~1{S) Π U is a submanifold of U of codimension
k. Since this is true for all ρ G /_1(5), the result follows. D
We can also define when a pair of maps are transverse to each other:
Definition 2.48. If /i : M\ —> N and /2 : M2 —> N are smooth maps, we
say that /1 and /2 are transverse at q G iV if
Tf(p)N = Γρ1/ι(ΓΡιΜ) + TP2f2(TP2M) whenever /г(рг) = f2(p2) = q.
(Note that /1 is transverse to /2 at any point not in the image of one of the
maps /1 and /2.) If /1 and /2 are transverse for all gGJV, then we say that
/1 and /2 are transverse and we write /1 iti /2.
One can check that if / : Μ —> N is a smooth map and S is a submanifold
of iV, then / and the inclusion l : S ^ N are transverse if and only if / iti S
according to Definition 2.47.
If /1 : Mi —> N and /2 : M2 —> N are smooth maps, then we can
consider the set
(Λ χ Λ)"1 (Δ) := {(рьрз) €M1xM2: h{Pl) = /2(p2)},
which is the inverse image of the diagonal Δ := {(gi, q2) e Ν χ Ν : qi = q2}.
Corollary 2.49 (Transverse pullbacks). If f\ \ M\ -* N and f2 : M2 -»
TV are transverse smooth maps, then (/1 χ /2)" (Δ) is a submanifold of
M\ χ M2. If g\ : Ρ —> M\ and g2 : Ρ —> M2 are any smooth maps with the
property /10 gi = /2o g2, then the map (31,32) · Ρ -» (/ι x Λ)"1 (Δ) ^en
^2/ {91^92) (#) — (si(#)>32(#)) ^5 smooth and is the unique smooth map such
that prx о (зьз2) = 3i anc? Pr2 ° (5ь5г) = 32-
Proof. We leave the proof as an exercise. Hint: /1 χ f2 is transverse to Δ
if and only if /1 iti f2. D
2.7. The Tangent and Cotangent Bundles
81
2.7. The Tangent and Cotangent Bundles
We define the tangent bundle of a manifold Μ as the (disjoint) union of
the tangent spaces; TM = [jpeM TpM. We show in Proposition 2.55 below
that Τ Μ is a smooth manifold, but first we introduce a couple of definitions.
Definition 2.50. Given a smooth map / : Μ —> N as above, the tangent
maps Tpf on the individual tangent spaces combine to give a map
Tf :TM -» TN
on the tangent bundle which is linear on each fiber. This map is called the
tangent map or sometimes the tangent lift of /.
For smooth maps f : Μ -ϊ Ν and д : TV —> Μ we have the following
simple looking version of the chain rule:
T(gof) = TgoTf.
If U is an open set in a finite-dimensional vector space V, then the
tangent space at χ G U can be viewed as {χ} χ V. For example, recall
that an element vp = (p, i>) corresponds to the derivation / ь->> vpf :=
$i\t=0f(p+tv). Thus the tangent bundle of U can be viewed as the product
U x V. Let U\ and U2 be open subsets of vector spaces V and W respectively
and let / : U\ —> U2 be smooth (or at least C1). Then we have the tangent
map Tf : TUi -» TU2. Viewing TUi as Ui χ V and similarly for TU2, the
tangent map Tf is given by (p, v) \-+ (/(p), Df(p) · v).
Definition 2.51. If / : Μ —> V, where V is a finite-dimensional vector
space, then we have the differential df(p) : TpM —> V for each p. These
maps can be combined to give a single map df : TM —> V (also called the
differential) which is defined by df(v) = df(p)(v) when ν G TpM.
If we identify TV with the product V χ V, then df = pr2 ο Τ/, where
pr2 : TV = V χ V —> V is the projection onto the second factor.
Remark 2.52 (Warning). The notation "df" is subject to interpretation.
Besides the map df : TM —> V described above it could also refer to the
map df : ρ ^ df(p) or to another map on vector fields which we describe
later in this chapter.
Definition 2.53. The map ктм ' Τ Μ —> Μ defined by ktm{v) = ρ if
ν € TpM is called the tangent bundle projection map. (The set TM
together with the map птм '· Τ Μ —> Μ is an example of a vector bundle
which is defined later.)
82
2. The Tangent Structure
Whenever possible, we abbreviate птм to π. For every chart (/7, x) on
M, we obtain a chart (f/,x) on Τ Μ by letting
Ε/:=Γί7 = π-1(ί7) CTM
and by defining χ on U by the prescription
x(vp) = (x1(p),...,xn(p),v1,...,vn), where vp e TpM,
and where г;1,..., vn are the (unique) coefficients in the coordinate expres-
siont; = E^sHp. Thusx-Hn1,...,^^1,...,^) = ^У ^r|x-i(tt)·
Recall that if vp = Σ,ν1 J=^\ , then г;г = dxl(vp). Prom this we see that
χ = (χ1 ο π,... , χη ο π, dx1,... , dxn).
For any (f/, x), we have the tangent lift Tx.TU -» TV where V = x (17).
Since F С Rn, we can identify TX^V with {χ (ρ)} χ Μη. Let us invoke this
identification. Now let vp G TpU and let 7 be a curve that represents vp so
that У (0) =υρ.
Exercise 2.54. Under the identification of TX^V with {χ (ρ)} χ Шп we have
Tpx · г;р = (χ (ρ), ^|ί=0 (х ° 7))· [Hint: Interpret both sides as derivations.]
If yp = -^L| ? then we can take 7(4) := x_1(x (p) + ie*), where e* is the
г-th member of the standard basis of Rn. Thus
д
TpX ' dx*
X{ph7t
(x(p) + ie<) ) = (x(p),ei).
i=0
Now suppose that г;р = ]Г г>г ^ | . Then
ipX * fp -tpX
From this we see that Tx is none other than χ defined above, and since
U = Т/У, we see that an alternative and suggestive notation for [U,x) is
(Ti/, Tx), and we adopt this notation below. This notation reminds one that
the charts we have constructed are not just any charts on TM, but are each
associated naturally with a chart on Μ and are essentially the tangent lifts
of charts on M. They are called natural charts.
Proposition 2.55. For any smooth η-manifold M, the set TM is a smooth
2n-manifold in a natural way and тттм '- TM —> Μ is a smooth map.
Furthermore, for a smooth map f : Μ -ϊ Ν, the tangent map Tf is smooth and
2.7. The Tangent and Cotangent Bundles
83
the following diagram commutes:
Tf
Τ Μ >TN
I , I
M—L-^N
Proof. For every chart (/7, x), let TU = π-1 (U) and let Tx be the map
Tx:TU -» χ(ί7) χ Rn. The pair (TU,Tx) is a chart on TM. Suppose that
(Tf/, Tx) and (TV, Ту) are two such charts constructed as above from two
charts (17, x) and (V,y) and that U П V φ 0. Then TV П TV φ 0 and on
the overlap we have the coordinate transitions Ту о Тх~х : (χ, υ) ь-> (у,гу)
where
j/ = yox"1(x),
w = D(yox_1)|xi;.
Thus the overlap maps are smooth. It is easy to see that Tx(TUDTV) and
Ty(TU Π TV) are open. Thus we obtain a smooth atlas on TM from an
atlas on Μ and this generates a topology. It follows from Proposition 1.32
that Τ Μ is Hausdorff and paracompact.
To test for the smoothness of π, we look at maps of the form χοπο (Tx) ~ .
We have
χοπο (Tx)~ (x, г;) = χ ο π Ι ν1 j—r
which is just a projection and so clearly smooth. The remainder is left for
the exercise below. D
In the above proof we observed that Τχ(Τί7 Π TV) and Ty(TU Π TV)
axe open. This must be checked because of (ii) in Definition 1.25 and is the
kind of detail we may leave to the reader as we move forward.
Exercise 2.56. For a smooth map f : Μ -* N, the map
Tf :TM -» TN
is itself a smooth map.
If ρ 6 U Π V and χ (ρ) = (χχ(ρ),... ,χη(ρ)), then, as in the proof
above, TyoTx-1 sends (xl(p),...,xn(p),vl,...,vn) to (yl(p),... ,yn(p),
w1,... ,ή/1), where
i v^^yox"1)* k
dxk
84
2. The Tangent Structure
If we abbreviate the г-th component of у ο χ-1 (χ1 (ρ),... ,χη(ρ)) to уг =
уг(х1{р), · · · ,χη(ρ)), then we could express the tangent bundle overlap map
by the relations
y* = yi(x1(p),...,xn(p)) and w* = Σ^"·
Since this is true for all ρ G x(J7n V), we can write the very classical looking
expressions
y* = yV, · · · ,*Ί and «/= £ f^,
where we now can interpret (x1,... , xn) as an η-tuple of numbers. Once
again we note that local expression could either be interpreted as living on
the manifold in the chart domain or equally, in Euclidean space on the image
of the chart domain. This should not be upsetting since, after all, one could
argue that the charts are there to identify chart domains in the manifold
with open sets in Euclidean space.
Definition 2.57. The tangent functor is defined by assigning to a
manifold Μ its tangent bundle TM and to any map / : Μ —> N the tangent
map Tf : TM —> TN. The chain rule shows that this is a covariant functor
(see Appendix A).
Recall that we also defined a "pointed" tangent functor.
We have seen that if U is an open set in a vector space V, then the
tangent bundle is often taken to be U χ V. Suppose that for some smooth
η-manifold M, there is a diffeomorphism F : Τ Μ —> Μ χ V such that the
restriction of F to each tangent space is a linear isomorphism TpM —> {ρ} χ V
and such that the following diagram commutes:
Τ Μ >- Μ χ V
Then for some purposes, we can identify Τ Μ with Μ χ V.
Definition 2.58. A diffeomorphism F : TM ^MxV such that the map
F\T M : TPM —> {ρ} χ V is linear for each ρ and such that the above
diagram commutes is called a (global) trivialization of TM. If a (global)
trivialization exists, then we say that TM is trivial. For an open set U С М,
a trivialization of TV is called a local trivialization of TM over U.
For most manifolds, there does not exist a global trivialization of the
tangent bundle. On the other hand, every point ρ in a manifold Μ is
contained in an open set U so that TM has a local trivialization over U. The
2.7. The Tangent and Cotangent Bundles
85
existence of these local trivializations is quickly deduced from the existence
of the special charts which we constructed above for a tangent bundle.
Next we introduce the cotangent bundle. Recall that for each ρ G M,
the tangent space TPM has a dual space T*M called the cotangent space at
Definition 2.59. Define the cotangent bundle of a manifold Μ to be the
set
T*M := (J T*M
рем
and define the map пт*м : T*M —> Μ to be the obvious projection taking
elements in each space T*M to the corresponding point p.
Remark 2.60. We will denote both the tangent bundle projection and the
cotangent bundle projection simply by π whenever no confusion is likely.
Remark 2.61. Suppose that / : Μ —> N is a smooth map. It is important
to notice that even though for each ρ G M, the map Tpf : TpM —> Tf^N
has a dual map (Tp/)* : (Tf^N)* —> (TPM)*, these maps do not generally
combine to give a map from T*N to T*M. In general, there is nothing like
a "cotangent lift". To see this, just consider the case where / is a constant
map.
We now show that T*M is also a smooth manifold. Let Λ be an atlas on
M. For each chart (f7,x) <E Л, we obtain a chart (T*/7,T*x) for T*M which
we now describe. First, T*U = тг^м({7) = \JpeUT*M. Secondly, T*x is a
map which we now define directly and then show that, in some sense, it is
dual to the map Tx. For convenience, consider the map pi : θρ ь-> ^ which
just peals off the coefficients in the expansion of any θρ eT*M in the basis
(dx1! dxn\p):
Pi (θρ) = Pi [Σ^ άχ1\Ρ) := &·
Notice that we have
д
θ
and so
Ρτ{θρ) ~-
Φ/
With this definition of the pi in hand, we can define
T*x = (χ1 ο π, ...,χηοπ,ρι,...,ρη)
on T*U. We call (Τ*ί7,Τ*χ) a natural chart. If χ = (χ1,... ,xn), then
for the natural chart (T*Z7, T*x), we could use the abbreviation T*x =
86
2. The Tangent Structure
(x1,..., xn,pi,... ,Pn)· Another common notation is ql := хг ο π. This
notation is very popular in applications to mechanics.
We claim that if we take advantage of the identifications of TxRn = Rn =
(Rn)* = T*Rn where (Rn)* is the dual space of Rn, then T*x acts on each
fiber Τ*Μ as the dual of the inverse of the map Tpx, i.e. the contragredient
of Tpx:
((ГрхГ1)*(вр).(т;) = вр((ГрхГ1.т;).
Let us unravel this. If θρ eT*M for some ρ € /7, then we can write
θρ = Σ&άχί\ρ
for some numbers & depending on #p which are what we have called Ρΐ(θρ).
We have
((ЗД-1)^) >) = 0P ((ВД-1 · г;)
= Σί*ώ:ΊΡ·((ΓρχΓ1·ν)
Σ*<4 Σ«*
д
дхк
= Σ^·
Thus, under the identification of Rn with its dual we see that ( (Tpx) l j (0P)
is just (ξι,...,in)· But recall that Τ*χ(θρ) = (x1(p),..., χη(ρ),ξ1,... ,ξη).
Thus for ΘΡΕΤ*Μ we have
Γ·χ(βρ) = (χ(ρ)>((Γρχ)-1)*(βρ)).
Suppose that (T*/7, T*x) and (Г* V, T*y) are the coordinates constructed
as above from two charts (/7, x) and (V, y) respectively with Uf)V φ 0. Then
on the overlap T*U DT*V we have
T*y ο (Γ*χ)_1 : x(U Π V) χ Rn* -» y(/7 Π V) χ Rn*.
This last map will send something of the form (я, ξ) € /7 χ Rn* to (я, ξ) =
(yox-1 (χ), D(xoy-1)* ·ξ), where D(xoy-1)* is the dual map to D(xoy-1),
which is the contragredient of the map D(y о x_1). If we identify Rn* with
Rn and write ξ = (ξι,..., ξη) and ξ = (ξι,..., ξη), then in the classical style
we have:
уг = уг(х\ ... ,xn) and & = Jj& —.
This should be compared to the expression (2.2). It is now clear that we
have an atlas on T*M constructed from an atlas on M. The topology of
T*M (induced by the above atlas) is easily seen to be paracompact and
Hausdorff.
2.8. Vector Fields
87
In summary, both Τ Μ and Γ* Μ are smooth manifolds whose smooth
structure is derived from the smooth structure on Μ in a natural way. In
both cases, the charts are derived from charts on the base Μ and are given
by the η coordinates of the base point together with the η components of
the element of Τ Μ (or Τ* Μ) in the corresponding coordinate frame.
2.8. Vector Fields
In this section we introduce vector fields. Roughly, a vector field is a smooth
assignment of a tangent vector to each point of a manifold.
Definition 2.62. If π : Μ —> N is a smooth map, then a (global) section
of π is a map σ : N —> Μ such that π ο σ = id. If σ is defined only on an
open subset U of N and π ο σ = idf/, then we call σ a local section. In
case the section σ is a smooth (or Cr) map, we call σ a smooth (or Cr)
section.
Clearly, if π : Μ —> N has a (global) section, then it must be surjective.
Definition 2.63. A smooth vector field on Μ is a smooth map X : Μ —>
Τ Μ such that X(p) £ TPM for all ρ e M. In other words, a vector field on
Μ is a smooth section of the tangent bundle π : TM —> M. We often
write Xp = X(p).
Convention: Obviously the notion of a section or field that is not
smooth makes sense. Sometimes one is interested in merely continuous
sections or measurable sections. In this book, by "vector field" or "section", we
will always mean "smooth vector field" or "smooth section" unless otherwise
indicated explicitly or by context.
A local section of TM defined on an open set U is just the same thing
as a vector field on the open manifold U. If (/7, x) is a chart on a smooth
η-manifold, then writing χ = (χ1,..., xn), we have vector fields defined on
U by
dxl 'P dxi
The ordered set of fields (gfr, · · ·, -^κ) is called a coordinate frame field
(or also "holonomic frame field"). If X is a smooth vector field defined on
some set including this chart domain /7, then for some smooth functions Хг
defined on U we have
or in other words
lf/ ^ дхг
2. The Tangent Structure
Notation 2.64. In this context, we will not usually bother to distinguish
X from its restrictions to chart domains and so we just write X = Y^Xl-J=^.
Lemma 2.65. If ν G TpM then there exists α vector field X such that
X(p)=v.
Proof. Write ν = J2yl ^ϊ\ · Define a field X\j by the formula Y^v1-^
where the vl are taken as constant functions on U. Let β be a cut-off
function with support in U and such that β(ρ) = 1. Then let X := βΧυ on
U and extended to zero outside of U. D
Let us unravel what the smoothness condition means for a vector field.
Let (TZ7, Tx) be one of the natural charts that we constructed for Τ Μ from
a corresponding chart (/7, x) on M. To test the smoothness of X, we look
at the composition Tx ο Χ ο χ-1. For χ Ε χ(ί7), we have
ΓχοΧοχ"1^)
д
-ΜΣ^Κ'ω
9
V ΟΧ \χ-ι{χγ
= (х,Гх-,мХ(£х-(*-(х))^| ,))
= (χ , Χ1 ο χ~\χ), ...,Χηο χ-^χ)) .
Our chart was arbitrary, and so we see that the smoothness of X is equivalent
to the smoothness of the component functions Xх in every chart of an atlas
for the smooth structure.
Exercise 2.66. Show that if X : Μ ->> Τ Μ is continuous and π ο Χ — id,
then X is smooth if and only if Xf : ρ ь-> Xpf is a smooth function for every
locally defined smooth function / on M. Show that it is enough to consider
globally defined smooth functions.
Notation 2.67. The set of all smooth vector fields on Μ is denoted by
X(M). Smooth vector fields may at times be defined only on some open set
U С Μ so we also have the notation X(U) = %m(U) for these fields.
We define the addition of vector fields, say X and У, by
(X + Y){p):=X(jp) + Y{p),
and scaling by real numbers, by
(cX)(p):=cX(p).
2.8. Vector Fields
89
Then the set X(M) is a real vector space. If we define multiplication of a
smooth vector field X by a smooth function / by
(fX)(p):=f(p)X(p),
then the expected algebraic properties hold making X(M) a module over
the ring C°°(M) (see Appendix D). It should be clear how to define vector
fields of class Cr on Μ and the set of these is denoted Xr(M) (a module
over Cr(M)).
The notion of a vector field along a map is often useful.
Definition 2.68. Let / : TV —>· Μ be a smooth map. A vector field along
/ is a smooth map X : TV —у Τ Μ such that птм ° X = /· A vector field
along a regular submanifold S С М is a vector field along the inclusion map
S <-»> M. (Note that we include the case where S is an open submanifold.)
We let Xf denote the space of vector fields along /.
It is easy to check that for a smooth map f : N —> M, the set Xf is a
C°°(JV)-module in a natural way.
We have seen how individual tangent vectors in TpM can be identified
as derivations at p. The derivation idea can be globalized. We explain how
we may view vector fields as derivations.
Definition 2.69. Let Μ be a smooth manifold. A (global) derivation on
C°°(M) is a linear map V : C°°(M) -> C°°(M) such that
4fg) = 4f)9 + f4g).
We denote the set of all such derivations of C°°(M) by Der(C°°(M)).
Notice the difference between a derivation in this sense and a derivation
at a point.
Definition 2.70. To a vector field I on M, we associate the map С χ :
C°°(M) -> C°°(M) defined by
(Cxf)(p):=Xpf.
€χ is called the Lie derivative on functions.
It is important to notice that (Cxf)(p) = Xp · / = df (Xp) for any ρ
ала so Cxf = df о X. If X is a vector field on an open set /7, and if /
is a function on a domain V С /7, then we take Cxf to be the function
defined on У by ρ и Xpf for all ρ G V. It is easy to see that we have
£aX+bY = aCx + ЬСу for a, b e Ш and Χ, Υ e X(M).
Lemma 2.71. Let U С Μ be an open set and X e X(M). If Cxf = 0 for
uUfeC°°(U), then X\u = Q.
90
2. The Tangent Structure
Proof. Let ρ G U be given. Working locally in a chart (V,x), let X =
Y^Xld/dxl. We may assume ρ e V С U. Using a cut-off function we
may find functions fl defined on U such that fl coincides with хг on a
neighborhood of p. Then we have Xl(p) = Xpxi = Xpf = (Cxf) (p) = 0.
Thus X(p) = 0 for an arbitrary ρ G U. D
The next result is a very important characterization of smooth vector
fields. In particular, it paves the way for the definition of the bracket of
vector fields which plays a central role in differential geometry.
Theorem 2.72. For X G X(M), we have Cx G Der(C°°(M)), and ifV G
Der(C°°(M)), then V = Cx for a uniquely determined X G X(M).
Proof. That Cx is in Der(C°°(M)) follows from the Leibniz law, in other
words, from the fact that Xp is a derivation at ρ for each p. If we are given
a derivation D, we define a derivation Xp at ρ (i.e. a tangent vector) by
the rule Xpf := (Vf) (p). We need to show that the assignment ρ ь-> Хр
is smooth. Recall that any locally defined function can be extended to a
global one by using a cut-off function. Because of this, it suffices to show
that ρ h-> Xpf is smooth for any / G C°°(M). But this is clear since
Xpf := (Vf) (p) and Vf G C°°(M). Suppose now that V = CXl = £χ2.
Notice that Cxx — Cx2 = Cxx-x2 and so Cxx-x2 is the zero derivation on
C°°(M). By Lemma 2.71, we have Χλ - X2 = 0. D
Because of this theorem, we can identify Der(C°°(M)) with X(M) and
we can and often will write Xf in place of Cxf:
Xf := Cxf.
The derivation law (also called the Leibniz law) Cx(fg) = gCx f + fCx g
becomes simply X(fg) = gXf + fXg- Another thing worth noting is that
if we have a derivation of C°°(M), then from our discussion above we know
that it corresponds to a vector field. As such, it can be restricted to any
open set U С Μ, and thus we get a derivation of C°°(U). If / G C°°(U) we
write Xf instead of the more pedantic X\v /.
While it makes sense to talk of vector fields on Μ of differentiability r
where 0 < r < oo and these do act as derivations on Cr(M), it is only in
the smooth case (r = oo) that we can say that vector fields account for all
derivations of Cr(M).
Theorem 2.73. IfV1,V2 G Der(C°°(M)), then [VUV2] G Der(C°°(M))
where
[Vi,V2] :=V1oV2-V2oV1.
2.8. Vector Fields
91
Proof. We compute
V1(V2(fg))=V1(V2(f)g + fV2(g))
= (V{D2f) g + V2fVig + VxfV2g + fVxV2g.
Writing out the similar expression for T>2 (V\ {fg)) and then subtracting we
obtain, after a cancellation,
[VuV2] (fg) = (ViV2f)g + fVxV2g - ((2>22?i/)g + fV2Vl9)
= ([VuV2]f)g + f[DuV2]g. D
Corollary 2.74. IfX,Y £ 3i(M), then there is a unique vector field [X,Y]
such that £\χ,γ] — C>x ° £γ — £γ ° £χ·
Since Cxf is also written Xf, we have [X, Y]f = X (Yf) - Υ (Xf) or
[X, Y) = XY - YX.
Definition 2.75. The vector field [X, Y] from the previous corollary is
called the Lie bracket of X and Y.
Proposition 2.76. The map (Χ, Υ) ι-» [Χ, Υ] is bilinear overR, and for
Χ,Υ,Ζ € X(M) we have
(i) [X,Y} = -[Y,X];
(ii) [Χ, [Ϋ, Ζ}}+ [Υ, [Ζ, Χ]} + [Ζ, [Χ, Υ]] = 0 (Jacobi Identity);
(iii) [fX, gY] = fg[X, Y]+f (Xg) Υ - g (Yf) X for all /, g € C°°(M).
Proof. These results follow from direct calculation and the previously
mentioned fact that СаХ+ьу = α£χ + bCy for a, b e R and Χ, Υ e X(M). D
The map (Χ, Υ) ь-> [X, Y] is bilinear over R, but by (iii) above, it is not
bilinear over C°°(M). Also notice that in (ii) above, Χ,Υ,Ζ are permuted
cyclically.
We ought to see what the local formula for the Lie derivative looks
like in conventional "index" notation. Suppose we have X = Σ Хг~§& ап<^
Υ = ^2Υ%·£-ζ- Then we have the local formula
Exercise 2.77. Verify this last formula.
The R-vector space X(M) together with the R-bilinear map (X, Y) \-+
[X, Y] is an example of an extremely important abstract algebraic structure:
92
2. The Tangent Structure
Definition 2.78 (Lie algebra). A vector space о (over a field F) is called a
Lie algebra if it is equipped with a bilinear map ο χ ο —> о (a multiplication)
denoted (v,w) ι-» [v,w] such that
[v,w] = —[w,v]
and such that we have the Jacobi identity
[s, [ϊΛ A] + [ϊΛ [ζ,χ]] + \z, [χ. У]] = О
for all ж, у, г Е о.
Definition 2.79. A Lie algebra o is called abelian (or commutative) if
[г;, w] = 0 for all г>, ги G о. A subspace f) of о is called a Lie subalgebra if it
is closed under the bracket operation, and it is called an ideal if [г;, w] G f)
for any υ G о and w G f). (We indicate this by writing [a, fj] С fj.)
Notice that the Jacobi identity may be restated as [ж, [у, г]] = [[ж, у], ζ] +
[у, [ж, г]], which just says that for fixed ж the map у ь-> [ж, у] is a derivation of
the Lie algebra o. This is significant mathematically and also an easy way to
remember the Jacobi identity The Lie algebra X(M) is infinite-dimensional
(unless Μ is zero-dimensional), but later we will be very interested in certain
finite-dimensional Lie algebras which are subalgebras of X(M).
Given a diffeomorphism φ : Μ —> Ν, we define the pull-back φ*Υ G
X(M) for Υ G X(N) and the push-forward φ*Χ G X(N) of X G X(M)
by φ by
φ*Υ = Τφ'1 οΥοφ and
φ*Χ = ΤφοΧοφ~\
In other words, (φ*Υ)(ρ) = Τφ~ιΎφ{ρ) and (φ*Χ)(ρ) = Тф-Хф-цр). Notice
that φ*Υ and 0*X are both smooth vector fields. Warning: Since many
authors use the notation /* for the tangent map Γ/, the notation f*X might
be interpreted to mean Tf ο Χ, which is actually a vector field along the
map f rather than an element of X(N). We shall not use /* as a notation
for Tf.
To summarize a bit, if / : Μ —> N is a smooth map, then for each ρ we
have the tangent map Tpf : TpM -> Tf(<p)N, the tangent lift Tf : TM -> TN
(a "bundle map"), and if / is a diffeomorphism, we have the induced maps
on the level of fields /* : X(M) -> X(N) and /* : X(N) -> X(M). Notice
that if φ : Μ —> N and φ : Ν —»· Ρ are diffeomorphisms, then we have
(ф о ф)^ = ф*оф*: X(M) -> Х(Р),
(φ о φ)* = 0* о ψ* : Х(Р) -> Х(М).
We have right and left actions of the diffeomorphism group Diff (M) on the
space of vector fields. The left action Diff(M) χ X(M) -> X(M) is given by
2.8. Vector Fields
93
{φ,Χ) н-> φ*Χ, and the right action X(M) χ Diff(M) -> X(M) is given by
\χ,φ) ^φ*Χ.
On functions, the pull-back is defined by ф*д := g ο φ for any smooth
map, but if φ is a diffeomorphism, then we can also define a push-forward
0* := (φ~1)*. With this notation we have the following proposition.
Proposition 2.80. The Lie derivative on functions is natural with respect
to pull-back and push-forward by diffeomorphisms. In other words, if φ :
Μ -+N is a diffeomorphism and f e C°°(M), g e C°°(N), X e X(M) and
Υ eX(N), then
£>φ*υΦ*9 = Ф*£уд
and
£>Φ*χΦ*ϊ = Φ*£χί-
Proof. We use Definition 2.19. For any ρ we have
(Сф-υΦΊ) (ρ) = (Φ*Υ)ΡΦΊ = {Τφ-1 οΥοφ)ρ[9ο ψ]
= {Τφ'1 ■ Υφ(ρ)) [доф}= Τφ {Τφ-%{ρ)) д
= Υφ{ρ)9 = (Суд) (φ(ρ)) = (ф*Суд) (р).
The second statement follows from the first since 0* = (0-1)*. D
Even if / : Μ —> N is not a diffeomorphism, it may still be that there
is a vector field Υ e X(N) such that
TfoX = Yof.
In other words, it may happen that Tf · Xp = Y/(p) for all ρ in M. In this
case, we say that Υ is /-related to X and write X ~f Y. It is not hard to
check that if Xi is /-related to Yi for г = 1,2, then aX\ + ЪХ\ is /-related
toali+6Yi.
Example 2.81. Let Μ and TV be smooth manifolds and consider the
projections prx : Μ χ TV -+ Μ and pr2 : Μ χ JV -> JV. Since Т(м) (Μ χ Ν) can be
identified with TpM χ TqN, we see that for Χ <Ε Χ(Μ), Υ e X(N) we obtain
a vector field X xY e X(M χ Ν) defined by (Χ χ Υ) (ρ, q) = (Χ (ρ), Υ (ρ)).
Then one can check that
Χ χ Υ and X are prx-related
and
Χ χ Υ and Υ are pr2-related.
Exercise 2.82. Let M, JV, Χ, Υ and Χ χ Υ be as in the example above.
Show that if iq : Μ —> Μ χ TV is the insertion map ρ ь-> (ρ, g), then X and
Χ χ Υ are ^-related if and only if Y(q) = 0.
94
2. The Tangent Structure
Lemma 2.83. Suppose that f : Μ —»■ JV is a smooth map, X € X(M) and
Υ € 3C(N). Then X and Υ are f-related if and only if X(g of) — (Yg) о f
forallg€C°°(N).
Proof. Let ρ € Μ and let g € C°°(N). Then
X(9 ° /)(P) = XP(9 о f) = (Tpf · Xp) g
and
(Y9of)(p)=Ymg
so that X(g о f) = (Yg) о f for all such g if and only if Tpf · Xp = У/(p). □
Proposition 2.84. If f : Μ —> N is a smooth map and X{ is f-related to
Yi for i = 1,2, then [Χχ,-Λ^] ^ f-related to [Υχ, Υ2]· ^n particular, if φ is a
diffeomorphism, then [ф*Х\,ф*Х<2\ = ф*[Х\,Х^ for all X\,X2 G X(M).
Proof. We use the previous lemma: Let # G C°°(N). Then ΧχΧ2(# ο /) =
*1((ВД о /) = (УхВД о /. In the same way, X2X1(g о f) = (Y2Yl9) о f
and subtracting we obtain
[*ь*2] (g°f) = XiX2(g о f) - X2Xi(g ° /)
= (YiY2g) о / - (Y2Yl9) о f
= ([YuY2}g)of.
Using the lemma one more time, we have the result. D
If S is a submanifold of Μ and X G X(M), then the restriction X\s G
X(S) defined by X\s(p) = X(p) for all ρ G S is ^-related to X where
l : S ^ Μ is the inclusion map. Thus for Χ, Υ G X(M) we always have
that [X\s, y\s\ is ^-related to [Х,У]. This just means that [X,Y] (p) =
[X|5,y|5](p)forallp.
We also have
Proposition 2.85. Let f : Μ —> N be a smooth map and suppose that X
~/ Y. Then we have Cx (f*g) = Г Суд for any g G C°°(N).
The proof is similar to what we did above and is left to the reader.
2.8.1. Integral curves and flows. Recall that if с : / —>· Μ is a smooth
curve, then the velocity at "time" t is
where gj is the standard field on R given at α G R as the equivalence class
of the curve t \-+ a + t or by the derivation gjj | / = /'(a)·
2.8. Vector Fields
95
Definition 2.86. Let X be a smooth vector field on M. A curve c: I —l· Μ
is called an integral curve for X if for all tG/, the velocity of с at time t
is equal to X(c(t)), that is, if
c = X о с
Thus if с is an integral curve for X and / is a smooth function, then
Xc{t)f = (foc)'(t)
for all t in the domain of c. If the image of an integral curve с lies in U for a
chart (/7,x), and if X = ΣXх-$^, then c — Xoc gives the local expressions
—xx о с = Xх о с for г = 1,..., η,
dt
which constitute a system of ordinary differential equations for the functions
хг о с. These equations are classically written as ^ = Xх.
A (complete) flow is a map Φ : R χ Μ —> Μ such that if Ф*(я) :=
Φ(£, χ) for each χ G Μ, then t ь-> Ф^ is a group homomorphism from the
additive group R to the diffeomorphism group of M. More generally, a flow
is defined similarly except that Φ(ί, χ) may not be defined on all of R χ Μ,
but rather on some open neighborhood of {0} χ Μ С R χ Μ, and so we
explicitly require that Φ^ ο φ5 = φί+5 and Φ^"1 = Φ_^ for all t and s such
that both sides of these equations are defined. Convention: We shall also
loosely refer to the map t \-> Φ^ as the flow.
Using a smooth flow, we can define a vector field Χφ by
Φ(ί,ρ) eTpM for ре Μ.
Ιο
If one computes the velocity vector c(0) of the curve с : t ь-> Φ(ί,ρ), one
gets Χφ(ρ). In fact, because of the two properties assumed above, we get
c(t) = Хф(с(£)) for any t for which Φ(ί,ρ) is defined.
We would like to start with a vector field and produce a flow at least
locally. Our study of the flows of vector fields begins with a quick recounting
of a basic existence and uniqueness theorem for differential equations
stated here in the setting of real Banach spaces. If desired, the reader may
take the Banach space to be a finite-dimensional normed space such as Rn.
Theorem 2.87. Let Ε be a Banach space and let F : U С Ε —> Ε be a
smooth map with open domain U. Given any xo G U, there is a smooth
curve с : (-б, б) —>· U with c(0) = xo such that d{t) = F(c(t)) for all
t € (-6, e). If c\ : (-61,61) —>· U is another such curve with ci(0) = xo
and di(t) = F(c(t)) for all t G (—61,61), then с — c\ on the intersection
(—€ΐ,€ι) Π (—6, б). Furthermore, given any fixed xo G U, there is an a > 0,
an open set V with xq G V С U, and a smooth map Φ : (—a, a) x V —> U
*·«-*
96
2. The Tangent Structure
such that t h-> cx(t) := Φ(ί,χ) is a curve satisfying c'x(t) = F(cx(t)) for all
t G (—a, a) and cx(0) = x.
Example 2.88. Consider the differential equation on the line given by
c'{t) = {c{t)fl\
There are two distinct solutions with initial condition c(0) = 0. Namely,
c(t) = 0 for all t
and
c(t) = —t3 for all t.
The reason uniqueness fails is the fact that the function F(x) = x2/3 is not
differentiate at χ = 0.
Now let X G X(M) and consider a point ρ in the domain of a chart
({7, x). The local expression for the integral curve equation c(t) = X(c(t)) is
of the form treated in the last theorem, and so we see that there certainly
exists an integral curve for X through ρ defined on at least some small
interval (—6, б). We will now use this theorem to obtain similar but more
global results on smooth manifolds. First of all, we can get a more global
version of uniqueness:
Lemma 2.89. If c\ : (—61,61) —> Μ and C2 : (—62,62) —> Μ are integral
curves of a vector field X with ci(0) = сг(0), then c\ — С2 on the intersection
of their domains.
Proof. Let К = {t e (-61, €1) Π (-€2, б2) : ci(t) = c2(i)}. The set К is
closed since Μ is Hausdorff. It follows from Theorem 2.87 that К contains
a (small) open interval (—e^e). Let to be any point in К and consider the
translated curves c^(t) = c\(to + t) and 4°(ί) = 02(^0 + t). These are also
integral curves of X and they agree at t = 0, and by Theorem 2.87 again we
see that c^0 = c^0 on some open neighborhood of 0. But this means that c\
and C2 agree in this neighborhood, so in fact this neighborhood is contained
in К implying that К is also open since to was an arbitrary point in K.
Thus, since / = (—61,61) Π (—62,62) is connected, it must be that I — К
and so c\ and C2 agree on / = (—61,61) Π (—62,62). □
Let X be a C°° vector field on M. A flow box for X at a point ρ G Μ
is a triple (/7, α, ψχ), where
(1) U is an open set in Μ containing p.
(2) ψχ : (-a, a) x U -> Μ is a C°° map and 0 < α < 00.
(3) For each ρ G /7, the curve t \-> Cp(t) = φχ(ί,ρ) is an integral curve
of X with Cp(0) = p.
2.8. Vector Fields
97
(4) The map φ* : U —>· Μ given by φ*(ρ) = φχ(ί,ρ) is a diffeomor-
phism onto its image for all t G (—a, a).
We sometimes refer to (/?* as a local flow for X. Before we prove
that flow boxes actually exist, we make the following observation: If we
have a triple that satisfies (l)-(3) above, then both c\ : t н> φ*+8{ρ) and
C2 : t \-> φ*(φ*(ρ)) are integral curves of X with ci(0) = сг(0) = φ*(p),
so by uniqueness (Lemma 2.89) we conclude that φ*(φ*{ρ)) = <#+5(p) as
long as both sides are defined. This also shows that
φ8 °Ψι = ^ί+5 = у* ° ^
whenever defined. This is the local group property, so called because if φ*
were defined for all ί G R (and X a global vector field), then t \-> φ* would
be a group homomorphism from R into the diffeomorphism group Diff(M).
Whenever this happens, that is, whenever ψχ is defined for all (ί,ρ), we
say that X is a complete vector field. In other words, a vector field is
complete if all its integral curves are defined on all of R. The local group
property also implies that φ* ο (p*t = id, and so in general φ* must at least
be a locally defined diffeomorphism with inverse φ*ν
Notice that whereas ^ is a complete vector field on R2 (using standard
coordinates x,y), this vector field restricted to R2\{0} is not complete on
the manifold R2\{0}. The reason is that integral curves starting at points
on the x-axis will run up against the missing origin in finite time. This
"running up to missing points" is not the only way a vector field can fail
to be complete. Consider the vector field (l + x2) J^ on R. The integral
curve that is at the origin at time zero is obtained by solving the initial
value problem x' = 1 + x2, x(0) = 0. The unique solution is x(t) = tani,
and since 1imt_>±n/2 tan t = ±oo, the solution cannot be extended beyond
±π/2.
Exercise 2.90. Show that on R2 the vector fields y2-§^ and х2щ are
complete, but y2-§£ + х2щ is n°t complete. In particular, the set of complete
vector fields is not generally a vector space (but this is true if Μ is compact).
Theorem 2.91 (Flow box). Let X be a C°° vector field on an n-manifold
Μ with r > 1. Then for every point po G Μ there exists a flow box for X
atpo. If (Ui,ai,(p*) and (f/2,a2,<^) are ^wo flow boxes for X atpo, then
Ψι = Ψ2 on (_ab ai) n (_a2,0.2) x U\ Π E/2.
Proof. First of all, notice that the U in the triple (/7, α, ψχ) does not have
to be contained in a chart or even be homeomorphic to an open set in Rn.
However, to prove that there are flow boxes at any point we can work in
the domain of a chart (Ϊ7, x) and so we might as well assume that the vector
field is defined on an open set in Rn. Of course, we may have to choose α
98
2. The Tangent Structure
to be smaller so that the flow stays within the range of the chart map x. In
this setting, a vector field can be taken to be a map U —> Rn, so Theorem
2.87 provides us with the flow box data (V, α, Φ), where we have taken α > 0
small enough that Vt = Φ(ί, V) С U for all t G (—a, a). Now the flow box is
transferred back to the manifold via x,
U = x-\V),
φΧ(ί,ρ)=χ-1[Φ(ί,χ(ρ))\-
If we have two such flow boxes (f/i, αϊ, ψχ) and ({/2, α2, φ*)·» then by Lemma
2.89, we see that for any χ G U\ Π /72 we must have tpx(t,x) = tpx(t,x) for
all ί G (-αϊ, αϊ) Π (-α2, α2).
Finally, since ψχ = φχ(ί , ·) and у>*4 = ψχ(—ί , ·) are both smooth and
inverses of each other, we see that ψχ is a diffeomorphism onto its image
Ut = x-1(Vt). D
Lemma 2.92. Suppose that X\,... ,X^ are smooth vector fields on Μ and
let po G Μ be given and О be an open set containing po. // ψΧι,..., ψΧ]ζ
are the local flows corresponding to flow boxes whose domains /7i,..., U^ all
contain po, then there is an open set U С U\ Π · · · Π Uk and an e > 0 such
that the composition
is defined on U and maps U into О whenever ii,..., i& G (—6, б).
Proof. If the flow box corresponding to ψΧϊ is (C/i, €i, φΧ), then by shrinking
U\ further we may arrange things so that ψι l maps U\ into О for all t G
(—€1, €1), and then inductively we arrange for ψΧί to map Щ into f/^-i for
all t G (—€», €j). Now let e = min{€i,..., б/J. D
Remark 2.93. When making compositions of local flows, we will not always
make careful statements about domains, but the previous lemma will be
invoked implicitly.
If Cp(i) is an integral curve of X defined on some interval (a, b) containing
0 and Cp(0) = p, then we may consider the limit
lim Cn(t).
If this limit exists as a point p\ G M, then we may consider the integral curve
cPi beginning at p\. One may now use Lemma 2.89 to combine t ь-> Cp(t)
with t н-> Cp^t — b) to produce an extended integral curve beginning at p. We
may repeat this process as long as the limit exists. We may do a similar thing
in the negative direction. This suggests that there is a maximal integral
curve defined on a maximal interval Jx := (Τ~χ,Τ*χ), where T~x might
be —00 and T*x might be +00. We produce this maximal integral curve
2.8. Vector Fields
99
as follows: Consider the collection Jv of all pairs (J, a), where J is an open
interval containing 0 and a : J —> Μ is an integral curve of X with a(0) = p.
Then let Jx = \Jua)ej J and define cmax(t) := a(t) whenever t G J for
{J,ol) G J^. By existence and uniqueness, this definition is unambiguous.
The curve cmax : Jx —У М is the desired maximal integral curve and is
easily seen to be unique.
Definition 2.94. Let X be a C°° vector field on M. For any given ρ G M,
let Jx := (Τ~χ,Τ~^χ) С R be the domain of the maximal integral curve
с : Jx —> Μ of X with c(0) = p. The maximal flow ψχ is defined on the
set (called the maximal flow domain)
рем
by the prescription that ί \-> φχ(ί,ρ) is the maximal integral curve of X
such that φχ(0,ρ) = p.
Thus by definition, X is a complete vector field if and only if Ί)χ =
R χ M. We will abbreviate (Tp~x,T+x) to (Тр~,Тр+).
Theorem 2.95. For X G X(M), ί/ie se£ Όχ is an open neighborhood of
{0} χ Μ in Ι χ Μ and the map ψχ : Τ>χ —>· Μ is smooth. Furthermore,
(2.4) ^x(t + 5,p) = (^x(^x(5,p))
whenever both sides are defined. If the right hand side is defined, then the
left hand side is defined. Suppose that t,s > 0 or t,s < 0. Then if the left
hand side is defined so is the right hand side.
Proof. Let q = (px(s,p). If the right hand side is defined, then s G
{T~,T+) and t G (T~,T+). The curve ψ : τ ь-> φχ(β + τ,ρ) is defined
for τ G (Τ~ — 5, Τ+ — s), and this is the maximal domain for ψ. We have
_d_
dr\
= Χ(φχ(8 + τ,ρ))=Χ(ψ(τ)).
We also have ψ(0) = ψχ(τ + s,p)|r=0 = φχ(8,ρ) so ψ is an integral curve
starting at q = 4>x{s,p). Thus (T~ - s ,T+ - s) С (Г",Г+) and ^ =
ψ*(·ιθ) on (^p~ ~~ 5?^ ~~ 5)· But the maximal domain for ψ is (7^" — s,
IJ+ - 5) and so in fact (T~ - 5, Γ+ - 5) = (T~, Γ+) for otherwise (/?*(·, g)
would be a proper extension. But then, since t G {T~ ,T+}, we have that
t € (T~ - s,T+ - s) and so ψ(ί) = <px(t + s,p) is defined and
φΧ^ + 8,ρ)=φΧ^φΧ(8,ρ)).
4 = -φ (* + r,p)=-
VX (и, Р)
100
2. The Tangent Structure
Now let us assume that t, s > 0 and that φx(s + t,p) is defined (the case
of i, s < 0 is similar). Then since s, ί > 0, we have
m,hsg (r-,r+).
Let c/ = (£X(s,p) as before and let 6(u) = φχ{β + г*,р) be defined for w
with 0 < и < t. But 9{u) is an integral curve with 0(0) = q. Thus we
have that φχ(ιι, q) must also be defined for и = t and 0(t) = <px{t, q). But
φχ{ί, q) φΧ(ί, ipx(s,p)), which is thereby defined, and we have
ψΧ{8 + t,p) :- 0(t) - </>X(i, ¥>*(s,p)).
Now we will show that Ί)χ is open in Μ χ Μ and that ψχ : Ί)χ —> Μ
is smooth. We carefully define a subset <S С Τ>χ by the condition that
(ί,ρ) Ε S exactly if there exists an interval J containing 0 and t and also an
open set U С Μ such that the restriction of ψχ to J χ U is smooth. Notice
that 5 is open by construction. We intend to show that S = Τ>χ. Suppose
not. Then let (£ο>Ρο) ^ ^* ^ *^c- We will assume that ίο > 0 since the case
ίο < 0 is proved in a similar way. Now let r := sup{i : (ί,ρο) £ £}♦ We
know that (0,po) is contained in some flow box and so r > 0. We also have
τ < to by the definition of ίο· Thus τ e Jx0 and we define qo := φχ(τ,ρο).
Now applying the local theory we know that qo is contained in an open set
Щ such that φχ is defined and smooth on (—6, e) χ Щ for some e > 0. We
will now show that φχ is actually defined and smooth on a set of the form
(S,r) χ Ο where О is open, 5, r > 0, and (τ,ρο) £ (—£,r) x O. Since this
contradicts the definition of r, we will be done.
We may choose t\ > 0 so that r G (ti, t\ + e) and so that φχ(ίι,ρο) Ε ί7ο·
Note that (ίι,ρο) € £ since ti < r. So on some neighborhood (—£,ίι+δ)χϊ/ι
of (ίι,Ρο) the flow φχ is smooth. By choosing U\ smaller if necessary we
can arrange that φχ{{ίι} xi/i)C Щ. Now consider the equation
φΧ(^ρ)-φΧ(ί-ίι,φΧ{ί1)ρ)).
If \t — ii < б and ρ G ί/ι, then both sides are defined and the right hand
side is smooth near such (ί,ρ). But the right hand side is already known
to be smooth on (—<J,ii + δ) χ [Д. We now see that φχ is smooth on
(—δ,t\ + e) χ C/i, which contains (τ,ρο) contradicting the definition of r.
Thus 5 £>*. D
Remark 2.96. In this text, ψχ will either refer to the unique maximal
flow defined on Τ>χ or to its restriction to the domain of a flow box. In
the latter case we call φχ a local flow. We could have introduced notation
such as <£max> but prefer not to clutter up the notation to that extent unless
necessary. We hope that the reader will be able to tell from context what
we are referring to when we write φχ.
2.8. Vector Fields
101
If φχ is a flow of X, then we write φ* for the map ρ н-> φ* (ρ). The
(maximal) domain of this map is Vlx — {p : t G {Τ~χιΊ^χ)}. Note that,
in general, the domain of φ* depends on t. Also, we have the tangent map
Τ0φ£ : T0R -> TpM and
— I
eft
<pf{t)=To<p* 9
t=o aw
— -Xp»
0
where ^|Q is the vector at 0 associated to the standard coordinate function
on R (denoted by и here).
Exercise 2.97. Let s and t be real numbers. Show that the domain of
φ* ο φ* is contained in Τ>8χι and show that for each ί, Ύ>ιχ is open. Show
Definition 2.98. The support of a vector field X is the closure of the set
{p : X(p) ф 0} and is denoted supp(X).
Lemma 2.99. Every vector field that has compact support is a complete
vector field. In particular} if Μ is compact, then every vector field is
complete.
Proof· Let cx be the maximal integral curve through ρ and J* = (Γ-, T+)
its domain. If X(p) = 0, then the constant curve, c(t) = ρ for all p, is the
unique integral curve through ρ and is defined for all t so J* = R. Now
suppose X(p) ф 0. If t Ε (Γ", Γ+), then the image point с*(t) must always
lie in the support of X. Indeed, since X{p) is not zero, c£ is not constant.
If (y{t) were in M\supp(X), then c£(t) would be contained in some ball on
which X vanishes and then by uniqueness this implies that c£ is constantly
equal to c*(t) for all time a contradiction. But we show that if T+ < oo,
then given any compact set К С Μ, for example the support of X, there is
an б > 0 such that for all t € (T+ - e, T+), the image c£(t) is outside K. If
not, then we may take a sequence U converging to T+ such that c*(ti) € K.
But then going to a subsequence if necessary, we have X{ := c^{U) -» χ € К.
Now there must be a flow box (Ϊ7, а, я), so that for large enough /с, we have
that tk is within distance a of T+ and x^ = c*{tk) is inside U. We are then
guaranteed to have an integral curve c£k(t) of X that continues beyond T+
and thus can be used to extend c£ which is a contradiction of the maximality
of T+. Hence we must have T+ = oo. A similar argument gives the result
that T~ = -oo. Π
Exercise 2.100. Let a > 0 be any fixed positive real number. Show that
if for a given vector field X, the flow φχ is defined on (—α, α) χ Μ, then in
fact the (maximal) flow is defined on R χ Μ and so X is a complete vector
field.
102
2. The Tangent Structure
Figure 2.3. Isolated vanishing points
•
Figure 2.4. Straightening
If a vector field is zero at some point but nonzero elsewhere in a
neighborhood of that point, then we say that the field has an isolated zero. The
structure of a vector field near such an isolated zero can be quite complex and
interesting. The qualitative structure of three such possibilities are show in
Figure 2.3 for dimension two. Near zeros that are not isolated, the situation
is also potentially complex. On the other hand, at non-vanishing points, all
vector fields are the same up to a local diffeomorphism. This is the
content of the following theorem which is sometimes called the straightening
theorem.
Theorem 2.101. Let X be α smooth vector field on Μ with X(p) φ 0 for
some ρ G M. Then there is α chart ([/, x) with ρ G U such that
x=-L°nu-
Proof. Since this is clearly a local problem, it will suffice to assume that
Μ = Rn and ρ = 0. Let (u1,... ,un) be standard coordinates on Rn. By
a rotation and translation if necessary, we may assume that X(0) = gfrL.
The idea is that there must be a unique integral curve through each point
of the hyperplane {u1 = 0}. We wish to arrange for the new coordinates of
q G Rn to be such that if an integral curve of X passes through (0, a2,..., an)
at time zero and hits q at time £, then хъ(р) = аг for г = 2,... ,n while
χ1 (ρ) = t. Let φ be a local flow for X near 0, and define χ in some sufficiently
2.8. Vector Fields
103
small neighborhood of 0 by
χ(α\...,αη):=φαι(0,α2 an).
For α = (a1, a2,..., αη) in the domain of χ, and / € C°°(M), we have
2*· —
/ dul
(foX)
= Yiml[f(x(a1+h,a2,...,an))-f(X(a1,a2,...,an))]
= lim 1 [/ [<pai+h(0, a2,..., an)) - f(x(a\a2,..., an))]
= Hm I [/ (<ph(a)) - /(χ(α)))] = (Χ/)(χ (a)).
h->0 η
In particular, Τχ ■ -j^L· = ^r|0· If i > 0, then at 0 we have
f°X
Τχ· —
диг
диг
= lim-[/(x(0,..,/i,...)-/(0)]
= limi[/(0,..)/i,...)-/(0)]=|?
/·
Thus Tqx = id and so by the inverse mapping theorem (Theorem 2.25)
we see that after restricting χ to a smaller neighborhood of zero, the map
x := χ-1 is a chart map. We have already seen that Τχ -£^\ = Χ ο χ. But
then for / G C°°{M) we have
д
дх1
so that •£T=X.
x(p)
/°x_1 =
д
ди1
χ(ρ)
f°x
=Γ*έ
*(p)
/ = (ΑΓοχ)(χ(ρ))/ = Αρ/,
D
2.8.2. Lie derivative. We now introduce the important concept of the
Lie derivative of a vector field extending the previous definition. The Lie
derivative will be extended further to tensor fields.
Definition 2.102. Given a vector field X, we define a map Cx : X(M) —>
X(M) by
CXY:=\X,Y].
This map is called the Lie derivative (with respect to X).
104
2. The Tangent Structure
The Jacobi identity for the Lie bracket easily implies the following two
identities for any Χ,Υ,Ζ e X(M) -> X{M):
CX[Y,Z] = ICXY,Z] + [Y,CXZ},
£[xy] = [£x»£y] (i-e- £[x,y\ = Cx о CY - Cy о Cx).
We will see below that CXY measures the rate of change of Υ in the
direction X. To be a bit more specific, (CXY) (p) measures how Υ changes
in comparison with the field obtained by "dragging" Yp along the flow of
X. Recall our definition of the Lie derivative of a function (Definition 2.70).
The following is an alternative characterization in terms of flows: For a
smooth function / : Μ ->· R and a smooth vector field X G X(M), the Lie
derivative Cx of / with respect to X is given by
£x/W-5
foifX(t,p).
0
Exercise 2.103. Explain why the above formula is compatible with
Definition 2.70.
We will also characterize the Lie derivative on vector fields in terms of
flows. First, we need a technical lemma:
Lemma 2.104. Let X G X{M) and f G C°°(U) with U open and ρ G U.
There is an interval 1$ := [—δ, δ] and an open set V containing ρ such that
φχ{Ιδ x V) С U and a function g G C°°{Is х V) such that
f{<pX{t,q)) = f{q) + tg{t,q)
for all (i, q) G Is x V and such that #(0, q) = Xqf for all q G V.
Proof. The existence of the set Is x V with φχ(Ι$ x V) С U follows from
our study of flows. The function r{r,q) := f(<px(r,q)) — f(q) is smooth on
I6xV and r(0,g) = 0. Let
fl dr
9(t,q) ·= / ^{st,q)ds,
so that
f1 dr f1 д
ЫЬя) = / -7^{st,q)tds= I d^r{st,q)ds = r(t,q).
Then f(<pX(t, q)) - f(q) + tg(t, q). Also
2.8. Vector Fields
105
Proposition 2.105. LetX andY be smooth vector fields onM. Let φ = ψχ
be the flow. The function t н* Τψ-t · Υφί(ρ) is differentiable at t = 0 and
d' Τψ-t ■ ΥφΜ = [Χ, Υ], = (СхУ) (р)-
<2'5>
t=0
Proof. Let / € C°°(U) with ρ € U as in the previous lemma. We have
d\ Tu} Y f _ ,. Τφ-г ■ Υ^ω -YP Ypf ~ (JVt · Υφ t{p)) f
by replacing t by —i. Thus we study the difference quotient in the second
limit above,
Ypf - {Τφι ■ Υψ t(p)) f _Ypf- Υφ_ί{ρ) (/ о <pt)
t t
_ΥΡί-ΥΨ t(p)(f + tgt)
t
where ρ is as in the lemma and gt{<l) = g(t, q). Continuing, we have
YPf-Y^(p)(f + tgt) (Yf)(P) - (Yf)(<P-t(p)) v n
Taking the limit as t —> 0 and recalling that go = Xf on V, the right hand
side above becomes
(У/)(р)-(У/)(^(р))
bo 1 jsy*-«w*
t-+o ί p J
= XpYf-YpXf=[X,Y]pf.
All told we have
(r^t.yvt(p))/ = [x,y]p/
eft
li=0
for all / G C°°(J7). If we let U be the domain of a chart (t/, x), then letting /
be each of the coordinate functions we see that each component of the TpM-
valued function t \-* Т(/?_гУ^(р) is differentiable at t = 0 and so the function
is also. Then we can conclude that ^\t=0 {Τψ-t ■ ^(p)) = [^»^]ρ· ^
We see from this characterization that in order for (CxY) (p) to make
sense, Υ only needs to be defined along the integral curve of X that passes
through p.
Discussion: Notice that if X is a complete vector field, then for each
t 6 R the map φ* is a diffeomorphism Μ —> Μ and
(2.6) И*У = (Г^Г1оУо^.
106
2. The Tangent Structure
<pf(i>)=<7
Figure 2.5. Lie derivative by flow
One may write
cxY-jt
(v?)'y-
On the other hand, if X is not complete, then there exists no t such that φχ
is a diffeomorphism of Μ since for any specific t there might be points of Μ
for which φχ is not even defined! For an X which is not necessarily complete,
it is best to consider the map φχ : (t,x) ι—> φχ{ί,χ) which is defined on
some open neighborhood of {0} χ Μ in Μ χ Μ which may not contain any
set of the form [-6, e] χ Μ unless 6 = 0. In fact, suppose that the domain of
ψχ contains such a set with e > 0. It follows that for all 0 < t < e the map
φχ is defined on all of Μ and φχ (ρ) exists for 0 < t < e independent of p.
But a standard argument (recall Exercise 2.100) shows that t \-^ φχ(ρ) is
defined for all i, which means that X is a complete vector field. If X is not
complete, then φ? is defined only on V*x. Thus (y>f )* : Χ(Όχ*) ->· Χ(νιχ).
(We will abbreviate (φ?) to φ**.) If Υ € 3£(M), then we may interpret
tp?*Y to mean φ?*(Υ v-t) G X(VX). Now Μ С Ц/о1'*' so both Vt<J>)
and (φ?*Υ) (ρ) make sense for any ρ as long as t is sufficiently small and
(φ*·Υ) (ρ) = {ΤφΧ)-1Υ{φϊ(ρ)).
In fact, one should take the last equation for the definition of (φ?*Υ) (ρ)
for the case of an incomplete X. At any rate, the function t н^ {φ**Υ)(ρ)
has a well-defined germ at t = 0. With this in mind we might still write
CxY = ^|0 (φχ*Υ) even for vector fields that are not complete as long as
we take the right interpretation of φχ*Υ*
Theorem 2.106. Let Χ, Υ be vector fields on a smooth manifold M. Then
- {φϊ*Υ) (ρ) = (φ?·£χΥ)(ρ)
dt
for any ρ G M.
2.8. Vector Fields
107
Proof. Let φ :— φχ. Suppressing the point ρ we have
dt
*v d
* XT d
0
{φ*Υ) = φΐεχΥ. D
s
0
Exercise 2.107. Show that ||0 (у£У)(р) = - (CXY) = -[X,Y].
Proposition 2.108. Let X G X(M) and Υ G X(N) be f-related vector
fields for a smooth map f : Μ -+ N. Then
f°<p? = <pl°f
whenever both sides are defined. Suppose that f : Μ —> N is a diffeomor-
phism and X G X(M). Then the flow of /*X = (/_1)*X is f ο φ? ο f'1
and the flow of f*X is f~l ο φ£ ο /.
Proof. For any ρ G M, we have |(/ ο φ?){ρ) = Tf - |¥>f (ρ) =T/oIo
ψ* (p) = У ° / ° Ψ% (ρ)· But / ° Ψο (ρ) = /(P) and so ί ь-> / о ^ (p) is
an integral curve of Υ starting at /(p). By uniqueness we have / ο φ*(ρ) =
ψϊ(ί(ρ))· The second part follows from the first. D
Theorem 2.109. For Χ, Υ G X(M), the following are equivalent:
(i) £xY=[X9Y]=0.
(ii) (φ*)*Υ = Υ whenever defined.
(Hi) The flows of X and Υ commute:
Ψ? ° Ψβ — Ψβ °Ψί whenever defined.
Proof. (Sketch) The equivalence of (i) and (ii) follows easily from
Proposition 2.105 and Theorem 2.106. Using Proposition 2.108, the equivalence of
(ii) and (iii) can be seen by noticing that φ* ο φζ = φζ ο φ£ is defined and
true exactly when φζ = (p*t ο φζ ο φ£ is defined and true, which happens
exactly when
φ3 = φϊ*
is defined and true. This happens, in turn, exactly when Υ = (φ*)*Υ. □
Example 2.110. On R2 we have the flows given by </>(£, (я, у)) = фг(х, у) :=
{x + ty,y) and ^(t,(x,j/)) = фь{х,у) := (x,y + t). We have ψι ο </>ι(0,0) =
(1,1) while ф\ ο ^ι(0,0) = (0,1). These noncommuting flows correspond to
the noncommuting vector fields X = уд/дх and Υ = д/ду.
Exercise 2.111. Find global noncommuting flows on S2.
108
2. The Tangent Structure
Figure 2.6. Bracket measures lack of commutativity of flows
The Lie derivative and the Lie bracket are essentially the same object
and are defined for local sections X G Xm(U) as well as global sections.
This is obvious anyway since open subsets are themselves manifolds. As is
so often the case for operators in differential geometry, the Lie derivative is
natural with respect to restriction, so we have the commutative diagram
X{U)
£χ\
__Ψ
X(U)
X(V) C-±¥ X(V)
where X\v denotes the restriction of X G X(M) to the open set U and Гу
is the map that restricts from U to V С U.
If X and Υ are smooth vector fields with flows φχ and φγ, then starting
at some ρ G Μ, if we flow with X for time \ft, then flow with Υ for time
y/ty and then flow backwards along X and then Υ for time \ft, we arrive at
a point a(t) given by
It turns out that a(t) is usually not ρ (see Figure 2.6). In fact, we have the
following theorem:
Theorem 2.112. With a(t) as above we have
d
dt
a(t) = [X,Y](p).
t=o
Proof. See Problem 8.
D
We know by direct calculation that if (x1,... ,xn) are the coordinate
functions of a chart, then
-0
2.8. Vector Fields
109
for alH, j = 1,..., n. The converse is also locally true. This follows from the
next theorem, which can be thought of as saying that we can simultaneously
"straighten" commuting local fields.
Theorem 2.113. Let Μ be an η-manifold. Suppose that on a
neighborhood V of a point ρ Ε Μ we have vector fields X\,..., Xk such that
Xi(x),..., Xk(x) Q>re linearly independent for all χ Ε V. If [Xi, X3) — 0 on
V for all i,j, then there exist a possibly smaller neighborhood U of ρ and a
chart χ : U —> Rn such that
д
dxi
Xi on U for г = 1,..., к
and such that for the corresponding flows we have
φ?1 о х~\и\ ... ,ti'",... ,un) = (u\ ...y+t,...,un)
fori = 1, ...,&.
Proof. Let ρ eV and choose a chart (0,y) centered at ρ with О С V. By
rearranging the coordinate functions if necessary and using a simple linear
independence argument, we may assume that the vectors
Xi(p),...,Xk(p),
д
„*i
dxk+l
д
' дхп
form a basis for TpM. Let φ^1,..., </?tfcfc be local flows for X\,..., Xk and let
W be an open neighborhood of ρ contained in О such that the composition
Xk
° Vf
is defined on W and maps W into О for all t\,..., t^ € (—e, e) as in Lemma
2.92. Define
S:={(ak+1,...,an):y-1(0,...,0,ak+\...,an)}eW,
and define the map ψ : (—e, e)fc χ S —> U by
Since [Xj, Xj] = 0 for 0 < i, j < k, we know that the flows in the composition
above commute. Hence for any α € у (W) and smooth function /, we have
for 1 < г < к,
д
™--L
/(*>:
ди
_ _а_
_ _а_
~ диг
= Хг&Ш,
ди1
хк
ик
fo^u1 ип)
•■•°<рХ11оу-1(0,...,0,иш,...,ип))
ί{φχ:°<ρ%°···°φχ;ο...οφ$ογ Но,...^,^1,...,^))
по
2. The Tangent Structure
where the caret indicates omission. Thus we have
= Хг(ф(а)) for 1 < г < к,
диг
and in particular,
диг
= Xi(p) for 1 <i < k.
10
For к + 1 < г < η, we have
V(0)...,0,u*+1>...,un) = y"1(0,...,0,ufc+1>---,tin),
so
Γ°*·έ
_9_
lo дУ1\
Thus Τοψ maps a basis of ToRn to a basis of TPM. We can use the inverse
mapping theorem (Theorem 2.25) to conclude that ψ is a diffeomorphism
from some open neighborhood of 0 € Mn onto an open neighborhood U of
pin M. It is now straightforward to check that χ = ψ~ι is a chart map of
the desired type. D
For later reference, we note that if (17, x) is a chart of the type produced
in the theorem above, then we can easily arrange that χ (17) is of the form
V χ W С Rk x Rn_/c, so that the fields ^r, ■ -., jf* are tangent to the
submanifolds of the form
S := {p e U : xk+\p) = afc+\ ... ,xk+\p) = an}.
The reader has probably surmised that the mathematics of vector fields
and flows can be applied to fluid mechanics. This is quite true, but one
needs to deal with time dependent vector fields. There is a trick that allows
time dependent vector fields to be treated as ordinary vector fields on a
manifold of one higher dimension, but it is not always best to think in those
terms. The author has included a bit about time dependent vector fields in
the online supplement [Lee, Jeff].
2.9. 1-Forms
Definition 2.114. A smooth (resp. Cr) section of the cotangent bundle is
called a smooth (resp. Cr) 1-form or also a smooth (resp. Cr) covector
field. The set of all Cr 1-forms is denoted by 3Cr*(M) and the set of smooth
1-forms is denoted by X*(M).
The set Xr*(M) is a module over Cr(M). Later we will have reason to
denote X*(M) also by Ω1(Μ). The analogue of Lemma 2.65 is true. That
is, if olp £ T£M, then there is a 1-form α such that a(p) = p. The proof is
again a simple cut-off function argument.
2.9. 1-Forms
111
Definition 2.115. Let / : Μ -> Ш be a Cr function with r > 1. The map
df : Μ —> Τ* Μ is defined by ρ к* df(p) where d/(p) is the differential at ρ
as defined in Definitions 2.20 and 2.22; df is a 1-form called the differential
of/.
As a consequence of the definitions of we have
d(fg) = gdf + fdg.
Indeed, for vp eTpM we have
d (fg) (p) 'Vp = vP (fg) = g(p)vP (/) + f(p)vp (g)
= g(p) (df) (p) · vP + f(p)dg(p) · vp
= (gdf + fdg) (p) · vp.
Three views on a 1-form: The novice may easily become confused
about what should be the argument of a 1-form. The reason for this is that
one can view a 1-form in at least three different ways. If α is a smooth
1-form, then we have the following interpretations of a:
(1) We may view α as a map a : Μ —> Τ*Μ (as in the definition)
so that a(.) takes points as arguments; a(p) G T*M. We also
sometimes need to write ap = a(p) just as for a vector field X we
sometimes write X{p) as Xp.
(2) We may view α as a smooth map a : Τ Μ —> Ш so that for vp G TPM
we make sense of a(vp) by a(vp) = ap(vp).
(3) We may view α as a map a : X(M) -» C°°(M), where for X e
X(M) we interpret a(X) as the smooth function ρ ^ ap(Xp).
The second and third interpretations are dependent on the first. The
third view is very important, and in that view, a : X(M) —> C°°(M) is a
C°°(M)-linear map. We have already mentioned that given a chart (/7, x)
with χ = (a;1,...,xn), the differentials dxl : ρ \-> dxl\ define 1-forms on U
such that dxl | ,..., dxn\ form a basis of T*M for each ρ G U. If α is any
smooth 1-form (covector field) defined at least on /7, then
а\ц = 2_]aidxl
for uniquely determined functions a^. In fact, ol(-J^) = бц (using the third
view from above). As with vector fields we will usually just write a =
Y^aidx1. In particular, df(-^) = ^, and so we have the following familiar
looking formula:
112
2. The Tangent Structure
which is interpreted to mean that at each ρ EUa we have
dxl\ .
Ρ
The covector fields dxl form what is called a coordinate coframe field or
holonomic coframe field 2 over U. Note that the component functions Хъ
of a vector field with respect to the chart above are given by Xх = dxl(X),
where by definition dxl(X) is the function ρ н-)> dxl\ (Xp). Thus
Note. If α is a 1-form on Μ and ρ G M, then one can always find many
functions / such that df(p) = a(p), but there may not be a single function
/ such that this is true for all points in a neighborhood, let alone all points
on M. If in fact df = a for some / € C°°(M), then we say that a is exact.
More on this later.
Let us try to picture 1-forms. As a warm up, let us recall how we might
picture a tangent vector vp at a point ρ G Rn. Let 7 be a curve with
7;(0) = vp. If we zoom in on the curve near p, then it appears to straighten
out and so begins to look like the curve t *-ϊ ρ + tvp. So one might say that
a tangent vector is the "infinitesimal representation" of a (parameterized)
curve.
At each point a 1-form gives a linear functional in that tangent space, and
as we know, the level sets of a linear functional are parallel affine subspaces
or hyperplanes. We should imagine level sets as being labeled by the values
of the function. A covector puts a ruling in the tangent space that measures
tangent vectors stretching across this ruling. For example, the 1-form df in
M3 gives a ruling in each tangent space as suggested by Figure 2.7a. For a
given p, the level sets of dfp are what we see if we zoom in on the level sets
of / near p. The fact that the individual dfps living in the tangent spaces
somehow coalesce into the level sets of the global function / as shown in
2.7b, is due to the fact that the 1-form df is the differential of the function
/·
A more general 1-form a is still pictured as straight parallel hyperplanes
in each tangent space. Because these level sets live in the tangent space,
we might call them infinitesimal level sets. These (value labeled) level sets
may not coalesce into the level sets of any global smooth function on the
manifold. There are various, increasingly severe ways coalescing may fail
to happen. The least severe situation is when a is not the differential of a
global function but is still locally a differential near each point. For example,
2The word holonomic comes from mechanics and just means that the frame field derives from
a chart. A related fact is that L·5 , —-] = 0.
2.9. l-Forms
113
Figure 2.7. The form df follows level sets
Figure 2.8. Level sets of overlapping angle functions
if Μ = R2\{0}, then the familiar 1-form a = (x2 + y2)~l(-ydx + xdy) is
locally equal to άθ for some "angle function" θ which measures the angle
from some fixed ray such as the positive χ axis. But there is no such single
smooth angle function defined on all of M2\{0}. Thus, globally speaking, a
is not the differential of any function. In Figure 2.8, we see the coalesced
result of "integrating" the infinitesimal level sets which live in the tangent
spaces. While these suggest an angular function, we see that if we try to
picture rising as we travel around the origin, we find that we do not return
to the same level in one full circulation, but rather keep rising. Locally,
however, we really do have level sets of smooth functions.
Now the second more severe way that a 1-form may fail to be the
differential of a function is where there is not even a local function whose
differential agrees with the 1-form near a point. The infinitesimal level sets
do not coalesce to the level sets of a smooth function, even in small
neighborhoods. This is much harder to represent, but Figure 2.9 is meant to at
least be suggestive. Nearby curves cross inconsistent numbers of level sets.
As an example consider the 1-form
β = —ydx + x dy.
The astute reader may object that surely radial rays do match up with the
114
2. The Tangent Structure
Figure 2.9. Suggestive representation of a form which is not closed
directions described by this 1-form. However, the point is that a covector
in a tangent space is not completely described by the level sets as such,
but rather the level sets are to be thought of as labeled according to the
values they represent in the individual tangent spaces. Here we have a case
where the infinitesimal level sets coalesce, but the values assigned to them
do not; they are 1-dimensional submanifolds that fit the 1-form but they
are not level sets of even a local smooth function whose differential agrees
with the 1-form. This brings us to the most severe case which only happens
in dimension 3 or greater. It can be the case that there is no nice family of
(n — l)-dimensional submanifolds that line up in any reasonable sense with
the 1-form (either globally or locally). This is the topic of the Frobenius
integrability theory for tangent distributions that we study in Chapter 11,
and we shall forgo any further discussion until then except to say that the
reader should be ready to understand much of that chapter, including the
Frobenius theorem, after finishing the next chapter.
Definition 2.116. If φ : Μ -+ N is a C°° map, the pull-back of a 1-form
a e X*(N) by φ is defined by
(φ*α)ρ·ν = αφ{ρ){Τρφ·ν)
for ν e TPM.
This extends the notion of the pull-back of a function defined earlier. If
we view a 1-form on Μ as a map TM —»■ R, then the pull-back is given by
φ*α = αοΤφ.
Exercise 2.117. The pull-back is contravariant in the sense that if φχ :
Mi -> M2 and φ2 : M2 -> N, then for a G X*(N) we have (φ2 ο φι)* =
ΦΙ ο φ*.
Next we describe the local expression for the pull-back of a 1-form. Let
([/, x) be a chart on Μ and (V, y) a coordinate chart on N with ф(и) С V. A
2.9. 1-Forms
115
typical 1-form has a local expression on V of the form a = Σ ctidy1 for ai G
C°°(V). The local expression for φ*a on U is φ*a = Σ (α» ο φ) d (уг о ф) =
Σ (fli ° 0) qxJ dxK Thus we get a local pull-back formula3 convenient for
computations:
(2.7) φ* (J2 <H <V) = Ε (at ο φ) Щ^-dxK
The pull-back of a function or 1-form is defined whether φ : Μ —ϊ Ν
happens to be a diffeomorphism or not. On the other hand, the pull-back
of a vector field only works in special circumstances such as where φ is a
diffeomorphism. Let φ : Μ —► TV be a C°° diffeomorphism with r > 1.
Recall that the push-forward of a function / G C°°(M) is denoted </>*/
and defined by φ*/(ρ) := ί(Φ~1(ρ))· We can also define the push-for ward
of a 1-form as φ*α = α о Тф~г.
Exercise 2.118. Find the local expression for φ+f and ф*а. Explain why
we need φ to be a diffeomorphism.
Lemma 2.119. The differential is natural with respect to pull-back. In other
words, if φ : Μ —>■ N is a C°° map and f : TV -» R a C°° function, then
dffif) = φ*άί. Consequently, the differential is also natural with respect to
restrictions.
Proof. We wish to show that
(φ*ά/)ρ = ά{φ*!)ρ
for all ρ € Μ. Let ν € TpM and write q = f(p). Then
&*df)\pv=df\g(T^-v)
= ((Tp0-t;)/)(e)
= »(/°0)(p) = d(07)lp«.
The second statement is obvious from local coordinate expressions, but also
notice that if U is open in Μ and ι : U <-»> Μ is the inclusion map (i.e. the
identity map idM restricted to J7), then f\v = L*f and df\jj = £*d/. So the
statement about restrictions is just a special case of the first part. D
The tangent and cotangent bundles Τ Μ and Τ* Μ are themselves
manifolds and so have their own tangent and cotangent bundles. Among other
things, this means that there exist 1-forms and vector fields on these
manifolds. Here we introduce the canonical 1-form on T*M. We denote this form
by 0Can and note that it is a section of T* (T*M). Let a G T*M and suppose
that a is based at ρ so that a G T*M. Consider a vector ua G Ta (T*M).
To ensure clarity we have not used the Einstein summation convention here.
116
2. The Tangent Structure
Notice that since π : T*M -> M, we have Ταπ : Τα (Τ*Μ) -+ ΤρΜ. Thus
Ταπ ■ ιια G ΤρΜ. We define
0can(^a) = α (Γα7Γ · Ua) .
The definition makes sense because а € T*M and Ταπ · ua e TPM. Let
(C/, x) be a chart containing ρ and let (ж1 ο π,... ,xn ο π,ρι,...,ρη) =
(ϊ1,..., 9η,Ρι,... ,Ρη) be the associated natural coordinates for Γ*Μ.
Exercise 2.120. It is geometrically clear that Ταπ · ^U =0 since τρ-
tangent to the fiber T*,sM along which π is constant. Deduce this directly
from the definitions. Hint: ^
can be represented by a curve in Τ*,^Μ
We wish to show that locally 0can = J^Pi d<?1. It will suffice to show that
) = Pi{o) and 0Сап(тг7 ) = 0 for all i. We have
Ρ Ια
в™Ш)=а(т°*-^\)=°
since in fact Γαπ · A = 0 by Exercise 2.120. Also, we have
д
6can\dq
= аг =Pi(a),
= a
where we have used the fact that Ταπ · ψλ = -^ | , which follows from the
definition ql — хг о π. Indeed, we know that Γαπ · -^
= ci^\pioTSOme
constants c^, but we have
-(■,··)(έϋ-*,(έϋ-*
This 1-form plays a basic role in symplectic geometry and classical
mechanics. For more about symplectic geometry see the online supplement
[Lee, Jeff].
2.10. Line Integrals and Conservative Fields
Just as in calculus on Euclidean space, we can consider line integrals on
manifolds, and it is exactly the 1-forms that are the appropriate objects to
integrate. First notice that all 1-forms on open sets in R1 must be of the
2.10. Line Integrals and Conservative Fields
117
form / dt for some smooth function /, where t is the coordinate function on
R1. We begin by defining the line integral of a 1-form defined and smooth
on an interval [a, b] С R1. If β = / dt is such a 1-form, then
f β:= f f(t)dt.
J[a,b] J а
Any smooth map 7 : [a, 6] —> Μ is the restriction of a smooth map on some
larger open interval (a — ε, b + ε), and so there is no problem defining the
pull-back 7*a. If 7 : [a, 6] -» Μ is a smooth curve, then we define the line
integral of a 1-form α along 7 to be
/<*:= /" Ί*α= ! f{t)dt
i,4
where 7*0: = f dt. Now if t = cf)(s) is a smooth increasing function, then we
obtain a positive reparametrization 7 = 7 ο φ : [с, d] —> Μ, where 0(c) = α
and 0(d) = 6. With such a reparametrization we have
JL* *
ψ η a
/ 7*<* = /
J[c,d\ J[c}d
= / 0*(/я)= / /оА,
= / /(0(5))0'(5)d5= / /(i)di,
J с Ja
where the last line is the standard change of variable formula and where
we have used cf)*(fdt) = ^3 'ds, which is a special case of the pull-back
formula mentioned above. We see now that we get the same result as before.
This is just as in ordinary multivariable calculus. We have just transferred
the usual calculus ideas to the manifold setting.
Definition 2.121. A continuous curve 7 : [a, 6] —> Μ into a smooth
manifold is called piecewise smooth if there exists a partition a = to < t\ <
-' <tk = b such that 7 restricted to [i$, ij+i] is smooth for 0 < г < к — 1
(in the sense of Definition 1.58).
It is convenient to extend the definitions a bit to include integration
along piecewise smooth curves. Thus if 7 : [a, b] -¥ Μ is such a curve, then
we define for a 1-form a,
Л" ^J\t
i=0 •'[Wt+i]
where ητ is the restriction of 7 to the interval [£j, ij+i].
118
2. The Tangent Structure
Just as in ordinary multivariable calculus we have the following:
Proposition 2.122. Let 7 : [a,b] —» Μ be α piecewise smooth curve with
7(a) = p\ and 7(6) — p2. If a = d/, i/ien
/a= fdf = f(p2)-f(Pl).
/7 </7
/n particular, J a is path independent in the sense that it is equal to Jc a
for any other piecewise smooth path с that also begins at p\ and ends at p2·
Definition 2.123. If a is a 1-form on a smooth manifold Μ such that
Jca = 0 for all closed piecewise smooth curves c, then we say that a is
conservative.
We will need a lemma on differentiability.
Lemma 2.124. Suppose f is a function defined on a smooth manifold M,
and let a be a smooth 1-form on M. Suppose that for any ρ € Μ, νρ € Τ Μ
and smooth curve с with c(0) = vp, the derivative ^|0/(c(*)) exists and
|| f(c(t)) = a(vp).
Then f is smooth and df = a.
Proof. We work in a chart (t/, x). If we take c(t) := х_1(х(р)+£е;), then the
hypotheses lead to the conclusion that all the first order partial derivatives
of / о х-1 exist and are continuous. Thus / is C1. But then also dfp · vp =
■jft\0f(c(t)) = olp{vp) for all vpy and it follows that df = a, and this also
implies that / is actually smooth. D
Proposition 2.125. If a is a 1-form on a smooth manifold M, then a is
conservative if and only if it is exact
Proof. We know already that if α = df, then a is conservative. Now
suppose a is conservative. Fix po G M. Then we can define f(p) = J a,
where 7 is any curve beginning at po and ending at p. Given any vp € TPM,
we pick a curve с : [—Ι,ε) with ε > 0 such that c(—1) = po, c(0) = ρ and
c'(0) = vp. Then
d 1./ / ч\ d \ Γ
-Α №τ)) = -Α / a
dT\0 dT\oJc[ 1,t]
d\ f d\ Γ
= τ\ I a+ τ\ a
dT\oJc\[-l,0] dT\oJc\[0,T]
d\ Γ * dl Γ ΜΙ
= 0+— / c*a- — / g{t)dt
= 9(0),
2.10. Line Integrals and Conservative Fields
119
where c*ct — gdt. On the other hand,
а(ур) = а(с'(0)) = а(т0с- —
= с-а(|[)=9(0)Л|0(||о)=9(0),
where t is the standard coordinate on R. Thus af|0/(c(r)) = a(vp) f°r
any vp € TpM and any ρ G M. Now the result follows from the previous
lemma. D
It is important to realize that when we say that a form is conservative
in this context, we mean that it is globally conservative. It may also be the
case that a form is locally conservative. This means that if we restrict the
1-form to an open set which is diffeomorphic to a Euclidean ball, then the
result is conservative on that ball. The following examples explore in simple
terms these issues.
Example 2.126. Let a = [x2 + y2)~ (—ydx + xdy). Consider the small
circular path с given by (x,y) = (xo + £cost,yo + ^sini) with 0 < t < 2π
and ε > 0. If (zo, yo) = (0,0), we obtain
г2тг
2тг j
~2 (— (esint) (—£sin£) + (εcost) (ecost))dt = 2π.
Thus a is not conservative and hence not exact. On the other hand, if
(яо?Уо) Ф (0,0), then we pick a ray Rq that does not pass through (хо.Уо)
and a function θ(χ: у) which gives the angle of the ray R passing through
(x,y) measured counterclockwise from Rq. This angle function is smooth
and defined on U = R2\Rq. If ε < \y/xl + Уо' ^^еп с ^^ ™а§е inside the
domain of θ and we have that a\ U = άθ. Thus /c a = 9(с(0))-в(с(2п)) = 0.
We see that a is locally conservative.
Example 2.127. Consider β = у dx — xdy on M2. If it were the case that
for some small open set U С Μ2 we had β\ν = df, then for a closed path с
with image in that set, we would expect that f β = f(c(2n)) — /(c(0)) = 0.
However, if с is the curve going around a circle of radius ε centered at
120
2. The Tangent Structure
(яо,2/о)> then we have
/*-/(
»m|-w£i*
/•27Г
= / ((xo + esini) (—esini) — (yo + ε cost) (ε cost)) dt
Jo
= -2ε2π,
so we do not get zero no matter what the point (xo, yo) and no matter how
small ε. We conclude that β is not even locally conservative.
The distinction between (globally) conservative and locally conservative
is often not made sufficiently clear in the physics and engineering literature.
Example 2.128. In classical physics, the static electric field set up by a
fixed point charge of magnitude q can be described, with an appropriate
choice of units, by the 1-form
-^x dx + -^y dy + -^zdz,
ρό ρό ρό
where we have imposed Cartesian coordinates centered at the point charge
and where ρ = \Jx2 + y2 + z2. Notice that the domain of the form is the
punctured space R3\{0}. In spherical coordinates (p, 0,</>), this same form
is
**-(?)·
so we see that the form is exact and the field is conservative.
2.11. Moving Frames
It is important to realize that it is possible to get a family of locally defined
vector (resp. covector) fields that are linearly independent at each point in
their mutual domain and yet are not necessarily of the form g~ (resp. dx1)
for any coordinate chart. In fact, this may be achieved by carefully choosing
n2 smooth functions flk (resp. af) and then letting Ek := J2ifk^ (resP·
Definition 2.129. Let Εχ,Εϊ,.. .,En be smooth vector fields defined on
some open subset U of a smooth η-manifold M. If E\(ρ), Ε<χ(ρ),..., Εη(ρ)
form a basis for TPM for each ρ € U, then we say that {Е^Еъ^..., En) is a
(non-holonomic) moving frame or a frame field over [7.
If Ει ^ £2,. ·., En is a moving frame over U С Μ and X is a vector field
defined on U, then we may write
X = Y^XlEi on U,
2.11. Moving Frames
121
for some functions Xх defined on U. If the moving frame (£Ί,..., En) is
not identical to some frame field (g|r,..., gjn) arising from a coordinate
chart on U7 then we say that the moving frame is non-holonomic. It is often
possible to find such moving frame fields with domains that could never be
the domain of any chart (consider a torus).
Definition 2.130. If £α, Ε2,..., En is a frame field with domain equal to
the whole manifold M, then we call it a global frame field.
Most manifolds do not have global frame fields.
Taking the basis dual to (Ei(p), ...}En(p)) in T*M for each ρ e U we
get a moving coframe field (01,..., 0n). The 0г are 1-forms defined on U.
Any 1-form α can be expanded in terms of these basic 1-forms as α = Σ а$г-
Actually it is the restriction of α to U that is being expressed in terms of the
0*, but we shall not be so pedantic as to indicate this in the notation. In a
manner similar to the case of a coordinate frame, we have that for a vector
field X defined at least on Ϊ7, the components with respect to (£4,..., En)
are given by θι(Χ):
X = Y^9i{X)EionU.
Let us consider an important special situation. If Μ χ Ν is a product
manifold and (t/, x) is a chart on Μ and (V, y) is a chart on TV, then we have
a chart (U χ V, χ χ у) on Μ χ Ν where the individual coordinate functions
are χ1 ο prb ..., xm о ргъ у1 о рг2,..., уп о рг2, which we temporarily denote
by x1,..., xm, y1,..., y*1. Now we consider what is the relation between
the coordinate frame fields (g|r,... ^г)5 (^т> - · · щж) and the frame field
(g|r,..., g£n )· The latter set of η + m vector fields is certainly a linearly
independent set at each point (p, q) G U x V. The crucial relations are
Exercise 2.131. Show that £-г\р = Трт,^^ and ^
dtf
{v,q)
Remark 2.132. In some circumstances, it is safe to abuse notation and
denote хг о prx by xl and уг о pr2 by уг. Of course we are denoting -^ by
gjr and so on.
A warning (The second fundamental confusion of calculus*): For a
chart (17, x) with χ = (χ1,... ,£η), we have defined ^ for any
appropriately defined smooth (or Cl) function /. However, this notation can be
In [Pen], Penrose attributes this cute terminology to Nick Woodhouse.
122
2. The Tangent Structure
ambiguous. For example, the meaning of ^jr is not determined by the
coordinate function x1 alone, but implicitly depends on the rest of the coordinate
functions. For example, in thermodynamics we see the following situation.
We have three functions P, V and Γ which are not independent but may
be interpreted as functions on some 2-dimensional manifold. Then it may
be the case that any two of the three functions can serve as a coordinate
system. The meaning of -^p depends on whether we are using the coordinate
functions (P, V) or alternatively (Ρ, Γ). We must know not only which
function we are allowing to vary, but also which other functions are held fixed.
To get rid of the ambiguity, one can use the notations {-φ)ν and (qp)t· In
the first case, the coordinates are (Ρ, ΐ0> and V is held fixed, while in the
second case, we use coordinates (Ρ, Γ), and Τ is held fixed. Another way
to avoid ambiguity would be to use different names for the same functions
depending on the chart of which they are considered coordinate functions.
For example, consider the following change of coordinates:
y1 = xl + x2,
y2 = xl -x2 + x3,
y3 = x3.
Here y3 — x3 as functions on the underlying manifold, but we use different
symbols. Thus -^ may not be the same as ^s- The chain rule shows
that in fact -^ = -^ + ifs- This latter method of destroying ambiguity
is not very helpful in our thermodynamic example since the letters P, V
and Τ are chosen to stand for the physical quantities of pressure, volume
and temperature. Giving these functions more than one name would only
be confusing.
Problems
(1) Show that if / : Μ —> N is a diffeomorphism, then for each ρ G Μ the
tangent map Tpf : TPM —> Tf^N is a vector space isomorphism.
(2) Let Μ and N be smooth manifolds, and / : Μ —> N a C°° map. Suppose
that Μ is compact and that N is connected. If / is injective and Tpf is
an isomorphism for each ρ Ε Μ, then show that / is a diffeomorphism.
(Use the inverse mapping theorem.)
(3) Find the integral curves in R2 of the vector field X — e~x-^ + ^- and
determine if X is complete or not.
(4) Which integral curves of the field X — x2-^. + y-§- are defined for all
times t?
Problems
123
(5) Find a concrete description of the tangent bundle for each of the
following manifolds:
(a) Projective space RPn.
(b) The Grassmann manifold G(k,n).
(6) Recall that we have charts on RP2 given by
[x, y, z] i-+ (uu u2) = (x/z, y/z) on f/3 = {ζ ψ 0},
[χ,у, ζ] \-> (νχ, ν2) = (χ/ι/, ζ/y) on U2 = {у Φ 0},
[χ,у, ζ] н> {wuw2) = (у/ж, г/ж) οη ^ι = {ж Φ 0}·
Show that there is a vector field on RP2 which in the last coordinate
chart above has the following coordinate expression:
д д
wi- w2~—.
OW\ OW2
What are the expressions for this vector field in the other two charts?
(Caution: Your guess may be wrong!).
(7) Show that the graph Г(/) = {(ρ, f(p)) G Μ χ Ν : ρ G Μ} of a smooth
map / : Μ —> N is a smooth manifold and that we have an isomorphism
Άρ,Κρ)) (MxN)^ ГШГ(/) Θ Tf(p)N.
(8) Prove Theorem 2.112.
(9) Show that a manifold supports a frame field defined on the whole of Μ
exactly when there is a trivialization of Τ Μ (see Definitions 2.130 and
2.58).
(10) Prove Proposition 2.85.
(11) Find natural coordinates for the double tangent bundle TTM. Show
that there is a nice map s : TTM —>· TTM such that s о s — ΊάττΜ
and such that Τπ ο s = ΤπτΜ and Ттгтм ° s = Τπ. Here π : Τ Μ -> Μ
and ктм : TTM —> TM are the appropriate tangent bundle projection
maps.
(12) Let N be the subset of Rn+1 χ Rn+1 defined by N = {(x,y) : ||x|| - 1
and χ · у — 0} is a smooth manifold that is diffeomorphic to TSn.
(13) (Hessian) Suppose that / G C°°(M) and that dfp = 0 for some ρ G M.
Show that for any smooth vector fields X and У on Μ we have that
Yp(Xf) = XP{Yf)· Let HLp(v,w) := Xp(Yf), where X and Υ are
such that Xp = ν and У^> = u/. Show that HfyP(v,w) is independent of
the extension vector fields X and У and that the resulting map Hfp :
TpM χ TpM —>■ R is bilinear. Hf%p is called the Hessian of / at p. Show
that the assumption dfp = 0 is needed.
(14) Show that for a smooth map F : Μ —> Ν, the (bundle) tangent map
TF : TM —> TN is smooth. Sometimes it is supposed that one can
124
2. The Tangent Structure
obtain a well-defined map F* : X (M) -> X (N) by thinking of vector
fields as derivations on functions and then letting (F*X) f = X (/ о F)
for / G C°°(JV). Show why this is misguided. Recall that the proper
definition of F* : X(M) -» X (N) would be F*X :=TFoXoF~l and is
defined in case F is a diffeomorphism. What if F is merely surjective?
(15) Show that if ψ : Mr —> Μ is a smooth covering map, then so also is
Τψ : TMf -> TM.
(16) Define the map / : MnXn(R) -> sym(MnXn(R)) by f(A) := ЛГЛ, where
MnXn(R) and sym(Mn Xn(R)) are the manifolds of η χ η matrices and
η χ η symmetric matrices respectively. Identify 7д (Mnxn(R)) with
Mnxn(R) and Tf(,4)Sym(Mnxn(R)) with sym(Mnxn(R)) in the natural
way for each A. Calculate Tjf : MnXn(R) —> sym(MnXn(R)) using these
identifications.
(17) Let /χ,..., /τν be a set of smooth functions defined on an open subset of
a smooth manifold. Show that if ci/i(p),..., df^ip) spans Τ* Μ for some
ρ G i7, then some ordered subset of {/i,..., /n} provides a coordinate
system on some open subset V of U containing p.
(18) Let ΔΓ be the vector space of derivations on Cr(M) at ρ G M, where
0 < r < oo is a positive integer or oo. Fill in the details in the
following outline which studies ΔΓ. It will be shown that ΔΓ is not finite-
dimensional unless r — oo.
(a) We may assume that Μ = W1 and ρ = 0 is the origin. Let mr :=
{/ G Cr(Rn) : /(0) = 0} and let m^ be the subspace spanned by
the functions of the form fg for f,g G mr. We form the quotient
space nv/m^ and consider its vector space dual (nv/m^)*. Show
that if δ G ΔΓ, then δ restricts to a linear functional on nv and is
zero on all elements of m^. Conclude that δ gives a linear functional
on mr/m^. Thus we have a linear map ΔΓ —> (rrv/tn^)*.
(b) Show that the map ΔΓ —>· (ην/ηχ^)* given above has an inverse.
Hint: For a λ G (πν/m^)*, consider 6\(f) := λ([/ - /(0)]), where
/ G Cr(Rn) and hence [/-/(0)] G nv/m^. Conclude that by
taking r — oo we have TqR4 = Δοο = (mr/m^)*. The case r < oo
is different as we see next.
(c) Let r < oo. The goal from here on is to show that nv/m^ and
hence (mr/m^) are infinite-dimensional. We start out with the
case Rn = R. First show that if / G nv, then f(x) = xg(x) for
g G C7-1^). Also if / G m?, then f(x) = x2g(x) for g G CT^R).
(d) For each r G {1,2,3,...} and each ε G (0,1), define
τι \ __ ί χΤ^ε f°r x > Oj
9eW—\ 0 for^<0.
Problems
125
Then gTe G mr, but gre i Cr+1(R). Show that for any fixed r G
{1,2,3,...}, the set of elements of the form [grE] := gr + m2 for ε G
(0,1) is linearly independent in the quotient. Hint: Use induction
on r. In the case of r = 1, it would suffice to show that if we are
given 0 < ει < · · · < ει < 1 and if Σί_ι aj9e ^ mn then clj = 0 for
all j.
(Thanks to Lance Drager for donating this problem and its solution.)
(19) Find the integral curves of the vector field on R2 given by X(x,y) :=
x Ш+ХУЩ-
(20) Show that it is possible that a vector field defined on an open subset of
a smooth manifold Μ may have no smooth extension to all of M.
(21) Find the integral curves (and hence the flow) for the vector field on R2
given by X{x,y) := -y£ + x^.
(22) Let N be a point in the unit sphere S2. Find a vector field on S2\{N}
that is not complete and one that is complete.
(23) Using the usual spherical coordinates (φ, θ) on S2, calculate the bracket
(24) Show that if X and Υ are (time independent) vector fields that have
flows ψ£ and φξ, then if [X, Y] = 0, the flow of X + Υ is φ? ο φξ.
(25) Recall that the tangent bundle of the open set GL(n,R) in Mnxn(R)
is identified with GL(n,R) χ MnXn(R). Consider the vector field on
GL(n,R) given by X : g H> (g,g2). Find the flow of X.
(26) Let
(cos t — sin t 0
sin t cos t 0
0 0 1
for t G R. Let <j)(t,P) := QtP, where Ρ is a plane in R3. Show that
this defines a flow on the Grassmann manifold G(3,2). Find the local
expression in some coordinate system of the vector field X® that gives
this flow. Do the same thing for the flow
(cos t 0 — sin t
0 10
sin t 0 cos t
and find the vector field XR. Find the bracket [XR, XQ].
(27) Develop definitions for tangent bundle and cotangent bundle for
manifolds with corners. (See Problem 21.) [Hint: A curve into an n-manifold
with corners should be considered smooth only if, when viewed in a
chart, it has an extension to a map into Rn. Similarly, a functions is
smooth at a corner (or boundary) point only if its local representative
126
2. The Tangent Structure
in some chart containing the point can be extended to an open set in
Rn.]
(28) Show that if p(x) = p(x\^..., xn) is a homogeneous polynomial, so that
for some m G Ν,
p{txu ..., txn) = tmp(xu ..., xn),
then as long as с ^ 0, the set p~l{c) is an (n — l)-dimensional subman-
ifold of Rn.
(29) Suppose that д : Μ ->· N is transverse to a submanifold W С N. For
another smooth map / : Υ —> Μ, show that / rtl g~l(N) if and only if
(gof)ft\W.
(30) Let Μ χ Ν be a product manifold. Show that for each X G X(M) there
is a vector field X G X(M x JV") that is prx-related to X and pr2-related
to the zero field on N. We call X the lift of X. Similarly, we may lift
a field on N to Μ χ N.
Chapter 3
Immersion and
Submersion
Suppose we are given a smooth map / : Μ —► N. Near a point ρ e M,
the tangent map Tpf : TpM —> TpN is a linear approximation of /. A very
important invariant of a linear map is its rank, which is the dimension of its
image. Recall that the rank of a smooth map / at ρ is defined to be the
rank of Tpf. It turns out that under certain conditions on the rank of / at
p, or near p, we can draw conclusions about the behavior of / near p. The
basic idea is that / behaves very much like Tpf. If L : V —> W is a linear
map of finite-dimensional vector spaces, then Ker L and L(V) are subspaces
(and hence submanifolds). We study the extent to which something similar
happens for smooth maps between manifolds. In this chapter we make heavy
use of some basic theorems of multivariable calculus such as the implicit and
inverse mapping theorems as well as the constant rank theorem. These can
be found in Appendix С (see Theorems C.l, C.2 and C.5). More on calculus,
including a proof of the constant rank theorem, can be found in the online
supplement to this text [Lee, Jeff].
3.1. Immersions
Definition 3.1. A map / : Μ ->· N is called an immersion at ρ Ε Μ
if Tpf : TPM -► Tf(p}N is an injection. A map / : Μ —» N is called an
immersion if it is an immersion at every ρ G M.
Note that Tpf : TpM —> Tf^N is an injection if and only if its rank
is equal to dim(M). Thus an immersion has constant rank equal to the
dimension of its domain.
127
128
3. Immersion and Submersion
Immersions of open subsets of R2 into R3 appear as surfaces that may
self-intersect, or periodically retrace themselves, or approach themselves in
various limiting ways. The map R2 —> R3 given by (it, v) н* (cos u, sin u, v) is
an immersion as is the map (u,v) \-l· (cos и sin υ, sin и sin ν,
(1 — 2 cos2 v) cos v). The map S2 —► R3 given by (x, y, z) \-> (ж, y,z — 2z3) is
also an immersion. By contrast, the map / : S2 —> R3 given by (ж,у, ζ) ^
(ж,у,0) is not an immersion at any point on the equator S2 Π {ζ = 0}.
Example 3.2. We describe an immersion of the torus T2 := S1 χ S1 into
R3. We can represent points in T2 as pairs (e**1, e**2). It is easy to see that,
for fixed a, b > 0, the following map is well-defined:
(e*l,e*2) h+ (^(егв1,е^2),у(ег01,ег02),г(е^,е%))5
where
i(e,fll,e*) = (a + 6cos0i)cos02,
y{eie\ei62) = (а + Ъсо8вх)8тв2,
г(е%$\е*в*) = Ъ8шв1.
Exercise 3.3. Show that the map of the above example is an immersion.
Give conditions on α and b that guarantee that the map is a 1-1 immersion.
Theorem 3.4. Let f : Μ —> N be α smooth map that is an immersion at
p. Then for any chart (x, U) centered at p, there is a chart (y, V) centered
at f (p) such that f (U) С V and such that the corresponding coordinate
expression for f is (ж1,..., xk) »->· (ж1,..., xk, 0,..., 0) € Rn. Here, η is the
dimension of N and к = dim(M) is the rank ofTpf.
Proof. Follows easily from Corollary C.3. D
Theorem 3.5. If f : Μ ->· N is an immersion (so an immersion at every
point), and if f is a homeomorphism onto its image f(M) (using the relative
topology on f(M)), then f(M) is a regular submanifold of N.
Proof. Let к be the dimension of Μ and let η be the dimension of N.
Clearly / is injective since it is a homeomorphism. Let f(p) Ε f{M) for a
unique p. By the previous theorem, there are charts (/7, x) with ρ e U and
(V, y) with / (p) e V such that the corresponding coordinate expression for /
is^1,...,^)^ (z\...,:rfc,0,...,0) ERn. We arrange to have f(U) С V.
But /([/) is open in the relative topology on /(M), so there is an open set
О С V in Μ such that f(U) - f(M) Π Ο. Now it is clear that (O, y|0) is a
chart with the regular submanifold property, and so since ρ was arbitrary,
we conclude that f(M) is a regular submanifold. D
If / : Μ —> N is an immersion that is a homeomorphism onto its image
(as in the theorem above), then we say that / is an embedding.
3.1. Immersions
129
Exercise 3.6. Show that every injective immersion of a compact manifold
is an embedding.
Exercise 3.7. Show that if / : Μ —> N is an immersion and ρ 6 Μ, then
there is an open U containing ρ such that f\v is an embedding.
Exercise 3.8. Recall the definition of a vector field along a map (Definition
2.68). Let X be a vector field along / : N —> Μ. Show that if / is an
embedding, then there is an open neighborhood U of f(N) and a vector
field X e X{U) such that X-Io/.
Recall that a continuous map / is said to be proper if /_1 (K) is compact
whenever К is compact.
Exercise 3.9. Show that a proper 1-1 immersion is an embedding. [Hint:
This is mainly a topological argument. You may assume (without loss of
generality) that the spaces involved are HausdorfF and second countable.
The slightly more general case of paracompact Hausdorff spaces follows.]
Definition 3.10. Let S and Μ be smooth manifolds. A smooth map / :
S -l· Μ will be called smoothly universal if for any smooth manifold TV,
a mapping g : N —> S is smooth if and only if / о д is smooth.
Definition 3.11. A weak embedding is a 1-1 immersion which is smoothly
universal.
Let / : S —> Μ be a weak embedding and let Λ be the maximal atlas
that gives the differentiable structure on S. Suppose we consider a different
differentiable structure on S given by a maximal atlas A<i. Now suppose
that / : S —ϊ Μ is also a weak embedding with respect to A2. Resorting to
seldom used pedantic notation, we are supposing that both / : (5, Λ) -> Μ
and / : (S, A2) -* Μ are weak embeddings. Prom this it is easy to show that
the identity map gives smooth maps (S, A) -» (5, A2) and (5, A2) —> (S, A).
This means that in fact A = A2, so that the smooth structure of S is
uniquely determined by the fact that / is a weak embedding.
Exercise 3.12. Show that every embedding is a weak embedding.
130
3. Immersion and Submersion
Figure 3.1. Figure eight immersions
In terms of 1-1 immersions, we have the following inclusions:
{proper embeddings} С {embeddings}
С {weak embeddings} С {1-1 immersions}.
3.2. Immersed and Weakly Embedded Submanifolds
We have already seen the definition of a regular submanifold. The more
general notion of a submanifold is supposed to realize the "subobject" in the
category of smooth manifolds and smooth maps. Submanifolds are to
manifolds what subsets are to sets in general. However, what exactly should be
the definition of a submanifold? The fact is that there is some disagreement
on this point. Prom the category-theoretic point of view it seems natural
that a submanifold of Μ should be some kind of smooth map I : S —> M.
This is not quite in line with our definition of regular submanifold, which is,
after all, a type of subset of M. There is considerable motivation to define
submanifolds in general as certain subsets; perhaps the images of certain
nice smooth maps. We shall follow this route.
Definition 3.13. Let S be a subset of a smooth manifold M. If S is a
smooth manifold such that the inclusion map ι : S —> Μ is an injective
immersion, then we call S an immersed submanifold.
Notice that in the above definition, S certainly need not have the sub-
space topology! Its topology is that induced by its own smooth structure.
The reader may rightfully wonder just how S could acquire such a smooth
structure in the first place. If / : JV -» Μ is an injective immersion, then
S := f(N) can be given a smooth structure so that it is an immersed
submanifold. Indeed, we can simply transfer the structure from N via the
bijection f : N —> f{N). However, this may not be the only possible smooth
structure on f(N) which makes it an immersed submanifold. Thus it is
imperative to specify what smooth structure is being used. Simply looking at
3.2. Immersed and Weakly Embedded Submanifolds
131
V
Figure 3.2. Immersions can approach themselves
the set is not enough. For example, in Figure 3.1 we see the same figure
eight shaped subset drawn twice, but with arrows suggesting that it is the
image of two quite different immersions which provide two quite different
smooth structures.
Suppose that S is a fc-dimensional immersed submanifold of a smooth
η-manifold M, and let ρ € S. Then using Theorem 3.4, we see that there is
a chart (O, x) on 5, and a chart (V, y) on M, with ρ e О С V, such that
yoto x_1 — у ο χ-1 : χ (Ο) -> у (V)
has the form (α1,..., ак) н-> (α1,..., α*, 0,..., 0). This means that у (О) =
у ox х(х(0)) is a relatively open subset of Rk χ {0}. Thus there is an
open subset W of у (V) С Rn such that у (О) = W Π (Rk χ {0}). Letting
U\ := y_1(W), we see that
у(С/1ПО) = у(/71)П(М/сх{0}).
Thus the chart (yl^ , U\) has the submanifold property with respect to О
(but not necessarily with respect to 5). The set О has a smooth structure as
an open submanifold of S. But this is the same smooth structure О has as
a regular submanifold. To see this note that the restrictions yl\Q ,..., yk\0
combine to give an admissible chart on 5. Indeed, using the functions we
obtain a bijection of О with an open subset of R*. We only need to show
that this bijection is smoothly related to the chart (0,x), and this amounts
to showing that y1^ ,..., yk\0 are smooth. But this follows immediately
from the fact that у о х-1 is smooth. Notice that unlike the case of a regular
submanifold, it may be that no matter how small V',
y(VnO)^y(V)n(Rkx{0}),
as indicated in Figure 3.2. So in summary, each point of an immersed
submanifold has a neighborhood that is a regular submanifold.
Proposition 3.14. Let S С М be an immersed submanifold of dimension
к and let f : N —> S be a map. Suppose that ι ο / : N —> Μ is smooth,
132
3. Immersion and Submersion
where ι : S <->· Μ is the inclusion. Then if f : JV —» S is continuous, it is
also smooth.
Proof. We wish to show that / : JV -» S is smooth if /_1(0) is open for
every set О С S that is open in the manifold topology on S. Let ρ € JV and
choose a chart (V, y) for Μ centered at ι ο /(ρ), so that
is an open neighborhood of ρ in 5 and such that y1 \v ,..., yk \v are
coordinates for S on U. By assumption /_1(i7) is open. Thus (l ο /) (/_1(ί7)) С
U. In other words, ю/ maps an open neighborhood of ρ into U. To test for
the smoothness of /, we consider the functions (y*)^) °/ on the set /_1(C/).
But
(y%) °f = y%oiof,
and these are clearly smooth by the assumption that ι ο / is smooth. D
Sometimes the previous result is stated differently (and somewhat
imprecisely): Suppose that / : N —> Μ is a smooth map with image inside an
immersed submanifold S\ then / is smooth as a map into S if it is continuous
as a map into S. The lack of a notational distinction between / as a map
into S and / as a map into Μ is what makes this way of stating things less
desirable.
Let S С Μ and suppose that S has a smooth structure. To say that an
inclusion S ^ Μ is an embedding is easily seen to be the same as saying
that S is a regular submanifold, and so we also say that S is embedded in
M.
Corollary 3.15. Suppose that S С Μ is a regular submanifold. Let f :
N —» S be a map such that ι о / : Ν —> Μ is smooth. Then f : JV —» S is
smooth.
Proof. The map ι : S «->· Μ is certainly an immersion, and so by the
previous theorem we need only check that / : N -» S is continuous. Let 0
be open in S. Then since S has the relative topology, О = U Π S for some
open set U in M. Then f~l{0) = f~l(UnS) = f-^r^U)) = (ι ο /)_1 ([/),
which is open since ι ο / is continuous. Thus / is continuous. D
Definition 3.16. Let S be a subset of a smooth manifold M. If S is a
smooth manifold such that the inclusion map ι : S —> Μ is a weak
embedding, then we say that S is a weakly embedded submanifold.
From the properties of weak embeddings we know that for any given
subset S С Μ there is at most one smooth structure on S that makes it a
weakly embedded submanifold.
3.2. Immersed and Weakly Embedded Submanifolds
133
Corresponding to each type of injective immersion considered so far we
have in their images different notions of submanifold:
{proper submanifolds} С {regular submanifolds}
С {weakly embedded submanifolds}
С {immersed submanifolds}.
We wish to further characterize the weakly embedded submanifolds.
Definition 3.17. Let S be any subset of a smooth manifold M. For any
χ Ε S, denote by CX(S) the set of all points of S that can be connected to
χ by a smooth curve with image entirely inside S.
It is important to be clear that CX(S) is not necessarily the connected
component of S with its relative topology since, for example, S could be
the image of an injective nowhere differentiable curve. In the latter case,
CX(S) = {x} for all xeSl
Definition 3.18. We say that a subset S of an η-manifold Μ has property
W(fc) if for each so € S there exists a chart (17, x) centered at so such that
x(CS0(i7 Π S)) = x(U) Π (Rk χ {0}). Here Rn = Rk χ Rn~fc.
Together, the next two propositions show that weakly embedded
submanifolds are exactly those subsets that have property W(fc) for some k.
Our proof follows that of Michor [Mich], who refers to subsets with property
W(fc), for some ft, as initial submanifolds. With Michor's terminology,
the result will be that the initial submanifolds are the same as the weakly
embedded submanifolds.
Proposition 3.19. // an injective immersion I : S —>· Μ is smoothly
universal, then the image I(S) has property W(k) where к = dim(S). In
particular·, if S С Μ is a weakly embedded submanifold of M, then it has property
W(k) where к = dim(S).
Proof. Let dim(5) — к and dim (M) = n. Choose so G S. Since / is an
immersion, we may pick a coordinate chart (W, w) for S centered at so and
a chart (V, v) for Μ centered at /(so) such that
vo/ow 1(2/) = (y,0)-(y1,...,yfc,0,...,0).
Choose an r > 0 small enough that Bk(0,2r) С v(W) and Bn(0,2r) С v(V).
Let U = v-1(Bn(Qir)) and Wx = ^(B^O.r)). Let χ := ν|^. We show
that the coordinate chart (17, x) satisfies the conditions of Definition 3.18:
x-\x(U) П (R* χ {0})) = x-^G/.O) : \\y\\ < r}
= /ow 1o(xolov 1)_1({(2/,0) : 112/11 < r-})
= /ow 1({y-b\\<r})=I(Wl).
134
3. Immersion and Submersion
Clearly I(W\) С I(S), but we also have
vo/(W1)cvo/ow_1(B*(0,r))
= Bn(0,r) П {Rk x {0}} С Bn(0,r),
so that I(Wi) С ν-^β^Ο,τ·)) = U. Thus I{WX) С Unl(S). Since /(Wi) is
smoothly contractible to I (so), every point of I(Wi) is connected to /(so) by
a smooth curve completely contained in I{W\) С Ui~]I(S). This implies that
I{WX) С C/(so)(C/n/(5)). Thus *~l(x(U) П (Κ* χ {0})) С CI{so)(Unl(S))
or
x(t/) П (Rfc χ {0}) С x(C/(so)(i/n/(5))).
Conversely, let ζ Ε Сцао)(и Π 7(5)). By definition there must be a smooth
curve с : [0,1] —> Μ starting at I (so) and ending at ζ with c([0,1]) С
U Π I(S). Since / : S —> Μ is injective and smoothly universal, there is a
unique smooth curve c\ : [0,1] —> S with Ι ο c\ = c.
Claim: ci([0,1]) С VFi. Assume not. Then there is a number t Ε [0,1]
with ci(i) € ν x({r < Цг/ll < 2r}). Therefore,
(vo J)(ci(t)) Ε (vo/ow"1)^ < ||y|| <2r})
= {(У,0) : r < \\y\\ <2r}c{zeRn:r< \\z\\ < 2r}.
This implies that (vo Joci)(i) = (voc)(i) Ε {г Ε Rn : r < ||г|| < 2r}, which
in turn implies the contradiction c(i) ^ i7. The claim is proven.
The fact that ci([0,l]) С W\ implies ci(l) = /_1(^) Ξ ^ъ and so
ζ Ε I(Wi). As a result we have C/(eo)(E/ Π I(S)) = I(W{) which together
with the first half of the proof gives the result:
J(Wi) = x-\x(U) П (Rfc χ {0})) С CI{so)(U П I(S)) = I{WX)
=* х"1^) П (Rfc χ {0})) = C/(so)(t/ П /(5))
=► x(C/) П (Rk χ {0}) = x(C/(so)(£7 П I(S))). D
Proposition 3.20. If S С Μ has property W(k)7 then there is α unique
smooth structure on S which makes it a k-dimensional weakly embedded
submanifold of M.
Proof. (Sketch) We are given that for every s Ε S, there exists a chart
(Us,xs) with xs(s) = 0 and with x(C3{Us Π S)) = x(Us) Π (Шк х {0}).
The charts on S will be the restrictions of the charts (i7s,xs) to the sets
C8(US Π 5). The overlap maps are smooth because they are restrictions of
overlap maps on Μ to subsets of the form V Π (Rk χ {0}) for V open in
Rn. If (Ε/β1,χβ1) and (i7g2,xS2) are two such charts with corresponding sets
Si := C3l(USl Π 5) and £2 ·— CS2(US2 Π 5), then we need to check that
x«i (Si Π 5χ) is open in Шк х {0} (recall the definition of smooth atlas). For
each ρ Ε U8l Π Ϊ7β2 Π 5 consider the set C(p) := Cp (i/Sl Π /7S2 Π 5). We
3.2. Immersed and Weakly Embedded Submanifolds
135
leave it to the reader to show that C(p) С SiD S2 and that if ρ Φ q then
C(p) Π C(q) = 0. Thus the sets C(p) form a partition of U8l Π US2 Π S. It is
not hard to see that each C(p) maps onto a connected path component of
Xsi (C^ei Π U82 Π S) and that every path component of this set is the image of
some C(p). But this implies that xSl (US1 Π US2 Π S) open. Notice however
that the topology induced by the smooth structure thus obtained on S is
finer than the relative topology that S inherits from M. This is because the
sets of the form C3(U Π S) are not necessarily open in the relative topology.
Since it is a finer topology, it is also Hausdorff.
It is clear that with this smooth structure on £, the inclusion l : S <->· Μ
is an injective immersion. We now show that the inclusion map t : S «->
Μ is smoothly universal and hence a weak embedding. By the comments
following Definition 3.11 the smooth structure on S is unique. Let g : N —>· S
be a map and suppose that ι ο g is smooth. Given χ 6 Μ, choose a chart
(f/s,xs) where s = g(x)· The set g~1(Us) is open since ι о д is continuous
and {tog)-1 {Ua) = g-^r^Us) = g-HUa). We may choose a chart
(V, y) centered at χ with V С <7-1(ί/β) and we may arrange that y(V) is a
ball centered at the origin. This means that iog(V) is smoothly contractible
in Ug{x) П S and hence g(V) С Cg{x)(Ug{x) Π S). But then
*s\cs{usnS) ° 9 ° У~Х = x5 ° 0- ° 9) ° У~\
and so ρ is smooth because ι ο ρ is smooth.
To be completely finished, we need to show that with the topology
induced by the atlas, each connected component of S is second countable. We
can give a quick proof, but it depends on Riemannian metrics which we have
yet to discuss. The idea is that on any paracompact smooth manifold, there
are plenty of Riemannian metrics. A choice of Riemannian metric gives a
notion of distance making every connected component a second countable
metric space. If we put such a Riemannian metric on M, then it induces
one on S (by restriction). This means that each component of S is also a
separable metric space and hence a second countable Hausdorff topological
space. D
We say that two immersions I\ : N1 —» Μ and /2 · N2 —>· Μ are
equivalent if there exists a diffeomorphism Φ : N\ —> N2 such that I2 ο Φ =
Ιχ\ i.e., such that the following diagram commutes:
iVi >N2
\ /
Μ
136
3. Immersion and Submersion
Figure 3.3. Tori converge to a point
If / : N —> Μ is a weak embedding (resp. embedding), then there is a
unique smooth structure on S = I(N) such that S is a weakly embedded
(resp. embedded) submanifold and / : N —> Μ is equivalent to the inclusion
t: S ^ Μ in the above sense.
We shall follow the convention that the word "submanifold", when used
without a qualifier such as "immersed" or "weakly embedded", is to mean
a regular submanifold unless otherwise indicated. What R. Sharpe [Shrp]
calls a "submanifold", refers to something more restrictive than the weakly
embedded submanifolds, but still less restrictive that the regular subman-
ifolds. Sharpe's definition of "submanifold" seems designed to exclude
examples like that shown in Figure 3.3. Here the tori converge to a point
on the plane (which is taken to be part of the manifold). Every
neighborhood of that point will contain an infinite number of tori. Such a behavior is
excluded by Sharpe's definition, but this is still a weakly embedded
submanifold. One can also imagine the tori flattening while only the holes converge
to a point. This example can easily be modified to be path connected and
yet, it could never be the maximal integral manifold of a tangent distribution
(see Chapter 11 for definitions).
The celebrated Whitney embedding theorem states that any second-
countable η-manifold can be embedded in a Euclidean space of dimension
2n. We do not prove the full theorem, but we will settle for the following
easier result.
Theorem 3.21. Suppose that Μ is an η-manifold that has a finite atlas,
Then there exists an injective immersion of Μ into R2n_hl. Consequently^
every compact η-manifold can be embedded into R2n+1.
Proof. Let Μ be a smooth manifold with a finite atlas. In particular, Μ
is second countable. Initially, we will settle for an immersion into RD for
3.2. Immersed and Weakly Embedded Submanifolds
137
some possibly very large dimension D. Let {Οι,φι}ίβΐ be an atlas with
cardinality N < oo. By applying Lemma B.4 twice, the cover {O;} may be
refined to two other covers {Ui}iei and {Vi}i^j such that U{ С V* С V% С О;.
Also, we may find smooth functions fi : Μ —> [0,1] with supp(/i) С О; and
such that /*(#) = 1 for all χ G U% and /г(я) < 1 for χ ^ V*. Next we write
φι = (a;J,..., xf) so that or- : O* —> Μ is the j-th coordinate function of the
г-th chart, and then form the product
Jij :— /i^j
which is defined and smooth on all of Μ after extension by zero.
Now we put the functions fi together with the functions /y to get a map
/ : Μ -> Rn+7Vn :
/ = (/l,...)/n,/lli/l2,---»/21j--->/iVn)-
Now we show that / is injective. Suppose that f(x) — /(y). Note that
Д(я) must be 1 for some к since χ e Uk for some fc. But then fk(y) = 1
also, and this means that у € Vk (why?). Since Д(я) = Д(у) = 1, it follows
that Д?(#) = fkj(y) f°r all J· Remembering how things were defined, we
see that χ and у have the same image under φ^ : Ok —> Rn and thus χ — у.
To show that Txf is injective for all χ € Μ, we fix an arbitrary such x\
then χ e Uk for some fc. But then near this x, the functions fkijk2, · · ·, /ы,
are equal to ж£,..., a?jj and so the rank of / must be at least η and in fact
equal to η since dimTxM = n.
So far we have an injective immersion of Μ into RD where D = η + Νη.
We show that there is a projection π : RD -> L С RD, where L = R2n+1
is a (2n + l)-dimensional subspace of RD such that π ο / is an injective
immersion. The proof of this will be inductive. So suppose that there is an
injective immersion f of Μ into Rd for some d with D > d > 2n + 1. We
show that there is a projection π^ : Rd ->> Ld~x = Rd_1 such that π^ ο / is
still an injective immersion. To this end, define a map h : Μ χ Μ xR ^ Rd
by h(x,y,t) := £(/(#) - /(y)). Since d > 2n + 1, Sard's theorem (Theorem
2.34) implies that there is a vector ζ 6 Rd which is neither in the image
of the map h nor in the image of the map df : Τ Μ —> Rd. This ζ cannot
be 0 since 0 is certainly in the image of both of these maps. If pv±z is
projection onto the orthogonal complement of z} then prl2 о / is injective;
for if prj_z ο /(χ) = pr_j_z о /(у), then f(x) — f(y) = az for some a € R.
But suppose χ ^ y. Since / is injective, we must have a ^ 0. This state of
affairs is impossible since it results in the equation h(x,y, 1/a) = z, which
contradicts our choice of z. Thus ρτ±ζ ο f is injective.
Next we examine Гх(рг^г о /) for an arbitrary χ £ Μ. Suppose that
Τχ{ρτ±ζ ο /)υ = 0. Then d(pr±z ° f)\xv = Q, and since prj_2 is linear, this
138
3. Immersion and Submersion
amounts to prj_2 о df\xv = 0, which gives df\xv = az for some number
a G R, and which cannot be 0 since / is assumed to be an immersion. But
then df\x ^v = z, which also contradicts our choice of z.
We conclude that pr±z о / is an injective immersion. Repeating this
process inductively we finally get a composition of projections pr : RD ->
R2n+1 such that pr о / : Μ —> R2n+1 is an injective immersion. The final
statement for compact manifolds follows from Exercise 3.6. G
3.3. Submersions
Definition 3.22. A map / : Μ —> N is called a submersion at ρ £ Μ if
Tpf : TPM -> Tf(p)N is a surjection. / : Μ -> N is called a submersion if
/ is a submersion at every ρ €. Μ.
Example 3.23. The map of the punctured space R3\{0} onto the sphere
S2 given by χ h-> χ/ \χ\ is a submersion. To see this, use any spherical
coordinates (ρ,φ,θ) on R3\{0} and the induced submanifold coordinates
(</>, Θ) on S2. Expressed with respect to these coordinates, the map becomes
(ρ,φ,θ) н-> (Φ,θ) on the domain of the spherical coordinate chart. Here we
ended up locally with a projection onto a second factor R χ 124 R2, but
this is clearly good enough to prove the point.
As in the last example, to show that a map is a submersion at some
ρ it is enough to find charts containing ρ and f(p) so that the coordinate
representative of the map is just a projection. Conversely, we have
Theorem 3.24. Let Μ be an m-manifold and N a k-manifold and let f :
Μ —> N be a smooth map that is a submersion at p. Then for any chart
(V,y) centered at f(p) there is a chart (U,x) centered at ρ with f(U) С V
such that у о / ο χ"1 is given by (ж1,.. .,£fc,... ,xm) ь-> (ж1,... ,хк) G Rfc.
Here к is both the dimension of N and the rank ofTpf.
Proof. Follows directly from Theorem C.4 of Appendix С. П
In certain contexts, submersions, especially surjective submersions, are
referred to as projections. We often denote such a map by the letter π.
Recall that if π : Μ —> N is a smooth map, then a smooth local section
of π is a smooth map σ : V —> Μ defined on an open set V and such that
π ο σ = idy. Also, we adopt the terminology that subsets of Μ of the form
ir~1(q) are called fibers of the submersion.
Proposition 3.25. If π : Μ -> N is a submersion, then it is an open map
and every point ρ Ε Μ is in the image of a smooth local section.
3.3. Submersions
139
Proof. Let ρ € Μ be arbitrary. We choose a chart (Ϊ7,χ) centered at ρ
and a chart (V,y) centered at π(ρ) such that у ο π ο χ-1 is of the form
(ж1,... ,xk,xk+1,... ,£m) «->· (я1,... , zfc), where dimM = m and dimiV =
k. By shrinking the domains if necessary, we can arrange that x(/7) has
the form Αχ Β <zRk χ Rm~k and y(V) = В с Rl. Then we may transfer
the section ц : α —ϊ (a, b) where b = x(p). More precisely, the desired local
section is σ := х-1 о^оуопА
We now use the existence of local sections to show that π is an open
map. Let О be any open set in M. To show that π(Ο) is open, we pick any
q Ε тг(О) and choose ρ e О with ρ Ε π_1(?). Now we choose a chart ({7, x)
as above with ρ G ί/ С О. Then g is in the domain of a section which is
open and contained in π(0). D
Proposition 3.26. Let π : Μ —> N be a surjective submersion. If f : N —t
Ρ is any map, then f is smooth if and only if f on is smooth:
Μ
I \ /0?г
Proof. One direction is trivial. For the other direction, assume that / ο π is
smooth. We check for smoothness of / about an arbitrary point q G N. Pick
ρ Ε 7Γ-1((/). By the previous proposition ρ is in the image of a smooth section
σ : V -> Μ. This means that / and (/ ο π) ο σ agree on a neighborhood of
g, and since the latter is smooth, we are done. D
Next suppose that we have a surjective submersion π : Μ —>· N and
consider a smooth map g : Μ —> Ρ which is constant on fibers. That is, we
assume that if pi,p2 £ K~l(q) for some q Ε Ν, then f(pi) = /(рг)· Clearly
there is a unique induced map / : N —> Ρ so that g = f οπ. By the above
proposition / must be smooth. This we record as a corollary:
Corollary 3.27. If g : Μ ->· Ρ is a smooth map which is constant on the
fibers of a surjective submersion π : Μ —> JV; then there is a unique smooth
map f : N —> Ρ suc/i £/mi ρ = / ο π.
The following technical lemma is needed later and represents one more
situation where second countability is needed.
Lemma 3.28. Suppose that Μ is a second countable smooth manifold. If
f: Μ —> N is a smooth map with constant rank that is also surjective, then
it is a submersion.
Proof. Let dimM = m, dim TV = η and rank(/) = к and choose ρ Ε
Μ. Suppose that / is not a submersion so that к < n. We can cover Μ
140
3. Immersion and Submersion
by a countable collection of charts (/7α,χα) and cover N by charts (^,уг)
such that for every a, there is an г = г(а) with / (Ua) С V\ and уг о / о
х~1(ж1,...,жп) = (xl,...,xk,0,...,Q). But this means that f (Ua) has
measure zero. However, f(M) = \Jaf (Ua) and so f(M) is also of measure
zero which contradicts the surjectivity of /. This contradiction means that
/ must be a submersion after all. D
Problems
(1) Let 0 < a < b. Show that the subset of R3 described by the equation
(V*2 + 2/2"b)
, 2 2
+ z — a
is a submanifold. Show that the resulting manifold is diffeomorphic to
S1 χ S1.
(2) Show that the map S2 ->· R3 given by (x, y, z) i-> (ж, у, ζ — 2г3) is an
immersion. Try to determine what the image of this map looks like.
(3) Show that if Μ is compact and N is connected, then a submersion
/ : Μ —>· N must be surjective.
(4) Let / : Μ -> N be an immersion.
(a) Let (17, x) be a chart for Μ with ρ 6 U, and let (V,y) be a chart
for N with f(p) eV such that
yo/ox 1(ο1>ο2,...,αη)~(α1>ο2,...,αη10ι...ι0).
Show that
д
dyi
for г — 1,... ,η.
/(ρ)
(b) Show that if / is as in part (a) and Υ e X(N) is such that Y(p) Ε
Tpf{TpM) for all p, then there is a unique X € 3t(M) such that X
is /-related to Y.
(5) Define a function s : Rn+1\{0} -> RPn by the rule that s(x) is the line
through χ and the origin. Show that s is a submersion.
(6) Show that there is a continuous map / : R2 ->> R2 such that /(£(0,1)) С
B(0,1),/(R2VB(0,1)) С/(R2\B(0,l))and/^(0,1) - idaB(0jl) and with
the properties that / is C°° on 5(0,1) and on R2\B(0,1), while / is not
C°° on R2.
(7) Construct an embedding of R χ Sn into Rn+1.
(8) Embed the Stiefel manifold of fc-frames in Rn into a Euclidean space R^
for some large N.
Problems
141
(9) Construct an embedding of G(n, k) into G(n, к + 1) for each / > 1.
(10) Show that the map / : RP2 -> R3 defined by f([x,y,z]) = (yz,xz,xy)
is an immersion at all but six points ρ Ε RP2. The image is called the
Roman surface, and nice images can be found on the web. Show that it
is a topological immersion (locally a topological embedding). Show that
the map g : RP2 -> R4 given by g([x, у, ζ]) = {yz, xz, xy, x2 + 2y2 + 3z2)
is a smooth embedding.
(11) Let h : Μ ->> Rn be smooth and let N С Rn be a regular submanifold.
Prove that for each ε > 0 there exists & ν € Rn, with \v\ < ε, such
that the map ρ ι-> h(p) + ν is transverse to N. (Think about the map
Μ χ Ν -> Rn given by (p, у) ^ у - /(ρ).)
(12) Define 0 : S1 -+ R by ег* н> б for 0 < θ < 2π. Define λ : R -+S1 by
θ ^ егв. Show that λ is an immersion, that λ ο φ is smooth, but that φ
is not differentiable (it is not even continuous).
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Chapter 4
Curves and
Hypersurfaces in
Euclidean Space
So far we have been studying manifold theory, which is foundational for
modern differential geometry. In this chapter we change direction a bit to
introduce some ideas from classical differential geometry. We do this for
pedagogical reasons. The reader will see several ideas introduced here that
will only be treated in generality later in the book. These ideas include
parallelism, covariant derivative, metric and curvature. We concentrate on the
geometry of one-dimensional submanifolds of Rn (geometric curves) and sub-
manifolds of codimension one in Rn, which we refer to as hypersurfaces.1
Recall that a vector in TpRn can be viewed as a pair (p, v) G {ρ} χ Rn, and
we sometimes write vp or (v1,..., vn)p for (p, v). The element ν is called the
principal part of vp. Recall also that in Rn we have a natural notion of what
it means for vectors in different tangent spaces to be parallel. By definition,
vp G TpRn is parallel to wq G TqRn if ν = w G Rn. For any wp G TpRn,
there is a unique vector wq G TqRn which is parallel to wp. We call wq
the parallel translate of wp to the tangent space TqRn. We often identify
TpW1 with Rn. If βχ,..., en is the standard orthonormal basis for Rn, then
?i,..., e"n will denote the standard global frame field on Rn defined by
^t(p) :~ g|r | for г = 1,..., n, where x1,..., xn are the standard coordinate
functions on Rn. The reason for this separate notation is to emphasize the
fiducial role of this frame field. If vp G TpRn and wq G TgRn, then vp is
1 Unless otherwise stated, we will assume that a hypersurface is a manifold without boundary.
143
144
4. Curves and Hypersurfaces in Euclidean Space
parallel to wq if vl — wl for all г, where
Vp = Σ ^(P) a]Qd ω9 = Χ] ™^(?)·
Recall that if с : / —> Μ is a smooth curve into an η-manifold M, then
a smooth vector field along с is a smooth map У : / —> TM such that
π ο У = с. The velocity of the map У is an element of Τ (ΤΜ) rather
than TM. On the other hand, if с : J -> Rn, then the special structure of
Rn allows us to take a derivative of a vector field along с and end up with
another vector field along c. If Υ is such a vector field along c, then it can
be written У = ^ Уг ^ о с for some smooth functions Y% : / —> R. In other
words, Y(t) = 53y*(t)ei(c(t)) f°r * € J- F°r such a field, we define
In notation that keeps track of base points, we can write Υ = (У1,..., Yn)c
so that Y(t) = (У1^)» · · · >^п(<))с(<)> and then
£<'>-&"<'> ^»(f,)i(1).
Since ^ is obviously also a vector field along c, we can repeat the process
to obtain higher derivatives ^- = Y^k\ In particular, we have the velocity
c', acceleration c", and higher derivatives c^k\ Except for the current
emphasis on the base point of vectors, these definitions are the usual definitions
from multivariable calculus.
Another thing that is special about Rn is the availability of the natural
inner product (the dot product) in every tangent space TpRn. This inner
product on ТрШ.п is denoted (·, ■) and is given by
(vP,Wp) i-> (vp,wp) = У%ггиг,
where vp = (p,v) and wp — (p,w) as explained above. This gives what is
called a Riemannian metric on Rn, and it is obtained from the canonical
identification of the tangent spaces with Rn itself. We study Riemannian
metrics on general manifolds later in the book. For smooth vector fields on
Rn, say X and У, the function (X, Y) defined by ρ ь-> (Xp, Yp) is smooth. If
X and У are fields along a curve с : / —> Rn, then (X, У) is a function on /
and we clearly have
|№<£,,) + <*,£).
Remark 4.1. We shall sometimes leave out the subscript ρ in the notation
vp if the base point is understood from the context. In fact, we often just
identify vp with its principal part ν € Rn.
4.1. Curves
145
An ordered basis v\,..., vn is positively oriented if det(t>i,..., vn) is
positive. For vp,wp Ε TPR3, we can obviously define a cross product vp χ
wp := (ν χ w) , which results in a vector based at ρ which is then orthogonal
to both vp and wp. If vp and wp are orthonormal, then (vp,wp) (ν χ w) )
is a positively oriented (right-handed) orthonormal basis for TPR3. The
following lemma allows us to do something similar in higher dimensions.
Lemma 4.2. Ifv\y..., νη ι € Rn, then there is a unique vector N(vi,...,
νη-χ) such that
(i) N(vi,..., vn-i) is orthogonal to each of v\,..., vn_i.
(ii) If {v\,... jVn-i} is an orthonormal set of vectors, then the list
(vi,..., υη_ι, iV) is α positively oriented orthonormal basis.
(Hi) iV(vi,..., vn_i) depends smoothly onvi,..., vn-i-
Proof. Let L be the linear functional defined by L(v):— det(vi,..., vn_i, v).
There is a unique iV 6 Rn such that L(v) = (N,v). Now (i), (ii) and (iii)
follow from the properties of the determinant. G
4.1. Curves
If С is a one-dimensional submanifold of Rn and ρ € С, then there is a chart
(V, y) of С containing ρ such that у (V) is a connected open interval I c R.
The inverse map y_1 : / —> V С Μ is a local parametrization. Thus
for local properties, we are reduced to studying curves into Rn which are
embeddings of intervals. We can be even more general and study immersions.
The idea is to extract information that is appropriately independent of the
parametrization. If 7 : I —> Rn and с : J —> Rn are curves with the same
image, then we say that с is a positive reparametrization of 7 if there is a
smooth function h: J —> I with Ы > 0 such that с— η oh. In this case, we
say that 7 and с have the same sense and provide the same orientation
on the image. We assume that 7 : I —> W1 has ||7'|| > 0, which is the case
of interest. Such a curve is called regular, which just means that the curve
is an immersion.
Definition 4.3. If 7 : I -> W1 is a regular curve, then T(t) := i(t)/ \\i{t)\\
defines the unit tangent field along 7. (Of course, we then have ||T|| = 1.)
We have the familiar notion of the length of a curve defined on a closed
interval 7: [tut2] -> Rn:
l= Γ||Υ(«)||λ·
Jti
146
4. Curves and Hypersurfaces in Euclidean Space
One can define an arc length function for a curve 7
to € / and, then defining
by choosing
s = h(t)
■■- f \W
J to
(r)\\dr.
Notice that s takes on negative values if t < to, so it does not always
represent the length in the ordinary sense. If the curve is smooth and regular,
then Ы — ||7;(τ)|| > 0, and so by the inverse function theorem, h has a
smooth inverse. We then have the familiar fact that if c(s) =70 h~l(s),
then ||c'|| (s) :— ||c'(s)|| = 1 for all s. Curves which are parametrized in
terms of arc length are referred to as unit speed curves. For a unit speed
curve, g!(s) = T(s). Since parametrization by arc length eliminates any
component of acceleration in the direction of the curve, the acceleration
must be due only to the shape of the curve.
Definition 4.4. Let с : I
function
Rn be a unit speed curve. The vector-valued
φ) := gw
is called the curvature vector. The function к defined by
k(s):=Ms)\\ =
dT
is called the curvature function. If k(s) > 0, then we also define the
principal normal
N(s) :=
dT
ds
(s)
-1
dT
so that ^ = kN.
Let 7 : 7 —> Rn be a regular curve. An adapted orthonormal moving
frame along 7 is a list (Εχ,... ,En) of smooth vector fields along 7 such
that Ei(i) = 7/(*)/11УМН and such that (Ei(t),..-,En(t)) is a basis of
T7(i)Mn for each t G I. Identifying Ei(i),... ,En(t) with elements of Rn
written as column vectors, we say that the orthonormal moving frame is
positively oriented if
Q(t) = [Ei(t),...,E„(t)]
is an orthogonal matrix of determinant one for each t.
Definition 4.5. A moving frame Εχ(ί),..., Έη(ί) along a curve 7 : I —> Rn
is a Frenet frame for 7 if 7M (t) is in the span of Εχ (i),..., Ε*(ί) for all t
and 1 < к < п.
As we have defined them, Frenet frames are not unique. However, under
certain circumstances we may single out special Frenet frames. For example,
4.1. Curves
147
0
к
0
—к
0
τ
0
—τ
0
if с : I -4 R3 is a unit speed curve with /s > 0, then the principal normal N
is defined. By letting
B = TxN
we obtain a Frenet frame Τ,Ν,Β. It is an easy exercise to show that we
obtain
fs = «N
f = -kT + rB
for some function r called the torsion. This is the familiar form presented
in many calculus texts. In matrix notation,
^[T,N,B]=[T,N,B]
Notice that for a regular curve, к > 0, while r may assume any real value.
Another special feature of the frame Τ, Ν, Β is that it is positively oriented.
If 7 is injective, then we can think of к and r as defined on the geometric
image 7(J). Thus, if ρ — 7(50), then κ(ρ) is defined to be equal to k(so).
Exercise 4.6. Let с : / —У R3 be a unit speed regular curve with к > 0.
Show that
((c-Wx^))y»W)fora|Ue/
Exercise 4.7. Let с : / —> R3 be a unit speed curve and so £ /■ Show that
we have a Taylor expansion of the form
7(5) - 7(50) = Hs - s0) - l(s - s0) V(s0) J T(*o)
+ Шв - 80)3фо)тЫ J B(s) + o((s - 50)3).
We wish to generalize the special properties of the Frenet frame Τ, Ν, Β
to higher dimensions thereby obtaining a notion of a distinguished Frenet
frame. For maximum generality, we do not assume that the curve is unit
speed. A curve 7 in Rn is called fc-regular if {jf(t)^/f(t),... ,7^(0} *s a
linearly independent set for each t in the domain of the curve. For an (n — 1)-
regular curve, the existence of a special orthonormal moving frame can be
easily proved. One applies the Gram-Schmidt process: If Εχ(£),..., Efc(£)
148
4. Curves and Hypersurfaces in Euclidean Space
are already defined for some к < η — 1, then
Efc+X(t) :=ck
7(fc+1)(0-E(7(/c+1)(0?E,(0)E,W
J=l
where Cfc is a positive constant chosen so that ||Efc+i(t)|| = 1. Inductively,
this gives us Ei(i),..., Εη_χ(ί), and it is clear that the Efc(t) are all smooth.
Now we choose En(i) to complete our frame by letting it be of unit length
and orthogonal to Ei(t),..., En_i(i). By making one possible adjustment
of sign on En(t) we obtain a moving frame that is positively oriented. In
fact, En(t) is given by the construction of Lemma 4.2, from which it follows
that En(t) is smooth in t. By construction we have a nice list of properties:
(1) For 1 < к < η, the vectors Ei(t),.. .,Efc(t) have the same linear
span as Y(t),... ,7^(0 so that there is an η χ η upper triangular
matrix function U(t) such that
W(t),...,^Ht)]U(t) = [E1(t),...,En(t)].
(2) For 1 < к < η — 1, the vectors Εχ(£),... ,Ε^(ί) have the same
orientation as 7'(£)> ■ · · j7^(*)· Thus J7(t) has diagonal elements
which are all positive except possibly the last one.
(3) (Ei(i),..., En(f)) is positively oriented as a basis of Tl{t)Rn Ξ Rn.
Exercise 4.8. Show that the moving frame we have constructed is the
unique one with these properties.
We call a moving frame satisfying the above properties a distinguished
Frenet frame along 7. For any orthonormal moving frame, the derivative
of each Ε»(ί) is certainly expressible as a linear combination of the basis
Ei(t),..., En(t), and so we may write
Of course, Uij(t) = (Ej(i),j£Ej(t)), but since (Εί(ί),Ε^(ί)) = iy, we
conclude that (Jij(t) = —Uji(t), i.e., the matrix u(t) = [^ij(^)] is
antisymmetric. However, for a distinguished Frenet frame, more is true. Indeed, if
(Ei(t),..., En(t)) is such a distinguished Frenet frame, then for 1 < j < η
we have Ej(t) — Σ£=ι ЭДу7^(*)> where U(t) = [Ukj(t)] is the upper
triangular matrix mentioned above. Using the fact that 17, gjt/, and f7_1 are all
upper triangular, we have
k=l ^ ' fc 1
4.1. Curves
149
But 7(fc+1)(0 = ЙЙ (U~\k+1 Br(i), and 7(fc)(<) = Σ"=ι (^"1)rfcEr(t)
so that
dt
^(t) = Σ (!"«) 7(Ч(*) + Σ ^7(fe+1)(i)
k=l ^ ' k=l
fe=l V" ' r=l
J fe+1
k=l r=l
From this we see that jgEj(t) is in the span of (Er(i))1<r<-+1. Thus
u)(t) = (cjy(t)) can have no nonzero entries below the subdiagonal. But
ω is antisymmetric, so we conclude that ω has the form
w(t) =
0 -ω2ι(ί) Ο
ω2ι(ί) Ο -ω32(ί)
0 ω32(ί) О
0 -ω„,„_ι(ί)
ωη,η-ι(<) 0
We define the г-th generalized curvature function by
г(} ·" IIVWII ·
Thus if 7 : / —> Mn is a unit speed curve, we have
0 -ki(s) 0 ··· 0
«1(5) 0 —«2(5)
/c2(5) 0
w(s) =
0 -κη-ι(«)
«n-i(5) 0
Note: Our matrix ω is the transpose of the ω presented in some other
expositions. The source of the difference is that we write a basis as a formal
row matrix of vectors.
Lemma 4.9. If η : I —> W1 (n > 3) is (n — 1)-regular, then for 1 < г < n—2,
the generalized curvatures K\ are positive.
150
4. Curves and Hypersurfaces in Euclidean Space
Proof. By construction, for 1 < г < η — 1
г
Ег(*) = Ес/л(*)7Ы(*)>
7(<)(*) = Ете1(№(*),
with 11гг > 0, and hence {U_1) . > 0. Thus if 1 < г < η — 2, we have
= ϋ"„ (Ει+1(ί),7(ί+1)(ΐ)) = ^ (^_1)i+lii+1 > 0.
In passing from the second to the third line above, we have used the fact
that Ег+1 is orthogonal to all 7^) for j < г since these are in the span of
{ЕЛ,=1 .· Π
The last generalized curvature function κη-\ is sometimes called the
torsion. It may take on negative values.
Exercise 4.10. If 7 : / —>· Rn is (n — l)-regular, show that Εχ = Τ and
E2 = N. If 7 is parametrized by arc length, then ^7 — Ei and ^7 = K1E2.
Conclude that κ\ κ (the curvature defined earlier).
The orthogonal group 0(Rn) is the group of linear transformations A :
Rn -> Rn such that {Av, Aw) = (v,w) for all v,w e Rn. The group 0(Rn)
is identified with the group of orthogonal η χ η matrices denoted O(n).
The Euclidean group Euc(Rn) is generated by translations and elements of
0(Rn). Every element φ € Euc(Rn) can be represented by a pair (A, 6),
where A G 0(Rn) and b G Rn and where φ(ν) = Αν + b. Note that in this
case, Όφ — A (the derivative of φ is A). The elements of the Euclidean group
are called Euclidean motions or isometries of Rn. If φ G Euc(Rn), then for
each ρ G Rn, the tangent map Τρφ : TpRn —> Тф^Жп is a linear isometry. In
other words, (Τρφ · νρ,Τρφ · wp),( ч — {vp,wp) for all vp:wp G TpRn. The
group SO(Rn) is the special linear group on Rn and consists of the elements
of 0(Rn) which preserve orientation. The corresponding matrix group is
SO(n) and is the subgroup of O(n) consisting of elements of determinant 1.
The subgroup 5Euc(Rn) С Euc(Rn) is the group generated by translations
and elements of SO(Rn). It is called the special Euclidean group.
4.1. Curves
151
Theorem 4.11. Let 7 : J -> Rn and 7 : I ->> Rn be two (n - l)-regular
curves with corresponding curvature functions Ki and Hi (1 < i <n- 1). If
VWII l7;(*)ll and Ki(t) = «i(*) for all t € I and 1 < г < η — 1, iften
iftere exists a unique isometry φ G 5Euc(Rn) such that
7 _ φ ο 7«
Proof. Let (Ei(t),...,En(t)) and (Ei(t),... ,En(£)) be the distinguished
Prenet frames for 7 and 7 respectively and let ωι>7· and ωτ3 be the
corresponding matrix elements as above. Fix to £ I and consider the unique
isometry φ represented by (A, 6) such that ^(7(^0)) = 7(<o) and such that
A(Ei(io))=Ei(to) for 1<г<п.
Since HVit)!! = ||7;(*)ΙΙ an(^ Ki(t) = /?t(*)» we ^ave *hat cjy(t) = 2y(t) for all
г, j and f. Thus we have both
d ~ n
and
d n
dtAE£(t) = 5]a;ji(t)AEJ(t).
Hence Ег and AE^ satisfy the same linear differential equation, and since
j4(Ei(i0)) - Ё»(*о), we conclude that A(Ei(t)) = %(t) for all t and 1 < г <
п. In particular, A7'(£) - ||7'(*)|| AEi(t) = W(t)\\ Ei(t) = η^ί). Thus
0(7(i))-0(7(«o))= / (0o7);(r)dr= / Z?0-y(r)dr
- / A7/(r)rfr= / 7'(r)dr = 7(i)-7(io),
J to Jto
from which we conclude that Φ(ί(Ϊ)) — 7(ί).
For uniqueness, we argue as follows. Suppose that φ о 7 = 7 for φ Ε
Euc(Rn) and suppose that ^ is represented by (5, c). The fact that Όφ must
take the Frenet frame of 7 to that of 7 means that A = Όφ — Ό φ — В.
The fact that φ(^(ίο)) — 7(^0) implies that b с and so φ φ. D
Conversely, we have
Theorem 4.12. // «χ,..., κη-ι are smooth functions on a neighborhood of
so Ε R such that кг > 0 fori < n — 1, then there exists an (n —1)-regular unit
speed curve 7 defined on some interval containing so such that «χ,..., κη χ
are the curvature functions of η.
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4. Curves and Hypersurfaces in Euclidean Space
Proof. We merely sketch the proof: Let
0 -κι Ο
A(s) :=
«ι Ο
0
0
K>n—1
о
: 0
0 ■·· 0 κη_ι
and consider the matrix initial value problem
X' = X A,
X(s0) = /.
This has a unique smooth solution X on some interval / which contains so.
The skew-symmetry of A implies that X{s) is orthogonal for all s € /. If
we let xi be the first column of X, then
7(e)
:= ΓΧι(ί)
•/so
dt
defines a unit speed (n — l)-regular curve with the required curvature
functions. D
Exercise 4.13. Fill in the details of the previous proof.
If η > 3, then a regular curve 7 need not have a Prenet frame. However,
a regular curve still has a curvature function, and if the curve is 2-regular,
then we have a principal normal N which is defined so that Τ and N
are the Gram-Schmidt orthogonalization of 7' and 7". We will sometimes
denote this principal normal by E2 in order to avoid confusion with the
normal to a hypersurface, which is denoted below by N.
4.2. Hypersurfaces
Suppose that Υ is vector field on an open set in Rn. For ρ in the domain of
Y, and vp e TpRn, let с be a curve with c(0) = ρ and c(0) = vp. For any t
near 0, we can look at the value of Υ at c(i). We let VVY := (Y о с)' (0),
which is defined since У о с is a vector field along c. Note that in this context
(Y о с)1 (0) is taken to be based at ρ = c(0). If X is a vector field, then a
vector field S/χΥ is given by VxY : ρ Μ· VxpY. In fact, it is easy to see
that if X = Σ X% and Υ = £ Yleu then
since (ΧΥτ) (ρ) = ΧρΥ1 — 2j |0 Yl о с. We have presented things as we have
because we wish to prime the reader for the general concept of a covariant
derivative that we will meet in later chapters. However, it must be confessed
4.2. Hypersurfaces
153
that under the canonical identification of Rn with each tangent space, VxpY
is just the directional derivative of Υ in the direction Xp.
The map (X,Y) H> VXY is C°°(Rn)-linear in X but not in Y. Rather,
it is R-linear in Υ and we have a product rule:
VxfY = (Xf) Υ + fVxY.
Because of these properties, the operator S/χ : Υ ι->· VxF, which is given
for any X, is called a covariant derivative (or Koszul connection). In
Chapter 12 we study covariant derivatives in a more general context. Since
we shall soon consider covariant derivatives on submanifolds, let us refer to
V as the ambient covariant derivative. Notice that we have VxY — VyX —
[X,Y].
There is another property that our ambient covariant derivative V
satisfies. Namely, it respects the metric:
Χ(Υ,Ζ) = (ψχΥ,Ζ) + (Υ^χΖ).
Similarly, if vp e TpRn, then vp (У, Z) = (VVpY, Zp) + (YpyVpZ).
Consider a hypersurface Μ in Rn. By definition, Μ is a regular (n - 1)-
dimensional submanifold of Rn. (If η = 3, then such a submanifold has
dimension two and we also just refer to it as a surface in R3.) A vector field
along an open set О С Μ is a map X : О —> ТШп such that the following
diagram commutes:
TRn
У\
О c ^ Rn
Here the horizontal map is inclusion of О into Rn. If X{p) € TpM for all
ρ Ε Ο, then X is nothing more than a tangent vector field on O. If JV is
a field along О such that (N(p),vp) = 0 for all vp e TPM and all peO,
then we call JV a (smooth) normal field. If JV is a normal field such that
(N(p)) N(p)) — 1 for all ρ e O, then N is called a unit normal field. Note
that because of examples such as embedded Mobius bands in R3, it is not
always the case that there exists a globally defined smooth unit normal field.
Definition 4.14. A hypersurface Μ in Rn is called orientable if there
exists a smooth global unit normal vector field N defined along M. We say
that Μ is oriented by N.
We will come to a more general and sophisticated notion of orientable
manifold later. That definition will be consistent with the one above.
Exercise 4.15. Show that for a connected orientable hypersurface, there
are exactly two choices of unit normal vector field.
154
4. Curves and Hypersurfaces in Euclidean Space
In this chapter, we study mainly local geometry. We focus attention
near a point ρ G M. We consider a chart (0,u) for Rn that is a single-slice
chart centered at ρ and adapted to M. Thus if u — (21,..., un), then the
restrictions of the functions u1,... , un_1 give coordinates for Μ on the set
О — Ο Π Μ = {ип = 0}. We denote these restrictions by it1,... ,un l.
Thus if we write u := (it1,..., it71-1), then (O, u) is a chart on M. We may
further arrange that u(O) is a cube centered at the origin in Rn and so,
in particular, О is connected and orientable. Let us temporarily call such
charts special. The coordinate vector fields -^ for г = 1,..., η are defined
on O, while the vector fields -^ are defined on O. We have
jfijiip) = д^&) for all ρ € О and i = 1,..., η - 1.
If X is a vector field on an open set O, then its restriction to О — Ο Μ is
a vector field along O, which certainly need not be tangent to M. If X is
a vector field along O, then there must be smooth functions Хг on О such
that
χ(Ρ) = Σχί(ρ)^{ρ)
г 1
for all ρ G O. Then X is a tangent vector field on О precisely when the last
component Xn is identically zero on О so that
П 1 гч
^ dul
г 1
Now if X is a field a/ong M, then it can be extended to a field X on О С Мп
by considering the component functions Хг as functions on О which happen
to be constant with respect to the last coordinate variable un. In other
words, if π : О -> О is the map (a1,..., an) н-> (a1,..., an_1,0), then
^ диг
г 1
where Xх = Xlou l οπο5. This last composition makes good sense because
(0,u) is a special single-slice chart as described above. We will refer to X
as an extension of X, but note that the extension is based on a particular
special choice of single-slice chart. The constructions on the hypersurface
that we consider below do not depend on the extension.
Given a choice of unit normal JV along a neighborhood of ρ Ε Μ, VUpiV
is defined for any vp G TPM by virtue of the fact that N only needs to be
defined along a curve with tangent vp. Alternatively, we can define VVpN
to be equal to Vv N for an extension TV of TV in a special single-slice chart
4.2. Hypersurfaces
155
Figure 4.1. Shape operator
adapted to Μ and containing p. We note that (Vv iV, N(p)) = 0. Indeed,
since (N,N) 1, we have
Thus ν^ΛΓ G TPM since TPM is exactly the set of vectors in TpRn
perpendicular to Np = N(p).
Exercise 4.16. Suppose we are merely given a unit normal vector at p.
Show that we can extend it to a smooth normal field near p. For dimensional
reasons, any two such extensions must agree on some neighborhood of p.
Definition 4.17. Given a choice of unit normal Np at ρ G M, the map
SNp : TPM -> TPM defined by
SNp(vP):=-VvpN
for any local unit normal field N with Np = N(p), is called the shape
operator or Weingarten map at p.
Prom the definitions it follows that if с is a curve with c(<o) = Ρ and
c(io) = vp, then Sjsjp(vp) = — (N о с)' (to). If О С М is an open set oriented
by a choice of unit normal field JV along O, then we obtain a map Sn '
TO -l· TO by Sn\t m := Snp- This map is also called the shape operator
(onO).
We have the inner product (·, ■) on each tangent space TpRn and TpM С
TpRn. For each ρ G M, the restriction of this inner product to each tangent
space TPM is also denoted by (·, ·) or just (·,·). We also denote this inner
product on TpM by gp so that gp{-y ■) = (·, ■) . The map ρ ь-> др is smooth
in the sense that ρ \-> gp(X(p),Y(p)) is smooth whenever X and У are
smooth vector fields on M. Such a smooth assignment of inner product to
the tangent spaces of Μ provides a Riemannian metric on M, a concept
156
4. Curves and Hypersurfaces in Euclidean Space
studied in more generality in later chapters. We denote the function ρ Η·
gp (X(p),Y(p)) simply by g (X, Y) or (X, У). In short, a Riemannian metric
is a smooth assignment of an inner product to each tangent space.
Definition 4.18. A diffeomorphism / : M\ —> Μ<ι between hypersurfaces
in R71 is called an isometry if Tpf : TpM\ -> Tf^M2 is an isometry of inner
product spaces for all ρ G M\. In this case, we say that M\ is isometric to
M2.
Proposition 4.19. Snp · TPM —> TPM is self-adjoint with respect to {·, ·) .
Proof. Let (0,y) be a special chart centered at ρ and let Ο = Ο Π Μ as
above. Let Xp and Yp be elements of TPM and extend them to vector fields
X and Υ on O. Then extend X and Υ to fields X and Υ on O. Similarly,
extend Np to N and then N. Note that Xp(N,X) = Xp (N,X) = 0 and
Yp(N}Y) = 0. Using this, we have
- (SnpXp, Yp) + (Xp, SnpYp)
= (VxpN,Yp) ~ (ΧΡ'VypN) = (VxAT,y)p- (X,VYN)p
= XP(N, Ϋ) - (iV, ψχΫ)ρ - Yp(Nt X) + (AT, ΨΫX)p
= (VYX-VxY,N)p = ([Yj],N^
= <[У,Х](р), JV(p)> = ([Y,X](p) ,N(p)) = 0
since [Γ,Χ](ρ) = [У, X](p) Ε ΤρΜ by Proposition 2.84 applied to the
inclusion map. D
Definition 4.20. The symmetric bilinear form IIp on TPM defined by
IIp{vp,wp) := (vp,SNpwp) = (SNpvP)wp)
is called the second fundamental form at p. If N is a unit normal
on an open subset of M, then for smooth tangent vector fields Χ, Υ, the
function II(X,Y) defined Ьури ΙΙρ(Χρ,Υρ) is smooth. The assignment
(X, Y) h> //(X, У) defined on pairs of tangent vector fields is also called
the second fundamental form.
The form II is bilinear over C°°(0), where О is the domain of N. As we
shall see, the shape operator can be recovered from the second fundamental
form i7 together with the metric g = (·, ·) (first fundamental form). If the
reader keeps the definition in mind, he or she will recognize that the second
fundamental form appears implicitly in much that follows. We return to the
second fundamental form again explicitly later.
Exercise 4.21. Let Xp € TpRn and let У be any smooth vector field
on Rn. Show that if / : Rn -> Rn is a Euclidean motion, then we have
4.2. Hypersurfaces
157
Tf-VxpY = Vr/.χ ДУ. More generally, show that this is true if / is affine
(i.e. if / is of the form f(x) — Ax + b for some linear map A and b G Rn).
Theorem 4.22. If M\ is α connected hypersurface in Rn and iff : Rn —> Rn
is a Euclidean motion, then M2 — /(Mi) is aZso a hypersurface and
(i) ί/ie induced map /|M : Μχ —> M2 is an isometry;
(ii) г/Μχ and M2 are oriented by unit normals N1 and N2 respectively,
then, after replacing N\ or N2 by its negative if necessary, we have
TfoSNl=SN2oTf.
Proof. That /(Mi) is a hypersurface is an easy exercise, which we leave to
the reader. After noticing that f\Ml : Mi —> M2 is smooth (why?), we argue
as follows: Let φ := f\M and notice that Τφ · ν = Τ/ · г? for all v tangent
to Mi. Since Tf preserves the inner products on TR71, we see that Τρφ is an
isometry for each ρ e M\. Since φ is clearly a bijection we conclude that (i)
holds.
Note that we must have Tpf-Ni(p) = ±N2(p) and, since Mi is connected,
one possible change of sign on unit normals gives
TfoNxof-l-N2>
Then by Exercise 4.21
Tf *SNlv = -Tf · V„7Vi - -Vrf-vftN! = -VTf.vN2 = SN2 (Tf · υ),
or Tf о sNl - sN2 о Tf. a
Notice that an arbitrary isometry Mi —> M2 need not be the restriction
of an isometry of the ambient Rn and need not preserve shape operators.
Definition 4.23. Let Μ be a hypersurface in Rn, take a point ρ G M,
and let Np be a unit normal at p. The mean curvature H(p) at ρ in the
direction Np is defined by
H(p) := trace{SN ).
η — 1
Notice that changing Np to — Np changes H(p) to —Я(р). If JV is a
unit normal field along an open set U С Μ, then the function рн Η (ρ) is
smooth. It is called a mean curvature function. If Μ is an orientable
hypersurface, then it has a global mean curvature function for each unit
normal field.
Definition 4.24. Let Μ be a hypersurface in Rn. Let a unit normal Np be
given at p. The Gauss curvature K(p) at ρ is defined by
K(p):=det(SNp).
158
4. Curves and Hypersurfaces in Euclidean Space
If TV is a unit normal field along an open set U С М, then the function
ρ *-* K(p) det(SN(p)) is smooth. It is called a Gauss curvature function
associated to the normal field, and always exists locally. If Μ is orientable,
then it has a global Gauss curvature function for every choice of unit normal
field. Notice that changing Np to —Np changes K(p) to (—l)n~ K(p). It
follows that if η — 1 is even, then there is a unique global Gauss curvature
function regardless of whether Μ is orientable or not. In particular, this is
the case for surfaces in M3.
The shape operator Sjyp encodes the local geometry of the submanifold
at ρ and measures the way Μ bends and twists through the ambient
Euclidean space. Since Sn is self-adjoint, there is a basis for TPM consisting
of eigenvectors of Sn . An eigenvector for Snp is called a principal vector,
and a unit principal vector is called a principal direction or a direction
of curvature. The eigenvalues are called principal curvatures at p. If
&i,..., kn-i are the principal curvatures at p, then
1 n—1 Ti—1
H(p) -Vi, and K(p) = T\ki.
η 1 f—' A-^
г 1 г 1
If и G TPM is a tangent vector with (щи) = 1, then k(u) = (Snpu,u) is
the normal curvature in the direction u. Of course, if и is a unit length
eigenvector (principal direction), then the corresponding principal curvature
к is just the normal curvature in that direction. Notice that k(—u) — k(u),
A vector ν G TPM is called asymptotic if (Sjsipv, v) — 0.
Proposition 4.25. Let Μ be α hypersurface and N a unit normal field. Let
7 : 7 —> Μ С К77, be a regular curve with image in the domain of N. Then
(SNpi(t)n'(t)) <N(7(t)),V(*)>.
Proof. Since {N(7(i)),7'(£)) — 0, differentiation gives
<ΛΓ(7(ί)),7"(ί)> + (jtNb(t)), jtlit)) = О-
Thus
<^p7/(i)!V(i)> = <-Vy(i)iV,7'(i)>
= (-|^(7(i)),^W> - <JV(7(i)),7"(i)>· □
In particular, if и is a unit vector at ρ and и = c(0) for some unit speed
curve c, then k(u) = (Np, c"(0)). This shows that all unit speed curves with
a given velocity и have the same normal component given by the normal
curvature in that direction. This curvature is forced by the shape of Μ and
we see how normal curvatures measure the shape of M.
4.2. Hypersurfaces
159
Corollary 4.26. Let с : / —> Μ С Rn be α unit speed curve. If k(sq) — 0
for some so € I, then k(c(so)) = 0. If k(s) > 0 and E2 is the principal
normal defined near s, then k(c(s)) — k(s)cos9(s), where 9(s) is the angle
between N(c(s)) and^is)-
Proof. Suppose /c(0) = 0. Then k(cf(0)) = (iV(c(0)),c"(0)) = 0. If «(e) >
0, then E2 is defined for an interval containing s. We have
k(c'(s)) = (N(c(s)),c"(s)) = (N(c(s)),kE2(c(s)))
= /φ) (JV(c(s)),E2(e)> = /c(e)cos0(s). D
If Ρ is a 2-plane containing ρ € Μ and such that JVp is tangent to P,
then for a small enough open neighborhood О of p, the set С = Ο Π Ρ Π Μ
is a regular one-dimensional submanifold of Rn. We parameterize С by a
unit speed curve с with c(0) = p. This curve с is called a normal section
at p. Notice that in this case £2(0) — ±Np. Then fc(c(0)) ±«i(0) where
the " " sign is chosen in case Np — —Е2(0). Thus we see that к(с(0)) is
positive if the normal section с bends away from Np.
Definition 4.27 (Curve types). Let Μ be a hypersurface in Rn and let
7 : / —> Μ be a regular curve.
(i) 7 is called a geodesic if the acceleration 7"(i) is normal to Μ for
all t Ε /.
(ii) 7 is called a principal curve (or line of curvature) if 7(i) is a
principal vector for all t Ε I.
(iii) 7 is called an asymptotic curve if ^(t) is an asymptotic vector
for all t e I.
Proposition 4.28. Let Μ be a surface in K3. Suppose that a regular curve
7 :1 —> Μ is contained in the intersection of Μ and a plane P. If the angle
between Μ and Ρ is constant along 7, then η is a principal curve.
Proof. The result is local, and so we assume that Μ is oriented by a unit
normal N. Let ν be a unit normal along P. Since Ρ is a plane, ν is constant.
By assumption, (TV, v) is constant along 7. Thus
0 = — (Ν ο 7, ν ο 7) = {V^iV, 1/07),
so V^iV is orthogonal to ν along 7. By the same token, V^JV is orthogonal
to N since (JV, N) = 1. Thus VjN must be collinear with 7. In other words,
Sjv7 — V-yiV = λ7 for some scalar function λ. D
160
4. Curves and Hypersurfaces in Euclidean Space
Example 4.29 (Surface of revolution). Let t \-> (g(t),h(t)) be a regular
curve in R2 defined on an open interval I. Assume that h > 0. Call this
curve the profile curve. Define χ : / χ R —> R3 by
x(u>v) = (g(u),h(u) cost;, h(u) sin v).
This is periodic in ν and its image is a surface. The curves of constant и
and curves of constant ν are contained in planes of the form {x — c} and
{z = my} (or {y = mz}) and so they are principal curves.
For a surface of revolution, the circles generated by rotating a fixed point
of the profile are called parallels. These are the constant и curves in the
example above. The curves that are copies of the profile curve are called the
meridians. In the example above, the meridians are the constant ν curves.
We know from standard linear algebra that principal directions at a
point in a hypersurface corresponding to distinct principal curvatures are
orthogonal. Generically, each eigenspace will be one-dimensional, but in
general they may be of higher dimension. In fact, if Sn is a multiple of the
identity operator, then there is only one eigenvalue and the eigenspace is all
of TPM. In this case, every direction is a principal direction. It may even
be the case that Sn = 0.
If с : I —> Μ is a unit speed curve into a hypersurface, then rather
than use the distinguished Frenet frame of c, we can use a frame which
incorporates the unit normal to the hypersurface:
Definition 4.30. Let с : I -¥ Μ be a unit speed curve into a surface in R3
and let N be a unit normal defined at least on an open set containing the
image of c. A frame field (Di,U2,D3) (along c) such that Όχ = Τ (= с),
D3 = N о с and D2 D3 χ Di is called a Darboux frame.
Exercise 4.31. Let с : I -¥ Μ be a unit speed curve into a surface Μ in
R3 and let Di,D2,U3 be an associated Darboux frame. Show that there
exist smooth functions g\, g^ and #3 such that
dDi/ds = £iD2 + 02D3,
dD2/ds = -g{Di + 33D3,
dO3/ds = -02D1 - 03D2.
Show that gi — 0 along с if and only if с is a geodesic. Show that 52 = 0
along с if and only if с is asymptotic, and дз = 0 along с if and only if с is
principal.
The function g\ from the previous exercise is called the geodesic
curvature function and is often denoted кд. Let us obtain a formula for кд.
Define J : TPM -> TPM by
Jvp := Np χ vp.
4.2. Hypersurfaces
161
Figure 4.2. Curvature vectors
Notice that || Jvp\\ = \\vp\\ and so J is an isometry of the 2-dimensional inner
product space (TPM, (·, ·) ). But J also satisfies J2 = — id, and in dimension
2 this determines J up to sign since it must be a rotation by ±π/2. We now
assume that N is globally defined so that the map J extends to a smooth
map Τ Μ —> TM. For a unit speed curve c: I ^ M, the geodesic curvature
is given by
(4.1) Kg = (c",Jc')-(^,JTy
Indeed, abbreviating N о с to N we have from the first equation in Exercise
4,31 the following:
— = c" = ngN χ c' + £2N = fyJc' + 52^·
Taking inner products with Jd gives formula (4.1).
Let с : / —¥ Μ be a curve in a surface that is parametrized by arc
length. The curvature vector k(s) can be decomposed into a component
Kg(s) tangent to Μ and a component Kn(s) normal to M:
k(s) = Kg(s) + Kn(s).
The curve will be a geodesic if and only if k9(s) = 0 for all s. The vector
Kp(s) is the geodesic curvature vector at c(s). It is easy to show that
κ9 = ±\\Kg(s)\\. Figure 4.2 depicts a sphere of radius R with a conical
"hat". The cone intersects the sphere in a curve of latitude. Since the cone
is tangent to the sphere, the vectors «, кд and кп apply equally well to
both surfaces at least along the curve. Using the Pythagorean theorem and
similar triangles, it is possible to show that \\кд\\ — Ι/α, where α is the
distance from the curve to the vertex of the cone. Now imagine cutting the
cone along a generating line and unrolling it as shown in Figure 4.3. The
result is a planar region with a circular arc of radius α and curvature whose
162
4. Curves and Hypersurfaces in Euclidean Space
magnitude is 1/a. The reader might want to try and give a reason why this
is to be expected. We will answer this later in this chapter.
Figure 4.3. Unrolling a cone
Definition 4.32. If Sjyp is a multiple of the identity operator, then ρ is
called an umbilic point. If Snp = 0, then ρ is called a flat point. If
every point of a hypersurface is umbilic, then we say that the hypersurface
is totally umbilic.
Example 4.33. If a\xl + a2%2 Η l· anxn = 0 is the equation of a hyper-
plane Ρ in Rn, then
η
Ν = Σ α&
is a normal field when restricted to P. Since clearly Sn = 0 for all pEP,
we see that every point of Ρ is flat and that the Gauss and mean curvatures
are identically zero.
Example 4.34. Let 5n_1 be the unit sphere in Rn. Then the map ρ =
(α1,..., αη) Μ" Ν(ρ) := Σ™=1 afe is a unit normal field along Sn~1. We can
calculate Sn(v) = —VVN. Let с be a curve in Sn~l with c(0) = v. Then
d
-VJV =
dt
η
■Σ
i=l
N(c(t))
t=0
de_
~dt
t=o
We are really just using the fact that up to a change in base point we have
N(c(t)) = c{t). Thus 5дг = — id and we see that every point is umbilic and
every principal curvature is unity. Also, К — \ and the mean curvature
Η — — 1 everywhere.
Exercise 4.35. What is the shape operator on a sphere of radius r?
Composition with a Euclidean motion preserves local geometry so if we
want to study a hypersurface near a point p, then we may as well assume
4.2. Hypersurfaces
163
that ρ is the origin and that Np = en(0) = gju|0· Locally, Μ is then the
graph of a function / : Εη_1 -> R with |£(0) - 0 for i == 1,...,η - 1. A
normal field TV which extends Nq = en(0) is given by
η 1 η /0 η—1 л л
(1+Е(а//ах')2)-1/2(г,-Е^)-
г=1 г—1
Let Ζ = ?η — Σ^ι.1 Ιΐ^ so that N = gZ, where g is the first factor in the
formula above. For ν = Σ?=ι уГ** tangent to Μ at the origin we have
Vv(gZ) = (vg)Z + gVvZ,
where vg means that ν acts on g as a derivation. But vg vanishes at the origin
since g takes on a maximum there. Since g(0) = 1, we have Vv (gZ) = VVZ.
Thus we may compute <SV0 using Z:
Sjv0*> = -V„Z.
We have
n-l
9/
i=l
n—1n—1
<=1 j=l
дх1дхэ
i,(0).
We conclude that the shape operator at the origin is represented by the
(n — 1) χ (n — 1) matrix
9/
[£>2/] (0)
0а:*сЬ»
(0)
J l<ij<n—1
This is only valid at the origin. In other words, [£>2/] does not give us a
representation of the shape operator except at the origin where Μ is tangent
to Rn_1. We can arrange, by rotating further if necessary, that ei,..., en ι
are directions of curvature. In this case, the above matrix is diagonal with
the principal curvatures fci,..., kn-\ down the diagonal.
Example 4.36. Let Μ be the graph of the function f(x, y) = xy. We look
at 0 G M. We have
which diagonalizes to [ο -ι] with corresponding eigenvectors 4= (ei 4- ег)
and 4= (ei — ег). Thus the Gauss curvature at the origin is —1 and the
mean curvature is 0. We can understand this graph geometrically. The
normal sections created by intersecting the graph with the planes у = 0 and
χ = 0 are straight lines, and so the normal curvatures in those directions
164
4. Curves and Hypersurfaces in Euclidean Space
are zero. However, the normal section given by intersecting with the plane
у = χ is concave up, while that created by the plane у = —χ is concave
down.
Proposition 4.37· Let Μ С Ж3 be α surface. If v\,v2 Ε ΤΡΜ are linearly
independent, then for a given unit normal Np we have
Snpvi x SNpv2 = K(p) (νι χ v2),
Snpv\ χν2 + νλχ SNpv2 = 2#(p) (vx χ ν2).
Proof. We prove the first equation and leave the second as an easy exercise.
Let (sl3) be the matrix of Swp with respect to the basis vi, v2. Then
Snvv\ x SNpv2
= ^sl + v2s\) χ (vls\ + v2sl)
- s\s\ - s\s\ = det(SNp). U
We remind the reader of the easily checked Lagrange identity:
(v x w} α χ 6) = ; ' ; /x
for any a, b>v}w Ε R3 (or in any TPR3). If N is a normal field defined over an
open set in the surface Μ, and if X and Υ are linearly independent vector
fields over the same domain, then we can apply the Lagrange identity and
the above proposition at each point to obtain
(SNX,X) (SNX,Y)
(SNY,X) (SNY,Y)
and also
Я=2
K =
(SnX, X)
(X,X)
(SNx,Y)
(Y,Y)
(x,Y)
(Y,Y)
+
{X,X)
(SNY,X)
(x,Y)
(sny,y)
(X,X) (X,Y)
(Y,X) (Y,Y)
These formulas show clearly the smoothness of К and Η over any region
where a smooth unit normal is defined. Also, the formula
fc* = η ± Vh2-k
gives the two functions fc+ and k~~ such that k+(p) and fc~(p) are principal
curvatures at p. These functions are clearly smooth on any region where
k+ > h~ and are continuous on all of the surface. Furthermore, if every
point of an open set in Μ is an umbilic point, then k(p) := k+(p) = fc~(p)
defines a smooth function on this open set. This continuity is important for
4.3. The Levi-Civita Covariant Derivative
165
obtaining some global results on compact surfaces. Notice that for a surface
in R3 the set of nonumbilic points is exactly the set where k+ > A;-, and so
this is an open set.
Definition 4.38. A frame field E\,... ,En \ on an open region in a hy-
persurface Μ is called an orthonormal frame field if E\{p),..., Εη„χ(ρ)
is an orthonormal basis for TpM for each ρ in the region. An orthonormal
frame field is called principal frame field if each Ei(p) is a principal vector
at each point in the region.
Theorem 4.39. Let Μ be α surface in R3. If ρ G Μ is a point that is not
umbilic, then there is a principal frame field defined on a neighborhood of p.
Proof. The set of nonumbilic points is open, and so we can start with any
frame field (say a coordinate frame field) on a neighborhood of p. We may
take this open set to be oriented by a unit normal N. We then apply the
Gram-Schmidt orthogonalization process simultaneously over the open set
to obtain a frame field jF\, F2. Since ρ is not umbilic, we can multiply by an
orthogonal matrix to assure that F\, F2 are not principal at ρ and hence not
principal in a neighborhood of p. On this smaller neighborhood we have
SNF1 = aF1 + bF2)
SNF2 = bFi + cF2
for functions a, 6, с with 6^0. Now define Gi, G2 by
Gi = 6Fi + (fc+-a)F2,
G2 = (AT - c)F1 + bF2
and check by direct computation that SpjGi — k+G\ and SjsiG2 = k+G2.
We have used a standard linear algebra technique for changing to an eigen-
basis. Since b φ 0, we see that ||Gi||and HG2II are not zero. Finally, let
J5i = Gi/||Gi|| andS2 = G2/||G2||. □
4.3, The Levi-Civita Covariant Derivative
The Levi-Civita covariant derivative is studied here only in the special case
of a hypersurface in Rn. The more general case is studied in Chapters 12
and 13. We derive some central equations, which include the Gauss formula,
the Gauss curvature equation and the Codazzi-Mainardi equation.
Let Μ be a hypersurface. For Xp G TPM, and Υ a tangent vector field
on Μ (or an open subset of M), define VxpY by
VxpY := projrpMVXpy,
where projT M : TpRn —> TPM is orthogonal projection onto TPM. For
convenience, let us agree to denote the orthogonal projection of a vector
166
4. Curves and Hypersurfaces in Euclidean Space
ν e TpRn onto TPM by vT and the orthogonal projection onto the normal
direction by ir1. We call vT the tangent part of ν and v1- the normal
part. Then Vxpy=(Vxpr)T.
Exercise 4.40. Show that for / a smooth function and Xp and Υ as above,
we have VXpfY = (Xpf) Y(p) + f(p)VXpY.
If N is any unit normal field defined near p, then
VXpr = VXpY + (Np,VXpYp) Np.
Since
0 = Xp (N, Y) = (SNXP, Y) + (N, VXpY) ,
we obtain the Gauss formula:
(4.2) VXpY = VXpY - (SNpXp, Yp) Np.
Notice that the right hand side of the above equation is unchanged if N
is replaced by —JV. It follows that if X and Υ are smooth tangent vector
fields, then ρ н-> VxpY is smooth and we may then define the field VxY by
(VXF) (p) :- VXpY. By construction (VXY) (p) = (VZY) (p) if X(p) =
Z(p). It is also straightforward to check that the map (Χ, Υ) ι->· VXY is
C°°(M)-linear in X, but not in Y. Rather, like V, it is R-linear in Υ and
we have the product rule
VxfY = (Xf)Y + fVxY}
which follows directly from Exercise 4.40. Thus V is a covariant derivative
on Μ and VXY is defined for Χ, Υ e X(M). We remind the reader that
X(M) is the space of tangent vector fields and not to be confused with vector
fields along Μ which may not be tangent to M.
Let Υ and Ζ be smooth tangent vector fields on Μ and take Xp € TPM.
We study the situation locally near p. Let (O, u) be a special chart centered
at ρ and let (0,u) be the chart obtained by restriction where Ο = Ο Π Μ
as before. If Υ and Ζ are extensions of Υ and Ζ to O, then since S^ is
self-adjoint, we have
(VyΖ - VZY) (p) = {VYZ - VZY) (p)
= {VyZ-VzY){p)
= [Y,Z]P=[Y,Z]P.
Thus VyZ - VZY = [У, Z] for all У, Ζ e X(M). This fact is expressed by
saying that V is torsion free. Also,
Xp (Y, Z) = Xp (Ϋ, z) = (VXpY, Zp) + (Ϋρ, VXpz)
= (VXpY,Zp) + (Yp,VXpZ),
4.3. The Levi-Civita Covariant Derivative
167
so that if X e £(M), then X (У, Z) = (VXY, Z) + (У, \7XZ). We express
this latter fact by saying that V is a metric covariant derivative on
M. There is only one covariant derivative on Μ that satisfies these last
two properties, and it is called the Levi-Civita covariant derivative (or
Levi-Civita connection). We prove the uniqueness later in this chapter.
With coordinates it1,..., it71"""*1 as above we have functions
_/A A\
If
X = Vl^ and Υ = YyJ°
then
n-l я n-l я
'S^andy = W'· A,
dul f-t dw>
i=l t=l
(Χ,Υ) =Σ9νΧΎ>\
h3
The length of a curve in Μ is just the same as its length as a curve in the
ambient Euclidean space. One may define a distance function on Μ by
dist(p,g) := inf{L(c)},
where the infimum is over all curves connecting ρ and q. This gives Μ a
metric space structure whose topology is the same as the underlying
topology. This will be proved in more generality in Chapter 13. For now, the
point is that metric aspects of Μ are determined by (·, ·) and locally by the
g%3 associated to each chart of an atlas for M. For example, the length of a
curve c:[a,5]^OcMis given by
(4.3) L{c)=j'nYgijic{t))^^dt,
Σ
where cl(t) :— и1 о с.
Exercise 4.41. Deduce the above local formula for length from the formula
for the length of the curve in the ambient Euclidean space.
Returning to our covariant derivative V, we have
dn—ι о
168
4. Curves and Hypersurfaces in Euclidean Space
for smooth functions Γ^- known as the Christoffel symbols of V. For X
and Υ expressed as above, we have
η 1 η—1 in—1 r\
г=1 г=1 V j 1
A^ I Z^ ^г ^ Z^ ^агР
n—In—1 лу^ о η—In—In—1 гч
fc-1 %=1 fc-1 г-l j 1
and so we have
n-l /n-l ^vfc n-l \ я
(4.4) v,y=s Σ^'+Σ^ έ·
fc=l уг 1 i,j=l }
Thus the functions Г£- determine V in the coordinate chart. Also note that
0
du*' dv?
л? dv? мдиг ^\ гз 3Ί
It follows that
(4.5) I* = Г*г for all i, j, к = 1,..., η - 1.
We also have
дж = A /A A\ = /v A A\ . / A v A\
n-l
s=l
The matrix (gij) is invertible, and it is traditional to denote the components
of the inverse by g1·7", so that J2r 9jr9ri ~ &)· One таУ solve to obtain
Exercise 4.42. Prove formula (4.6) above. [Hint: First write the formula
-jjtji = Σ3 ^ki9sj + ^ij9si two more times, but cyclically permuting г, j, к to
obtain three expressions. Subtract the second expression from the sum of
the first and third. Use equality (4.5).]
Proposition 4.43. The Levi-Civita connection on a hypersurface is
determined uniquely by the properties of being a torsion free metric connection.
4.3. The Levi-Civita Covariant Derivative
169
Proof. In deriving the local formula for the Christoffel symbols, we only
used the fact that V is a torsion free metric connection. We do this again
in Chapter 13 in a more satisfying way. D
If an object is determined completely by the metric on M, then we say
that the object is intrinsic. Equivalently, if all local coordinate expressions
for an object can be written in terms of the metric coefficients дц (and
their derivatives, etc.), then that object is intrinsic. We have just seen that
the connection V is intrinsic. It follows that if / : M\ —> M<i is an isome-
l 2
try of hypersurfaces and V and V are the respective Levi-Civita covariant
derivatives, then
ubXY=bUxUY
for all vector fields Χ, Υ G X(M). One way to see this is to examine the
situation using a chart (V,u) on M\ and the chart (/(F), u о / l) on M2.
2
A better way is to show that (X,Y) *-> f*Vf%xf*Y defines a torsion free
metric connection on M\ and then use Proposition 4.43. (Exercise!)
If Υ : / —> Τ Μ is a vector field along a curve с : / —> Μ С Rn, then we
define
Va, У :=projTMY'(*<>) = П*о)Т
dt\tQ CW
for any ίο G I and then define V a Y by
at
(Ve.y)(t):=Vei rfori€/.
\ at / at It
Suppose that с : I —> Μ is such that c(J) dV for some subinterval J С I
and some chart (V^u). We can then write
for all t e J.
c(t)
Let us focus attention on a to such that c(to) φ 0 and let J be an open
interval with to £ J- By restricting J if necessary we may also assume that
с is an embedding. Then there is a smooth field У on a neighborhood of
c(J) such that Υ о с = Y. In fact, a simple partition of unity argument
shows that we may arrange that Υ be defined on all of V (in fact, on all of
170
4. Curves and Hypersurfaces in Euclidean Space
M). For ί G J we have
(у^у)(*) = г'(^ = ((уоС)'(0)Т
= V,
(v0(t)y)T = V0(t)F
n— 1 In—1
n-1
- ΣΙΣ τ£ <««»£ + Σ i%wd)^wvwo)
We arrive at
fc=l \г=1
dt
dt
ct
-Ete + .S(i%«)<w^«>
47=1
c(t)
(4.7)
at
= £(^£>И^;
*J=1
<9ufe"
We would like to argue that the above formula holds for general curves. If
έ(ίο) φ 0, then there is an interval around to so that the formula holds as we
have just seen. If c(to) = 0, we consider two cases. If there exists a sequence
U converging to ίο such that c(U) = 0 for all г, then the formula holds for
each U and hence by continuity at to* Otherwise there must be an interval
J containing to such that с is constant on J, say c(t) = ρ for all t £ J, and
then in this interval Υ is just a map into the vector space TpM. In this case
we have
(νΑκ)(ί)=η*)τ=(|Σ^) a'4T
дик
dYk
= Σ^(*)
д
дик
Φ)
Μ and a
But this agrees with the formula since ^ = 0.
Exercise 4.44. Show that for a vector field X along с : /
smooth function h € C°°(I) we have
Vd/dthX = hVd/QtX + h'X.
Exercise 4.45. Show that for vector fields Χ, Υ along с we have
| (X, Y) = (Vd/dtX, Y) + (X, Vd/dtY) = 0.
The operator V^/^ involves the curve с despite the fact that the latter is
not indicated in the notation.
4.3. The Levi-Civita Covariant Derivative
171
We pause to consider again the question posed earlier about the circular
arc on the unrolled cone in Figure 4.3. The key lies in the fact that the
absolute geodesic curvature \кд\ is intrinsic. We remarked earlier that
the operator J is intrinsic up to sign (the latter being determined by
orientation). On the other hand, in Problem 11 the reader is asked to derive the
formula
_ (V, Л)
K° Wit '
But
_ |(7",±JV)I _ |(v^V,±</Y)|
ΙβΙ" ΙУII3 " b'll3 '
and since both V and the pair ±J are intrinsic, we see that \кд\ is intrinsic.
A little thought should convince the reader that if a curve in one surface
is carried to a curve in another surface by an isometry, then the curves
will have equal absolute geodesic curvatures at corresponding points. The
unrolling of the cone in Figure 4.3 can be thought of as inducing an isometry
between the cone (minus a line segment) and a region in a planar surface
in R3. Thus we expect \кд\ to be the same for both curves. But \кд\ for a
circular arc in a plane is just the reciprocal of the radius.
Definition 4.46. A tangent vector field Υ along a curve с : I —> Μ is said
to be parallel (in M) along с if V aY — 0 for all t G /.
dt
If с is self-parallel, i.e. V#/^c(£) = 0 for all t G /, then it is easy to see
that с is a geodesic (in M) and in fact, this could serve as an alternative
definition of geodesic curve, which will be the basis of later generalizations.
Notice that if с is a curve in Rn, then Υ can be considered as taking values in
TRn. However, Υ being parallel in Μ is not the same as Υ being parallel as a
TRn-valued vector field along с In particular, a geodesic in Μ certainly need
not be a straight line in Rn. For example, constant speed parametrizations
of great circles on S2 С R3 are geodesies.
Exercise 4.47. Given a smooth curve с : I —l· M, show that d' = 0 if and
only if с is a geodesic in Μ such that (5jvc(t), c(t)) = 0 for all t and choice of
unit normal at c(t). Suppose that c" is never zero. Show that с is a geodesic
in Μ if and only if c" is normal to Μ (i.e. c"(i) ±TC^M for all t e I).
The following simple result follows from the preceding exercise:
Proposition 4.48. Let Mi and M2 be hypersurfaces in Kn. Suppose that
с : / -> Rn is such that c(t) G M\ Π Μ2 for all t. If c" is not zero on any
subinterval of I, then Τφ^Μχ = TC^M2 for all t Ε I.
Proof. By Exercise 4.47, Tc{t)Mx = c"^)1 = Tc{t)M2 for all t. Π
172
4. Curves and Hypersurfaces in Euclidean Space
Proposition 4.49. If vector fields Χ, Υ along a curve с : I —> Μ are
parallel, then (X,Y)(t) := (X(t),Y(t)) is constant in t. In particular, a
parallel vector field has constant length.
Proof. ft (X, Y) = (Vd/dtX, Y) + (X, Vd/dtY) = 0. D
Corollary 4.50. A geodesic has a velocity vector of constant length.
Definition 4.51. Suppose Υ G X(M) is a smooth vector field on M; then
Υ is called a parallel vector field on Μ if VXY = 0 at all points of Μ
and for all smooth vector fields X G X(M).
Obviously, У is a parallel field if and only if У о с is parallel along с for
all curves c.
We now move on to prove two basic identities and introduce the
curvature tensor. It is easy to check by direct computation that for vector fields
Χ, Υ, Ζ on an open subset of Rn, we have
(4.8) ΨχΨγΖ - νγνχΖ - V[iif]Z = 0.
If Μ is a hypersurface and X, У, Ζ are tangent vector fields on a
neighborhood of an arbitrary ρ G M, then we may extend these to fields Χ, Υ Ζ on
a neighborhood О in Rn. Then we have
(VxVyZ - VyVXZ - V[X%Y]Z) (p)
= fixVyZ - VyV^Z - V[jt%Y]z) (p) = 0,
and so
(4.9) VXVYZ - VyVXZ - V[x,y]Z = 0
wherever the fields are all defined on M. Suppose that N is a unit normal
field defined on the same domain in M. We apply the Gauss formula (4.2) to
equation (4.9) above and then decompose it into tangent and normal parts:
0 = Vx (VyZ + (SNY, Ζ) Ν) - Vy (VXZ + (SNX, Ζ) Ν) - V[X,Y]Z
= VxVyZ + (SNX, VyZ) N + X {SNY, Z)N- (SNY, Z) SNX
- VyVxZ - (SNY, VXZ) N-Y (SNX, Ζ) Ν + (SNX, Z) SNY
-V[X>Y]Z-(SN[X,Y],Z)N.
Equating the tangential parts of the above gives the Gauss curvature
equation:
(4.10) VXVYZ - VYVXZ - V[X<Y]Z = (SNY, Z) SNX - (SNX, Z) SNY.
The normal parts give
0 = (SNX, VYZ) + X (SNY, Z) - (SNY, VXZ)
-Y(SNX,Z)-(SN[X,Y],Z),
4.3. The Levi-Civita Covariant Derivative
173
or {4XSNY, Z) - (VYSNX, Z) - (SN[X, Y], Z) = 0 for all Z. From this we
obtain the Codazzi-Mainardi equation:
(4.11) VXSNY - VYSNX - SN[X, Y] = 0.
Let us give an application of the Codazzi-Mainardi equation and then
return to the Gauss curvature equation.
Proposition 4.52. Let Μ be α connected hypersurface in Rn oriented by
a unit normal field N. // every point of Μ is umbilic (i.e. Μ is totally
umbilic), then the normal curvatures are all equal and constant on Μ.
Furthermore, Μ is an open subset of a hyperplane or a sphere according to
whether the normal curvatures are zero or nonzero. In particular, if Μ is a
closed subset o/Mn, then it is a sphere or a hyperplane according to whether
it is compact or not.
Proof. There is a function к such that Sn — kl, where / is the identity on
each tangent space. This function is continuous since к = ~j trace Sjv· Let
Xp G TPM and pick Yp e TPM so that Xp and Yp are linearly independent.
Extend these to tangent fields X and У on a neighborhood of p. By the
Codazzi-Mainardi equation we have
0 = VxkY - VYkX - k[X} Y]
= {Xk) Υ + kVxY - ((Yk) X + kVYX) - k[X, Y]
= {Xk)Y-(Yk)X,
where we have used VxY — S/γΧ = [X, Y]. In particular, at ρ we have
(Xpk) Yp—(Ypk) Xp = 0. Since Xp and Yp are linearly independent, Xpk = 0.
Since ρ and Xp were arbitrary and Μ is connected, we see that к is in fact
constant.
If the constant is к = 0, then Sn = 0 on M, and this means that
N = No is constant along M. This implies that Μ is in a hyperplane
normal to Nq.
If к φ 0, then (changing N to — N if necessary) we may assume that
к > 0. Define a function / : Μ -> Rn by f(p) -p+ ^N{p). We identify
tangent spaces with subspaces of W1 and calculate Df(p). Let ν G TPM and
choose a curve с : (—α, α) —» Μ with c(0) = v. Then we have
Df(p)-v=±
•/„N 1 d
N oc
о
ln 1,
= ν — t£>nv = ν — тку — 0.
к к
Thus Df(p) = 0 for all ρ G M, and since Μ is connected, / is constant
(Exercise 2.23). Thus ρ + jiV(p) = q for some fixed q and all p. In other
174
4. Curves and Hypersurfaces in Euclidean Space
words, all ρ G Μ are at a distance l/k from gGln, and so Μ is contained
in that sphere of radius l/k. D
The left hand side of the Gauss curvature equation (4.10) is given its
own notation
R(X, Y)Z := VxVyZ - VyVXZ - V[x,y]Z,
and looking at the right side of (4.10) we see that (R(X,Y)Z) (p) depends
only on the values of X, У, Ζ at the point p, which means that we obtain a
map Rp : TpM χ ΤΡΜ χ TpM -> TPM defined by the formula
R9(Xp,Yp)Zp:=(R(X:Y)Z)(p).
We say that R is a tensor since it is linear in each variable separately. We
study tensors systematically in Chapter 7. The tensor R is called the Rie-
mannian curvature tensor and the map ρ \-¥ Rp is smooth in the sense
that if X, У, and Ζ are smooth tangent vector fields, then ρ н* Rp(Xp, YP)ZP
is a smooth vector field. We often omit the subscript ρ and just write
R(XPiYp)Zp. (Notice that equation (4.8) just says that the curvature of W1
associated to the ambient covariant derivative V is identically zero.) Using
(4.10) again, it is also easy to check that Rp is linear in each slot separ
rately on TpM. In particular, for fixed Xp, Yp G TPM we have a linear map
R(Xp,Yp):TpM->TpM.
Theorem 4.53. If Μ is α surface in R3 and (XP,YP) is an orthonormal
basis for TPM, then
(R(Xp,Yp)Yp,Xp) = K(p).
Proof. Using 4.10, and abbreviating Snp to 5, we have
\R\Xp) YpjYp, Xp) :== {οΥρ,Υρ) [οΧρ,Χρ) — {οΧρ,Υρ/ [οΥρ,Χρ)
= det S = K{p). U
We have shown that V is intrinsic and thus R is also intrinsic. Thus the
previous theorem implies that the Gauss curvature К for a surface in E3
is intrinsic. This is the content of Gauss's Theorema Egregium, which we
prove (again) below using local parametric notation.
If Μ is a fc-dimensional submanifold in Rn, and (V,u) is a chart on Μ
with U = u (V), then u_1 : U -> Μ is a parametrization of a portion of Af,
and we denote this map by
χ : U -> Μ.
This notation is traditional in surface theory. Composing with the inclusion
l : Μ <ч· Rn, we obtain an immersion ι ο χ : U —> Rn, but we normally
identify ι ο χ and χ when possible. One may study immersions that are not
4.3, The Levi-Civita Covariant Derivative
175
necessarily one-to-one. A reason for this extension is that it might be the
case that while χ : U —> Rn is not one-to-one, its image is a submanifold Μ
and so we can still study Μ via such a map. For example, consider the map
χ: R2 -> R3 given by
(4.12) x(w, ν) = ((α + 6cosu) cost?, (a + bcosu)sinv,bsinu)
for 0 < b < a. This map is periodic and its image is an embedded torus.
Notice that the restriction of χ to sets of the form (щ — π/2, щ + π/2) χ
(vq — π/2, νο + π/2) are parametrizations whose inverses are charts on the
torus. Another example is the map χ : R2 —>· S2 С R3 given by
x((£,0) = (cos θ sin φ, sin θ sin φ, cosy?).
The restriction of this map to (Ο,π) χ (0,2π) parametrizes all of S2 but a
set of measure zero.
If χ : U —> Μ is a parametrization of a hypersurface in Rn, then for
и G t/, the vectors §^(u) are tangent to Μ at ρ = x(u) and in fact form the
coordinate basis at ρ in somewhat different notation. In fact, if we abuse
notation and write ul for иг о х-1, we obtain a chart (V,u) with x(/7) = V.
Then g(u) is essentially just gjr|x(u)- We have (0, J£) = 5<i ο χ, but
in the current context of viewing things in terms of the parametrization,
we just change our notations slightly so that (J^> J^j) = 9ij- These gi3
are the components of the metric with respect to the parametrization. In
these terms, some of the calculations look a bit different. For example, if
7 : [a, 6] -» Μ is a curve whose image is in the range of x, then the length of
the curve can be computed in terms of the g^, which are now functions of
the parameters иг. First, 7 must be of the form t \-> х(г^(£),... ,un_1(£)),
where и : t »->· (u1^),..., un_1(£)) is a smooth curve in U. Then we have
= /вЕ*>(*>)~л,
which is only notationally different from formula (4.3) due to our current
parametric viewpoint.
Exercise 4.54. Show that there always exists a local parametrization of
a hypersurface Μ С Rn around each of its points such that g%3 (0) — δ%3
and ^£(0) = 0 f°r aH *»J»fc· [Hint: Argue that we may assume that Μ
is written as a graph of a function / : R71""1 —> R with /(0) = 0 and
Df(0) = 0. Then let x : Rn_1 -> W1 be defined by (и1,...^""1) ■->
(tz1,...,^-1,/^1,...,^"1)).]
176
4. Curves and Hypersurfaces in Euclidean Space
Theorem 4.55 (Gauss's Theorema Egregium). Let Μ be α surface in R3
and let ρ € Μ. There exists a parametrization χ : U —> Μ with x(0,0) = ρ
such that gij = 5ij to first order at 0 and for which we have
π(*λ d2gi2,n, 1 d2g22 ,M 1 d2gn ,M
dudv
2 du2
2 dv2
Proof. In the coordinates of the exercise above, which give the
parametrization (u, ν) ι-» (u, г?, /(u, г?)), where ρ is the origin of R3, we have
9n(u,v) gi2{u,v)
92i{u,v) д22{щу)
i + φ2
dudv
du dv
i+di)2j'
from which we find, after a bit of straightforward calculation, that
#9ΐ2(ο)-1^(ο)-^(ο)
dudvK~' 2 du2
d2fd2f d2f
du2 dv2 dudv
= det S(p) = K(p).
2 dv2
= detD2f(0)
D
Let us introduce some traditional notation. For χ : U -¥ Μ С R3, and
denoting coordinates in U again by (u, v), and in R3 by (ж, у, г), we have
(дх ду дх\
ди',ди',ди')^
дх ду дх\
dvf dv' dv J x
**> I ο ι α > о
_ / d2a Э2у д2г \
\ dudv' 5г&с?г>' dudv J x'
and so on. In this context, we always take the unit normal to be given, as
a function of и and г>, by
Xw X "X.v
N(u,v) =
||Xu X X„||
A careful look at the definitions gives
dN
(based at x(u,v)).
dN
du = Sn^ and "^7 = Sn(*v)'
Recall that the second fundamental form is defined by II(y, w) = (5дгг?,ги),
and this makes sense if the tangent vectors v, w are replaced by fields along χ
defined on the domain of N. The traditional notation we wish to introduce
is
Ε = \Хш xtt/ j F = \X^, Хг;/ > G — \Хгм Xu/ ?
1 = (5τνΧΐί? Xu) j m = (Sn*-ui χυ) , П = (Sn*v, Xv) -
4.3. The Levi-Civita Covariant Derivative
177
Thus the matrix of the metric (·, ·) (sometimes called the first fundamental
form) with respect to χη,χυ is
9ii 912 1 = Γ Ε F
921 922 \ [ F G
while that of the second fundamental form II is
1 m
m η
The reader can check that ||xu χ χυ||2 = EG — F2. The formula for the length
of a curve written asi^ x(it(i), v(t)) on the interval [a, b] is
■(*)'
ι · *
For this reason the classical notation for the metric or first fundamental
form is
ds2 = Ε du2 + 2F dudv + F dv2,
where ds is taken to be an "infinitesimal element of arc length".
Consider the map g : R3 —> (R3) given by ν κ->· (ν,·). With respect
to the standard basis and its dual basis, the matrix for this map is ["].
Similarly, we can consider the second fundamental form as a map II: R3 —>·
(R3)* given by ν >-> II (v, ·), and the matrix for this transformation is [i J}].
Then since // (v, w) = (SW^> w), we have II = g о Sn and so
SN = g-1 о //.
We conclude that Syv is represented by the matrix
i-l г
Ε F
F G
1 m
m η
This matrix may not be symmetric even though the shape operator is
symmetric with respect to the inner products on the tangent spaces. Taking the
determinant and half the trace of this matrix we arrive at the formulas
nl-m2
K =
H =
EG - F2'
Gl + En - 2Fm
2(EG-F2) '
Exercise 4.56. Show that 1 = (JV, xww), m = (N,tcuv) and η = (TV, xvv).
Exercise 4.57. Consider the surface of revolution given parametrically by
χ(ιι,ν) = (д(и),к(и)со8У,}1(и)вту)
178
4. Curves and Hypersurfaces in Euclidean Space
with h > 0. Denote the principal curvature for the meridians through a
point with parameters (u,v) by /ζμ and that of the parallels by kn. Show
that these are functions of и only given by
Κμ —
Κπ —
h'
h"
((gf + (h')2)3/2''
Ч(д>У + (Л'П1/а
4.4. Area and Mean Curvature
In this section we give a result that provides more geometric insight into
the nature of mean curvature. The basic idea is that we wish to deform a
surface and keep track of how the area of the surface changes. Let Mel3
be a surface and let χ : U —l· V С Μ be a parametrization of a portion V of
M. We suppose that V has compact closure. The area of V is defined by
A(V)
-L
||xu x JCull dudv.
The total area of Μ (if it is finite) can be obtained by breaking Μ up
into pieces of this sort whose closures only overlap in sets of measure zero.
However, our current study is local and it suffices to consider the areas of
small pieces of Μ as above. Suppose χ : U χ (—ε, ε) —)► R3 is a smooth
map such that for each fixed t G (—ε, ε), the partial map x(·, ·, i) : (u, ν) Η
x(u,v,t) is an embedding and such that ^ is normal to xu and xv for all
t. For each £, the image Vt = x(i7, t) is a surface. We have in mind the
case where x(·, -,0) is a parametrization of a portion of a given surface Μ
so that Vq = V С Μ (see Figure 4.4). The normal N = xw χ x„/ ||xu χ χν
depends on t and at time t provides a unit normal to the surface Vt. Thus
Vt is a one parameter family of surfaces.
Theorem 4.58. Let χ : U χ (—ε, ε) —>· К3 and let Vt = x(l/,i) be as above
so that ^ is normal to the surface Vt. Let H(t) denote the mean curvature
of the surface Vt. Then
iAw
—a/IIIе
к II0*
H(t) dA.
Proof. Since |s is parallel to N, we have fjf = ||fjr|| -W· Thus
<9xu _ д дх _ д (
~~дТ~дй~д1~ дй V
дх
\dt\
И-1
дх|
\Щ
SN (χ„) + £
дх
dt
N
4.4. Area and Mean Curvature
179
Figure 4.4. Deformation of a patch
and so
Similarly,
dxu
(»·%*-)--
dx
at
(N, SN (xu) χ x„).
N, xu χ η^] = - \\-£\\ (N, xu χ SN (xr)).
Now we calculate using the second formula of Proposition 4.37:
dt
A = — / ||xu x x„|| dudv = — I (N,xu χ χ„) dudv
= / (-—N,xu χ xv\dudv + / (N, -^ χ x„ + xu χ -^-jdudv
= Ην,-^-xxv + XuX -?£■} dudv
I Sxl
= - / l^rll {|| Sn (xu) χ χν + χω χ 5jv (xv)ll} <&<&>
= -2 JH{p) Η ||xu χ xj dudv = -2У ^
#dA.
D
have
In particular, we can arrange that |||||| = 1 at time t = 0, and then we
dt
f=0
-2JH
dA.
Thus, Я is a measure of the rate of change in area under perturbations of
the surface.
Definition 4.59. A hypersurface in Rn for which Η is identically zero
is called a minimal hypersurface (or minimal surface if η — 3 so that
dimM = 2).
180
4. Curves and Hypersurfaces in Euclidean Space
" Ε
F
r 1
L m
F "
G
ш
11
Example 4.60. For с > 0, the catenoid is parametrized by
x(u,v) = (и, с cosh (it/c) cos?;, с cosh it sin г?)
and is a minimal surface. Indeed, a straightforward calculation gives
c2 cosh2 (и/с) О
0 cosh2 (u/c)
-с О
0 1/c
It follows that Η = 0.
Example 4.61. For each 6^0, the map x(u,v) = (bv,ucosv,usinv) is
a parametrization of a surface called a helicoid and this is also a minimal
surface. The reader may enjoy plotting this and other surfaces using a
computer algebra system such as Maple or Mathematica.
4.5. More on Gauss Curvature
In this section we construct surfaces of revolution with prescribed Gauss
curvature and also prove that an oriented compact surface of constant curvature
must be a sphere. Let
x(u,v) = (g{u),h(u) cos v,h(u) sin v)
be a parametrization of a surface of revolution. By a reparametrization of
the profile curve (g{u), h(u)) we may assume that it is a unit speed curve so
that ((/) + (hf) = 1. When this is done, we say that we have a canonical
parametrization of the surface of revolution. A straightforward calculation
using the results of Exercise 4.57 shows that for any surface of revolution
given as above we have
K =
h"
A(CiOa + (fc')2)
If we assume that it is a canonical parametrization, then we have
-Wfti' + sfffh*
K =
h
J\2
l/\2
On the other hand, differentiation of {g'Y + {h')z = 1 leads to g'g" = — Л'Л",
and so we arrive at
which shows the expected result that К is constant on parallels (curves
along which и is constant). Now suppose we are given a smooth function
К defined on some interval /, which we may as well assume to contain 0.
4.5. More on Gauss Curvature
181
We would like К to be our Gauss curvature and so we wish to solve the
differential equation h" + Kh = 0 subject to h(Q) > 0 and |/г'(0)| < 1. The
first condition simplifies the analysis, while the second condition allows us
to obtain the canonical situation (</') + (hf)2 = 1. In fact, we let
g(u) = JUy/i-(h'(t))2dt,
and this will give the desired solution defined on the largest open subinterval
J of / such that h > 0 and \h'\ < 1. With this solution, our surface of
revolution will be defined, but only for и G J.
Suppose we try to obtain a surface of revolution with constant positive
Gauss curvature К = 1/c2 for some constant c. Then a solution of the
equation for h will be h(u) = acos(u/c) for an appropriate a > 0. Then
fu I a2
g(u) = / yi--^sin2(£/c)<&,
and with the resulting profile curve, we obtain a surface of revolution with
Gauss curvature 1/c2 at all points. There are three cases to consider. First,
if a = c, then the interval J on which h > 0 and \h'\ < 1 is easily seen to be
(-7гс/2,7гс/2) and we have
h{u) = ccos(u/c) and g(u) = csm(u/c).
This gives a semicircle which revolves to make a sphere minus the two points
on the axis of revolution. We already know that these two points can be
added to give the sphere of radius с = \j\[K, which is a compact surface of
constant positive Gauss curvature К. It turns out that the spheres are the
only compact surfaces (without boundary) of constant positive Gauss
curvature. Now consider the case 0 < a < c. The interval J is the same as before,
but the surface extends in the χ direction between x\ = \\ти^_жс/2 д(и) and
X2 = У\ти_+ъс/2 9{и)- It is easy to show that x\ < —a and X2 — —x\ > a.
Since the maximum value of h is now smaller than c, the profile curve is
shallower and wider than the semicircle of the a = с case above. Although
linv+±7rc/2fr(w) = 0 as before, we now have the profile curve tangents at
the endpoints given by
lim (g'(u),h'(u)) = lim I \ 1 ~ sin2(u/c), — sm(u/c) I
u->±ttc/2V ' ч->±тгс/2 I V С2 С J
This shows that the revolved surface is pointed at its extremes and forms
an American football shape. In this case, there is no way to add in the
missing points on the axis of revolution to obtain a smooth surface. The
two principal curvatures are no longer equal, but their product is still 1/c2.
182
4. Curves and Hypersurfaces in Euclidean Space
The third case а > 0 gives a surface that has an extension to a surface
with boundary.
Exercise 4.62. Analyze the case а > 0.
Prom above we see that there is an infinite family of surfaces with
constant curvature (only one extending to a compact surface). The situation for
constant negative curvature is similar, but we obtain no compact surfaces.
One particular constant negative curvature surface is of special interest:
Example 4.63 (Bugle surface). This surface has Gauss curvature —1/c2
and is given by
x(u,u) = (u, h(u) cos ν,h(u) sin v),
where h is the solution of the differential equation
-h
ti =
sfe^W'
subject to initial condition limu_>o h(u) = c. The function h is defined on
(0,oo).
Lemma 4.64. Let Μ be a surface in R3. Let ρ 6 Μ be a nonumbilic point
and E\,E2 a principal frame on a neighborhood of ρ (oriented by N) so that
SnE\ = k+Ei and S^E2 = k~E2. If we define functions
. -E2k+ Eik'
k+-k- k+-k~'
then
К = -Eih2 - E2hx -hi hj.
Proof. Since (E2,E2) = 1, we have (VElE2,E2) = 0 and so there is some
function hi such that VElE2 = h\E\. Similarly, V E2E\ = h2E2 for some
function h2. We find formulas for each of these functions. We have
Q = El(EuE1) = 2(VElEuE1)
and
0 = #i (EUE2) = {ЪЕ1ЕЪЕ2) + (EuVElE2),
from which it follows that V ElE\ = —h\E2. Similarly, VE2E2 = —h2E\.
We have
[EUE2] = VElE2 - νΕ2Ελ = кгЕг - h2E2.
4.5. More on Gauss Curvature
183
We now apply the Codazzi-Mainardi equations (4.11):
0 = VEiSnE2 - VEbSpjEi — Sn[Ei, E2]
= VElk~E2 - VEik+Ex - SN (hxEx - h2E2)
= {Eik~ E2 + k~VElE2) - (E2k+ Ex + k+VE2Ex)
- (hxk+Ex - h2k~E2)
= (Exk~ E2 + k~hxEx) - (E2k+ Ex + k+h2E2)
- (hik+Ei - h2k~E2)
= (k~hx - E2k+ - hik+) Ex + (Exk~ + h2k~ - k+h2)E2.
Setting the coefficients of Ex and E2 equal to zero we obtain
, -E2k+ Eik-
η,χ = r-r—— and n2 =
k+-k~ k+- k~
We compute as follows:
R{EU E2)E2 = VEiVe2E2 - VE2VElE2 - V[EuE2]E2
= V^! (-h2Ei) - VE2 (hiEi) - V^hlEl h2e2)E2
= - (Eih2) Ει - h2VElEi - (E2hi) Ελ - hxVe^Ei
- hiVElE2 + h2VE2E2
= - (Exh2) Ει - (E2hi) Ει - h2 (-hiE2) - hi (h2E2)
- hi (hiEi) + h2 (-h2Ei)
= - (EM Ει - (E2hx) Ει - h\Ei - h\Ei.
Using the Gauss curvature equation (4.10) we arrive at
(4.13) К = (R(EUE2)E2, Ex) = -Exh2 - E2hi - h\ - hi О
Corollary 4.65. If ρ is α nonumbilic point for which both k+ and k~ are
critical, then
Proof. Using equation (4.13), we have
К = -Exh2 - E2hx -h\-h\
~»(*ед-*(й£)-(й^-гз*
- \ 2
k-J \k+-k
At ρ we have E2k+ = Exk~ = 0, and so (suppressing evaluations at p) we
obtain
_ -(k+-k~)Elk- _ -(к+-к~)Е%к+ _ Е%к+ - Е\к~
K(p)~ {k+-k-f (k+-k~)2 ~ k+-k~ ■
184
4. Curves and Hypersurfaces in Euclidean Space
Corollary 4.66 (Hilbert). Let Μ С R3 be α surface. Suppose that К is
a positive constant on M. Then k+ cannot have a relative maximum at
a nonumbilic point Similarly, k~ cannot have a relative minimum at a
nonumbilic point.
Proof. Since К = к+к~ > 0 is constant, k+ has a relative maximum exactly
when k" has a relative minimum, so the second statement follows from the
first. For the first statement, suppose that fc+ has a relative maximum at p.
Notice that X2k+ < 0 and X2k~ > 0 for any tangent vector field defined
near p. Near p, there is a principal frame as above, and if we use Corollary
4.65, we have
contradicting our assumption about K. 0
We are now able to use the above technical results to obtain a nice
theorem.
Theorem 4.67. // Μ CR3 is an oriented connected compact surface with
constant positive Gauss curvature K, then Μ is a sphere of radius 1/у/К.
Proof. The hypotheses give us the existence of a global unit normal field
N. Note that fc+ > \[K at every point, and since Μ is compact, k+ must
have an absolute maximum at some point p. By Corollary 4.66, ρ is an
umbilic point and so fc+(p) = k~(p). But then (fc+(p)) = k+ (p)k~(p) = K)
and thus the maximum value of fc+ is л/К. We have both k+ > \[K and
fc+ < \[К on all of Μ, and so k^ = k~ = \[Κ everywhere (M is totally
umbilic). By Proposition 4.52 we see that Μ is a sphere of radius \j\[K. D
We already know from our surface of revolution examples that there are
many noncompact surfaces of constant positive Gauss curvature, but now
we see that the sphere is the only compact example. Actually, more is true:
If a surface of constant positive Gauss curvature is a closed subset of K3,
then it is a sphere. This follows from a theorem of Myers (see Theorem
13.143 in Chapter 13).
4.6. Gauss Curvature Heuristics
The reader may be left wondering about the geometric meaning of the Gauss
curvature. We will learn more about the Gauss curvature in Section 9.9 and
much more about curvature in general in Chapter 13. For now, we will
simply pursue an informal understanding of the Gauss curvature.
4.6. Gauss Curvature Heuristics
185
Negative Curvature Zero Curvature Positive Curvature
Figure 4.5. Curvature bends geodesies
On a plane, geodesies are straight lines parametrized with constant
speed. If two insects start off in parallel directions and maintain a
policy of not turning either left or right, then they will travel on straight lines,
and if their speeds are the same, then the distance between them remains
constant. On a sphere or on any surface with positive curvature, the
situation is different. In this case, geodesies tend to curve toward each other.
Particles (or insects) moving along geodesies that start out near each other
and roughly parallel will bend toward each other if they travel at the same
speed. The second derivative of the distance between them will be negative.
For example, two airplanes traveling due north from the equator at constant
speed and altitude will be drawn closer and, if they continue, will eventually
meet at the north pole. It is the curvature of the earth that "pulls" them
together. On a surface of negative curvature, initially parallel motions along
geodesies will bend away from each other. The second derivative of the
distance between them will be positive. These three situations are depicted
in Figure 4.5. For a more precise formulation of these ideas it is best to
consider a parametrized family of curves, and we will do this in Chapter 13.
It should be mentioned that according to Einstein's theory, it is the
curvature of spacetime that accounts for those aspects of gravity (such as
tidal forces) that cannot be nullified by a choice of frame (accelerating frames
cause gravity-like effects even in a flat spacetime). For example, an initially
spherical cluster of particles in free fall near the earth will be deformed into
an egg shape. For a wonderful popular account of gravity as curvature, see
[Wh].
Another way to see the effects of curvature is by considering triangles
on surfaces whose sides are geodesic segments. These geodesic triangles are
affected by the Gauss curvature. For instance, consider the geodesic triangle
on the sphere in Figure 4.6. The angles shown are actually measured in the
tangent spaces at each point based on the tangents of the curves at the
endpoints of each segment. We have the following Gauss-Bonnet formula
©
186
4. Curves and Hypersurfaces in Euclidean Space
Figure 4.6. Curvature and triangle
involving the interior angles:
01+02 + /%=*+ / KdS,
JD
where D is the region interior to the geodesic triangle. This formula is
true for geodesic triangles on any surface M. For a sphere of radius a, this
becomes 0i + 02 + 03 — π + A/a2, where A is the area of the region interior
to the geodesic triangle. The fact that the sum of the interior angles for a
triangle in a plane is equal to π regardless of the area inside the triangle is
exactly due to the fact that a plane has zero Gauss curvature. If one starts
at ρ and moves counterclockwise around the triangle, then at the corners
one must turn through angles αχ, 0:2,^3. In terms of these turning angles,
the statement is
Υ"α; = 2π·~ / KdS.
ы\ Jd
By an argument that involves triangulating a surface, it can be shown that if
one integrates the Gauss curvature over a whole surface (without boundary)
McR3, then something amazing occurs. We obtain
/ KdS = 27rX(M).
JM
This result is called the Gauss-Bonnet theorem. Here χ(Μ) is a
topological invariant called the Euler characteristic of the surface and is equal
to 2 — 2<?, where g is the genus of the surface (see [Arm]). Thus while the
left hand side of the above equation involves the shape dependent curvature
if, the right hand side is a purely topological invariant! A presentation of a
very general version of the Gauss-Bonnet theorem may be found in [Poor]
(also see [Lee, Jeff]).
Problems
187
Problems
(1) Let 7 : / -► R2 be a regular plane curve. Let J : R2 -» R2 be the
rotation (я, у) \-> (—у, х). Show that the signed curvature function
IIY'WII3
determines 7 up to reparametrization and Euclidean motion.
(2) Let /, g : (a, 6) -» R be differentiable functions with f2 + g2 — 1. Let
to £ (a, 6) and suppose that /(to) — cos#o and g(to) = sin#o f°r some
00 £ R· Show that there is a unique continuous function θ : (α, b) —> R
such that #(to) = #0 and
/(t) = cos <9(t), p(t) = sin0(t) for t e (a,b).
(3) Let 7 : (a, 6) —> R2 be a regular plane curve.
(a) Given to € (a, 6) and #0 with
π „. хм = (cos<90,sin<90),
show that there is a unique continuous turning angle function
ΘΊ : (α, b) -> R such that θ (to) = #o and
Щ| = (cos07(f),sin07(t)) for t G (α,6).
(b) Show that θ'Ί(ί) = ||7;(ί)|| «2(*)» where «2 is as in Problem 1.
(4) Show that if K2 for a curve 7 : (a, 6) -> R2 is 1/r, then 7 parametrizes a
portion of a circle of radius r.
(5) Let 7 : R —» R3 be the elliptical helix given by t \-> (acos t, bsint, ct).
(a) Calculate the torsion and curvature of 7.
(b) Define a map F : R —> SO(3) by letting F have columns given by
the Prenet frame of 7. Show that F is a periodic parameterization
of a closed curve in SO(3).
(6) Calculate the curvature and torsion for the twisted cubic 7(4) :=
(t,t2,t3). Examine the behavior of the curvature, torsion, and Prenet
frame as t —> ±00.
(7) Find a unit speed parametrization of the catenary curve given by c(t) :—
(acosh(t/a), t). Revolve the resulting profile curve to obtain a canonical
parametrization of a catenoid and find the Gauss curvature in these
terms.
188
4. Curves and Hypersurfaces in Euclidean Space
(8) (Four vertex theorem) Show that the signed curvature function Κ2 for
a simple, closed plane curve is either constant or has at least two local
maxima and at least two local minima.
(9) If 7 : J —>· R3 is a regular space curve (not necessarily unit speed), then
show that B{t) = ^, r = ^f and к = Ь£р.
(10) With 7 as in Problem 1, show that 7" = (| ||7' ||) Τ + ||У ||2 «Ν.
(11) Let 7 : / —> Μ С R3 be a curve which is not necessarily unit speed and
suppose Μ is oriented by a unit normal field N. Show that кд — ή'Λ·
(12) Show that if a surface Μ С R3, either the image by an immersion of an
open domain in R2 or is the zero set of a real-valued function / (such
that df φ 0 on M) then it is orient able.
(13) Calculate the shape operator at a generic point on the cylinder {(x, y, z):
x2 + y2 = r2}.
(14) (Euler's formula) Let ρ be a point on a surface Μ in R3 and let и\,щ
be principal directions with corresponding principal curvatures kifa at
p. If и = (cos#) u\ + (sin0) г&2, show that k(u) = ki cos2 Θ + &2δίη20.
(15) Show that a point on a hypersurface in Rn is umbilic if and only if there
is a constant fco such that k(u) = ko for all и € TpM with \\u\\ = 1.
(16) Let Ζ be a nonvanishing (not necessarily unit) normal field on a surface
Μ С R3. If X and Υ are tangent fields such that Χ χ Υ = Ζ, then
show that
„ (Z,VxZxVYZ) (Z,VxZxY + XxVYZ)
К = -л and Я = — 5 -.
\\Zf 2||Z||3
(17) Find the Gauss curvature К at (x, г/, ζ) on the ellipsoid x2/a2 +y2/62 +
y2l<? = 1.
(18) Show that a surface in R3 is minimal if and only if there are orthogonal
asymptotic vectors at each point.
(19) Compute E, F, G, l,m,n as well as Η and К for the following surfaces:
(a) Paraboloid: (w, v) 1—> (гх, ν, au2 + bv2) with a, b > 0.
(b) Monkey Saddle: (u,v) 1—> (u,v,uz - 3uv2).
(c) Torus: (u,v) 1—> ((a + bcos v) cos u, (a + b cos u) sin u,b sin г?) with
α > Ь > 0.
(20) (Enneper's surface) Show that the following is a minimal but not 1-1
immersion:
/ ч / u3 2 v3 2 2 i\
x(u, г?) := ( и - — + uv', v + — - "nt nt - "'
Chapter 5
Lie Groups
One approach to geometry is to view it as the study of invariance and
symmetry. In our case, we are interested in studying symmetries of smooth
manifolds, Riemannian manifolds, symplectic manifolds, etc. The usual
way to deal with symmetry in mathematics is by the use of the notion of a
transformation group. The wonderful thing for us is that the groups that
arise in the study of geometric symmetries are often themselves smooth
manifolds. Such "group manifolds" are called Lie groups.
In physics, Lie groups play a big role in connection with physical
symmetries and conservation laws (Noether's theorem). Within physics,
perhaps the most celebrated role played by Lie groups is in particle physics
and gauge theory. In mathematics, Lie groups play a prominent role in
harmonic analysis (generalized Fourier theory), group representations,
differential equations, and in virtually every branch of geometry including
Riemannian geometry, Cartan geometry, algebraic geometry, Kahler geometry,
and symplectic geometry.
5.1. Definitions and Examples
Definition 5.1. A smooth manifold G is called a Lie group if it is a
group (abstract group) such that the multiplication map μ : G χ G -> G
and the inverse map inv : G —> G, given respectively by /i(<7,/b) = gh and
inv(g) = g-1, are C°° maps. If the group is abelian, we sometimes opt to
use the additive notation g + h for the group operation.
We will usually denote the identity element of any Lie group by the same
letter e. Exceptions include the case of matrix or linear groups where we
189
190
5. Lie Groups
use the letter 7 or id. The map inv : G -¥ G given by g н·» д l is called
inversion and is easily seen to be a diffeomorphism.
Example 5.2. R is a one-dimensional (abelian) Lie group, where the group
multiplication is the usual addition +. Similarly, any real or complex vector
space is a Lie group under vector addition.
Example 5.3. The circle S1 = {z € С : \z\2 = 1} is a 1-dimensional
(abelian) Lie group under complex multiplication. It is also traditional to
denote this group by U(l).
Example 5.4. Let R* = R\{0}, C* = C\{0} and H* = И\{0} (here Η is
the quaternion division ring discussed in detail later). Then, using
multiplication, R*, C, and H* are Lie groups. The Lie group H* is not abelian.
The group of all invertible real η χ η matrices is a Lie group denoted
GL(n,R). A global chart on GL(n,R) is given by the n2 functions aj,
where if A 6 GL(n,R) then xlj(A) is the ij-th entry of A. We study this
group and some of its subgroups below. Showing that GL(n,R) is a Lie
group is straightforward. Multiplication is clearly smooth. For the inversion
map one appeals to the usual formula for the inverse of a matrix, A~l =
adj(A)/det(A). Here adj(A) is the adjoint matrix (whose entries are the
cofactors). This shows that A~l depends smoothly on the entries of A.
Similarly, the group GL(n, C) of invertible η χ η complex matrices is a Lie
group.
Exercise 5.5. Let Я be a subgroup of G and consider the cosets gH, g eG.
Recall that G is the disjoint union of the cosets of Я. Show that if Я is
open, then so are all the cosets. Conclude that the complement Hc is also
open and hence Я is closed.
Theorem 5.6. If G is a connected Lie group and U is a neighborhood of
the identity element e, then U generates the group. In other words, every
element of g is a product of elements ofU.
Proof. First note that V = inv(17) Π U is an open neighborhood of the
identity with the property that inv(F) = V. We say that V is symmetric.
We show that V generates G. For any open W\ and W2 in G, the set
W1W2 = {W1W2 : w\ e W\ and W2 € W2} is an open set being a union of
the open sets \Jgew19^r^t Thus, in particular, the inductively defined sets
Vn = VVn-\ n = l,2,3,...,
are open. We have
e e V С V2 С · · · Vn С · · · .
5.1. Definitions and Examples
191
It is easy to check that each Vй is symmetric and so also is the union
oo
71=1
Moreover, V°° is not only closed under inversion, but also obviously closed
under multiplication. Thus V°° is an open subgroup. Prom Exercise 5.5,
V°° is also closed, and since G is connected, we obtain V°° = G. D
In general, the connected component of a Lie group G that contains
the identity is a Lie group denoted Go, and it is generated by any open
neighborhood of the identity. We call Go the identity component of G.
Definition 5.7. For a Lie group G and a fixed element jeG, the maps
Lg : G ->· G and Rg : G -» G are defined by
Lgx = gx for χ G G,
Rgx = xg for χ G G,
and are called left translation and right translation (by g) respectively.
It is easy to see that Lg and Rg are diffeomorphisms with L~l = Lg ι
and Rg1 = Rg ι.
If G and Я are Lie groups, then so is the product manifold GxH, where
multiplication is (<7i,hi) · (52,^2) = (ffi^uifo)- The Lie group G χ Я is
called the product Lie group. For example, the product group S1 χ Sl
is called the 2-torus group. More generally, the higher torus groups are
defined by Tn = S1 χ - - · χ S1 (η factors).
Definition 5.8. Let Я be an abstract subgroup of a Lie group G. If Я is
a Lie group such that the inclusion map Η ^ G is an immersion, then we
say that Я is a Lie subgroup of G.
Proposition 5.9. If Η is an abstract subgroup of a Lie group G that is also
a regular submanifold, then Η is a closed Lie subgroup.
Proof. The multiplication and inversion maps, Η χ Η —>· Η and Η —> Я,
are the restrictions of the multiplication and inversion maps on G, and since
Я is a regular submanifold, we obtain the needed smoothness of these maps.
The harder part is to show that Я is closed. So let xo G Я be arbitrary.
Let (U, x) be a single-slice chart adapted to Я whose domain contains e.
Let δ : G χ G —> G be the map δ (g\ ,32) = 9^92^ and choose an open set
V such that e € V С V С U. By continuity of the map δ we can find an
open neighborhood О of the identity element such that О х О С 5_1(V).
Now if {hi} is a sequence in Я converging to #o € Я, then х$1Ы —>> e and
Zq1/^ G О for all sufficiently large г. Since h"1/^ = (^o^1^) Ζο^υ we
192
5. Lie Groups
have that hj h% G V for sufficiently large z,j. For any sufficiently large fixed
j, we have
lim h~lhi = Η~λχο e V С U.
г-юо ^ J
Since /7 is the domain of a single-slice chart, U Π Η is closed in t/. Thus
since each h ■ lhi is in U Π Η, we see that h7lxo € /7 Π if С i? for all
sufficiently large j. This shows that xq € if, and since zo was arbitrary, we
are done. D
By a closed Lie subgroup we shall always mean one that is a regular
submanifold as in the previous theorem. It is a nontrivial fact that an
abstract subgroup of a Lie group that is also a closed subset is automatically
a closed Lie subgroup in this sense (see Theorem 5.81).
Example 5.10. S1 embedded as Sl χ {1} in the torus Sl χ S1 is a closed
subgroup.
Example 5.11. Let S1 be considered as the set of unit modulus complex
numbers. The image in the torus T2 = S1 χ Sl of the map R1 -> S1 χ Sl
given by t >-> (e*27ri, el27vat) is a Lie subgroup. This map is a homomorphism.
If α is a rational number, then the image is an embedded copy of S1 wrapped
around the torus several times depending on a. If a is irrational, then the
image is still a Lie subgroup but is now dense in T2.
The last example is important since it shows that a Lie subgroup might
actually be a dense subset of the containing Lie group.
5.2. Linear Lie Groups
Let V be an η-dimensional vector space over F, where F = R or С The
space L(V, V) of linear maps from V to V is a vector space and therefore
a smooth manifold. A global chart for L(V,V) may be obtained by first
choosing a basis for V and then defining n2 functions {x)}i<i.j<n by the
rule that if Л € L(V, V), then хг3(А) is the ij-th entry of the matrix that
represents A with respect to the chosen basis. If the field is R, then these
are the coordinate functions of a global chart. If the field is C, then we
simply take the real and imaginary parts of the xxz and thereby obtain 2n2
coordinate functions. The various choices of basis give compatible charts,
and the reader may check that these are charts from the smooth structure
that L(V, V) has by virtue of being a finite-dimensional vector space.
The determinant of an element A e L(V, V) is given as the determinant
of any matrix which represents A with respect to some basis. The group
GL(V) of all linear automorphisms of V is an open submanifold of L(V, V
given by the condition of nonvanishing determinant, and the restrictions of
5.2. Linear Lie Groups
193
the coordinate functions just introduced provide a global chart for GL(V).
Let GL(n, F) denote the group of invertible matrices with entries from F.
We obtain an isomorphism of GL(V) with the matrix group GL(n,F) by
choosing a basis and then simply sending each element of GL(V) to its
matrix representative with respect to that basis. If we write m&t(A) for
the matrix that represents A 6 GL(V) with respect to a fixed basis, then
A -* mat (A) is a group isomorphism and is clearly smooth. It follows that
GL(V) is a Lie group, and for each choice of basis we have an isomorphism
of Lie groups GL(V) — GL(n,F). In practice it is common to work with the
matrix group GL(n, F) and its subgroups. The Lie group GL(V) is called the
general linear group of V and is also denoted GL(V, F) when we want to
make the field apparent. The matrix group GL(n,F) is also referred to as
a general linear (matrix) group and is often identified with GL(Fn). More
specifically, GL(n,R) is called the real general linear group, and GL(n, C)
is called the complex general linear group. Lie groups that are subgroups of
GL(V) for some vector space V are referred to as linear Lie groups and
are often realized as matrix subgroups of GL(n, F) for some n.
Definition 5.12. Let V be an η-dimensional vector space over the field F
which we take to be either R or С Then the group SL(V) defined by
SL(V) = SL(V,F) := {A e GL(V) : det(A) - 1}
is called the special linear group for V.
A bilinear form β : V x V —» F on an F-vector space V is called
nondegenerate if the maps V -¥ V* given by /?я : ν ι-> β (ν, ·) and
βΐ : ν и-» /3(· , υ), are both linear isomorphisms. If V is finite-dimensional,
then Дя is an isomorphism if and only if βι is an isomorphism and then
β is nondegenerate provided it has the property that if β{ν,νύ) = 0 for all
w £ V, then ν = 0.
Definition 5.13. A (real) scalar product on a (real) finite-dimensional
vector space V is a nondegenerate symmetric bilinear form β : V χ V —> R.
A (real) scalar product space is a pair (V,/3) where V is a real vector
space and β is a scalar product. (As usual we refer to V itself as the scalar
product space when β is given)
Definition 5.14. Let V be a complex vector space. An R-bilinear map β :
V χ V -> С that satisfies β(αν, w) — αβ(ν, w) and β(ν, aw) = αβ(ν, w) for all
a £ С and v, w € V is called a sesquilinear form. If also β(ν, w) = β(ιν, ν),
we call β a Hermitian form. If a Hermitian form is nondegenerate, we
call it a Hermitian scalar product, and then (V, β) a Hermitian scalar
product space.
194
5. Lie Groups
The sesquilinear conditions imposed are described by saying that β is
to be conjugate linear in the first slot and linear in the second slot. Non-
degeneracy for a sesquilinear form is defined as for bilinear forms. Many
authors define sesquilinear and Hermitian forms to be conjugate linear in
the second slot and linear in the first. Obviously, β(ν,ν) is always real for a
Hermitian form.
Definition 5.15. Let β be a (real) scalar product or Hermitian scalar
product on a vector space V. Then
(i) β is positive (resp. negative) definite if β(ν,ν) > 0 (resp. β(ν,ν)
< 0) for all ν e V and β(ν, υ) = 0 => υ = 0;
(ii) β is positive (resp. negative) semidefinite if β(ν>ν) > 0 (resp.
β(ν,ν) <0) for all ν G V.
What is called an inner product (a term we have already used) is a
positive definite scalar product (or positive definite Hermitian scalar product
in the complex case). In this book the term "inner product" always implies
positive definiteness. Let us generalize the notion of orthonormal basis for
an inner product space to include indefinite scalar product spaces. A basis
(ei,..., en) for a scalar product space (V, β) is called an orthonormal basis
if /3(e*,ej) = 0 when i ^ j, and β(βΐ,βι) = ±1 for all i. An orthonormal
basis always exists for a finite-dimensional scalar product space.
Definition 5.16. Let V be an η-dimensional vector space over the field F
which we take to be either R or С If β is a bilinear form or sesquilinear
form on V, then Aut(V, β) is the subgroup of GL(V) defined by
Aut(V, β) := {A e GL(V) : β{Αυ, Aw) - β(υ, w) for all v, w € V}.
If β is a scalar product, then the elements of Aut(V, β) are called isome-
tries of V.
Theorem 5.17. SL(V) is a closed Lie subgroup o/GL(V). If β is a
bilinear or sesquilinear form as above, then Aut(V,/3) and 5Aut(V,/?) :~
Aut(V, β) nSL(V) are closed Lie subgroups o/GL(V). (The form β is most
often taken to be nondegenerate.)
Proof. It is easy to check that the sets in question are subgroups. They are
clearly closed. For example, Aut(V,/3) = f)vwFViW, where
FVtW := {A e GL(V) : β(Αν,Ανυ) = β{ν,νυ)}.
The fact that they are Lie subgroups follows from Theorem 5.81 below.
However, as we shall see, most of the specific cases arising from various
choices of β can be proved to be Lie groups by other means. That they
are Lie subgroups follows from Proposition 5.9 once we show that they axe
5.2. Linear Lie Groups
195
regular submanifolds of the appropriate group GL(V,F). We will return to
this later, once we have introduced another powerful theorem that will allow
us to verify this without the use of Theorem 5.81. D
Let dim V = n. After choosing a basis, SL(V) gives the matrix version
SL(n,F) := {A e Mnxn(¥) : det A - 1}. Notice that even when F = C,
it may be that β is only required to be R-linear. Depending on whether
F = С or R and on the nature of /3, the notation for the linear groups takes
on special conventional forms introduced below. When choosing a basis in
order to represent one of the groups associated to a form β in a matrix
version, it is usually the case that one uses a basis under which the matrix
that represents β takes on a canonical form.
Example 5.18 (The (semi) orthogonal groups). Let (V, β) be a real scalar
product space. In this case we write Aut(V,/3) as 0(V,/3) and refer to
it as the semiorthogonal group associated to β. With respect to an
appropriately ordered orthonormal basis, β is represented by a diagonal
matrix of the form
Чрл —
1
0
0
0
0
0
1
-1
0
0
0
0
0
-1
where there are ρ ones and q minus ones down the diagonal. The group of
matrices arising from 0(V, β) with such a choice of basis is denoted 0(p, q)
and consists exactly of the real matrices Q satisfying Qrfo^Q1 — ηΡι4. These
groups are called the semiorthogonal matrix groups. With such an
orthonormal choice of basis as above, the bilinear form (scalar product) is given as
a canonical form on Rn where (p + q = n):
г—1 i=p+l
and we have the alternative description
0(p,g) = {Q e GL{n) : (Qx,Qy) = (x,y) for all x}y e Rn}.
If β is positive definite, we then have q = 0, and 0(V, β) is referred to as a
real orthogonal group. We write 0(n, 0) as O(n) and refer to it as the
real orthogonal (matrix) group; Q e 0(n) <=^> Q*Q = I.
196
5. Lie Groups
Example 5.19. There are also complex orthogonal groups (not to be
confused with unitary groups). In matrix representation, we have 0(n, C) :=
{QeGL^QiQ'Q^J}.
Example 5.20. Let (V, β) be a Hermitian scalar product space. In this
case, we write Aut(V,/?,C) as C/(V,/3) and refer to it as the semiunitary
group associated to /?. If /? is positive definite, then we call it a unitary
group. Again we may choose a basis for V such that β is represented by
the Hermitian form on Cn given by
ρ p+q=n
(x,y):= jVy*- Σ *V·
г—1 г— p+1
We then obtain the semiunitary matrix group
U(p,9) = {Ae GL(n,C) : (Ax, Ay) = (x,y) for all x,y e Rn}.
We write U(n, 0) as U(n) and refer to it as the unitary (matrix) group. In
particular, U(l) = S1 - {z e С : \z\ = 1}.
Definition 5.21. For (V, β) a real scalar product space, we have the special
orthogonal group of (V, β) given by
SO(V,/?) = 0(V,/?)nSL(V,R).
For (V,/3) a Hermitian scalar product space, we have the special unitary
group of (V, β) given by
SU(V, β) = U(V, β) П SL(V, С).
Definition 5.22. The group of η χ η complex matrices of determinant
one is the complex special linear matrix group SL(n,C). We also
have the similarly defined real special linear group SL(n,M). The
special orthogonal and special semiorthogonal matrix groups, SO(n) and
SO(p, <?), are the matrix groups defined by SO(n) = 0(n) Π SL(n,R) and
SO(p,#) = 0(p,g)nSL(ra,R). The special unitary and special
semiunitary matrix groups SU(n) and SU(p, q) are defined similarly.
The group SO(3) is the familiar matrix representation of the proper
rotation group of Euclidean space and plays a prominent role in classical
physics. Here "proper" refers to the fact that SO(3) does not contain any
reflections. In the problems we ask the reader to show that SO(3) is the
connected component of the identity in 0(3).
Exercise 5.23. Show that SU(2) is simply connected while SO(3) is not.
Example 5.24 (Symplectic groups). We will describe both the real and
the complex symplectic groups. Suppose that β is a nondegenerate skew-
symmetric C-bilinear (resp. R-bilinear) form on a 2n-dimensional complex
5.2. Linear Lie Groups
197
(resp. real) vector space V. The group Aut(V, β) is called the
complex (resp. real) symplectic group and is denoted by Sp(V, C) (resp.
Sp(V,R)). There exists a basis {/г} for V such that β is represented in the
canonical form by
η η
(ν, ™) = ς vi™n+i - ΣvU+Ju,j·
г=1 J=l
The symplectic matrix groups are given by
Sp(2n,C) : ={A € M2„x2n(C) : (Av,Aw) = (v,w)},
Sp(2n,R) := {A € M2nx2n(R) : (Av,Aw) = (v,w)},
where (v, w) is given as above.
Exercise 5.25. For F = С or E, show that A G Sp(2n,F) if and only if
A* J A — J, where
Much of the above can be generalized somewhat more. Recall that the
algebra of quaternions Η is a copy of R4 endowed with a multiplication
described as follows: First let a generic elements of R4 be denoted by χ =
(ж0,;*;1,я2,я3), у = (у0,у1,у2,у3), etc. Thus we are using {0,1,2,3} as
our index set. Let the standard basis be denoted by l,ij,k. We define a
multiplication by taking these basis elements as generators and insisting on
the following relations:
•2 ·2 ι 2 t
ι -J =kz = -1,
U = -ji = К
jk = -kj = i,
ki= —ik = j.
Of course, Η is a vector space over R since it is just R4 with some extra
structure. As a ring, Η is a division algebra which is very much like a field,
lacking only the property of commutativity. In particular, we shall see that
every nonzero element of Η has a multiplicative inverse. Elements of the
form αϊ for α € R are identified with the corresponding real numbers, and
such quaternions are called real quaternions. By analogy with complex
numbers, quaternions of the form xl\ + x2} + x3k are called imaginary
quaternions. For a given quaternion χ = ^°l+a:1i+^2j+^3k, the quaternion
x4 + x2} + x3k is called the imaginary part of x, and x°l =x° is called the
real part of x. We also have a conjugation defined by
χ \-^ χ := x°l — xli - x2} — #3k.
198
5. Lie Groups
Notice that xx = xx is real and equal to (я0) + (я1) + (я2) + (я3) . We
ι 19
denote the positive square root of this by \x\ so that xx = \x\ .
Exercise 5.26. Verify the following for x, у Ε Η and α, b G R:
ax + by = ax + by, (x) = x,
\xy\ = \x\\y\, \x\ = |s|,
xy = yx.
Now we can write down the inverse of a nonzero xGH:
Notice the strong analogy with complex number arithmetic.
Example 5.27. The set of unit quaternions is /7(1,H) := {\x\ = 1}. This
set is closed under multiplication. As a manifold it is (diffeomorphic to)
S3. With quaternionic multiplication, S3 = /7(1, H) is a compact Lie group.
Compare this to Example 5.3 where we saw that /7(1, C) = S1. For the
future, we unify things by letting /7(1, R) := Ъч = 5° С R. In other words,
we take the 0-sphere to be the subset {—1,1} with its natural structure as
a multiplicative group.
J7(1JH) = 53>
t/(l,C) = 51,
C7(1,R):=Z2 = 5°,
Exercise 5.28. Prove the assertions in the last example.
We now consider the η-fold product Hn, which, as a real vector space
(and a smooth manifold), is R4n. However, let us think of elements of Hn
as column vectors with quaternion entries. We want to treat Hn as a vector
space over Η with addition defined just as for Rn and Cn, but since Η is not
commutative, we are not properly dealing with a vector space. In particular,
we should decide whether scalars should multiply column vectors on the right
or on the left. We choose to multiply on the right, and this could take some
getting used to, but there is a good reason for our choice. This puts us into
the category of right Η-modules were elements of И are the "scalars". The
reader should have no trouble catching on, and so we do not make formal
definitions at this time (but see Appendix D). For v, w € Hn and a, b Ε Η,
we have
v(a + b) = va + vb
(v + w)a — va + wa
(va) b = ν (ab).
5.2. Linear Lie Groups
199
A map A : Ип -> Ип is said to be Η-linear if A(va) = A{v)a for all ν еШп
and α € И. There is no problem with doing matrix algebra with matrices
with quaternion entries, as long as one respects the noncommutativity of H.
For example, if A = {a1·) and В = (Ъ1-) are matrices with quaternion entries,
then writing С = AB we have
but we cannot expect that Σα\^ = Σ^α1· F°r апУ А = (αι)> the map
jjri _^ jjn define(j by v ,_> ^ is Η-linear since Α (να) = (Αν) a.
Definition 5.29. The set of all m χ η matrices with quaternion entries is
denoted Mmxn(H). The subset GL(n,H) is defined as the set of all Q e
Mmxn(H) such that the map ν ь-> Qv is a bijection.
We will now see that GL(n, И) is a Lie group isomorphic to a subgroup of
GL(2n, C). First we define a map ι: С2—» Η as follows: For (21,22) € С with
z\ = x° + xli and z2 = x2 + x\ we let t(zl, z2) — (x° + xH) + [x2 + x3i)j
where on the right hand side we interpret i as a quaternion. Note that
(x° + xli) + (x2 + x3i) j = x° + x1! + #2j + x3k . It is easily shown that this
map is an R-linear bijection, and we use this map to identify C2 with H.
Another way of looking at this is that we identify С with the span of 1 and
i in И and then every quaternion has a unique representation as z1 + 22j
for zl} z2 € С С Н. We extend this idea to square quaternionic matrices;
we can write every Q e Mmxn(W) in the form A + Bj for А, В G Mmxn(C)
in a unique way. This representation makes it clear that Mmxn(H) has a
natural complex vector space structure, where the scalar multiplication is
z(A + Bj) =zA + zBy Direct computation shows that
(A + Bj) (C + Dj) = (AC - BD) + (AD + ВС)}
for A + Bj еМтхп(Ш) and С + D] еМпх^(Ш), where we have used the fact
that for Q G Mmxn(C) we have Qj = jQ. From this it is not hard to show
that the map tfmXn : MmXn(H) -> M2mx2n(C) given by
timxn : A + Bj
is an injective R-linear map which respects matrix multiplication and thus is
an R-algebra isomorphism onto its image. We may identify Mmxn(H) with
the subspace of M2mx2n(C) consisting of all matrices of the form (^в д),
where A,Be CmXn. In particular, if m = n, then we obtain an injective R-
linear algebra homomorphism ϋηχη : МпХп(Ш1) —> M2nX2n(C), and thus the
image of this map in M2nX2n(C) is another realization of the matrix algebra
Mn п(И). If we specialize to the case of η = 1, we get a realization of Η as
the set of all 2 χ 2 complex matrices of the form (Jw ™). This set of matrices
U*
200
5. Lie Groups
is closed under multiplication and forms an algebra over the field R. Let us
denote this algebra of matrices by the symbol 7£4 since it is diffeomorphic
to Η = R4. We now have an algebra isomorphism ϋ : Ш —> TZ4 under which
the quaternions 1, i, j and к correspond to the matrices
(ί!)·(ί-)■(-. i)-(!i)
respectively. Since Η is a division algebra, each of its nonzero elements has
a multiplicative inverse. Thus 7£4 must contain the matrix inverse of each
of its nonzero elements. This can be seen directly:
(z w \~ _ 1 ( ζ —w \
-w ζ J ~~ |z|2 + |w|2 \w г )'
Consider again the group of unit quaternions £7(1, H). We have already
seen that as a smooth manifold, £7(1, H) is 53. However, under the
isomorphism Η —t 7£4 С М2х2(С) just mentioned, 17(1, H) manifests itself as
SU(2). Thus we obtain a smooth map £7(1, H) —> SU(2) that is a group
isomorphism. We record this as a proposition:
Proposition 5.30. The map f7(l,H) -> SU(2) given by
where χ = x° + x1! + x2j + ж3к, ζ — x° + xli and w — x2 + xH, is a group
isomorphism. Thus Ss = 17(1, H) ~ SU(2).
Proof. The first equality has already been established. Notice that we then
have
and so χ G £7(1, H) if and only if (Jw ^) has determinant one. But such
matrices account for all elements of SU(2) (verify this). We leave it to the
reader to check that the map £7(1, H) —► SU(2) is a group isomorphism. D
Exercise 5.31. Show that Q G GL(n,H) if and only if det(tfnxn(Q)) Φ 0.
The set of all elements of GL(2n, C) which are of the form (^B ^) is
a subgroup of GL(2n, C) and in fact a Lie group. Using the last exercise,
we see that we may identify GL(n, Ш) as a Lie group with this subgroup of
GL(2n,C). We want to find a quaternionic analogue of U(n,C), and so we
define b : Шп χ ΕΓ -> Η by
b(v,w) = vlw.
5.3. Lie Group Homomorphisms
201
Explicitly, if
ν =
and w =
iir
w"
then
b(v, w) = [ v1 · · · νη ]
iir
w"
= £VV.
Note that 6 is obviously R-bilinear. But if a E H, then we have Ь(ш, it;) =
6(г>,ги)а and b(v,wa) = &(г>,ги)а. Notice that we consistently use right
multiplication by quaternionic scalars. Thus b is the quaternionic analogue
of an Hermitian scalar product.
Definition 5.32. We define U(n,H):
U(n, M) := {Q e GL(n, И) : b{Qv, Qw) = b{v, w) for all w, w e W1}
U(n, H) is called the quaternionic unitary group.
The group U(n, H) is sometimes called the symplectic group and is
denoted Sp(n), but we will avoid this since we want no confusion with the
symplectic groups we have already defined. The group U(n,H) is in fact
a Lie group (Theorem 5.17 generalizes to the quaternionic setting). The
image of U(n, H) in M2nx2n(C) under the map ϊ?ηχη is denoted USp(2n, C).
Since it is easily established that #nxnlu(n,H) ^s a S1011? homomorphism, the
image USp(2n, C) is a subgroup of GL(2n, C).
Exercise 5.33. Show that ϋηχη(Αι) = ii?nxn(i)) . Show that USp(2n, C)
is a Lie subgroup of GL(2n, C).
Exercise 5.34. Show that USp(2n,C) = U(2n) П Sp(2n5C). Hint: Show
that
#пхп(Мпхп(Ш)) = {AE GL(2n,C) : 3ΑΓλ = A} ,
where J = ( _°id ^). Next show that if A e U(2n), then JAJ 1 = A if and
only if A* J A = J.
5.3. Lie Group Homomorphisms
Definition 5.35. Let G and Η be Lie groups. A smooth map / : G —ϊ Η
that is a group homomorphism is called a Lie group homomorphism. A
Lie group homomorphism is called a Lie group isomorphism in case it has
an inverse that is also a Lie group homomorphism. A Lie group isomorphism
G -> G is called a Lie group automorphism of G.
202
5. Lie Groups
If / : G —> Η is a Lie group homomorphism, then by definition /(3152) =
/(91)1(92) for all pi, 32 £ G, and it follows that /(e) = e and also that
f(9-1) = f(9)-1foT*llg€G.
Example 5.36. The inclusion SO(n,R) «->· GL(n,R) is a Lie group
homomorphism.
Example 5.37. The circle S1 С С is a Lie group under complex
multiplication and the map
ζ = егв ι-»·
cos(0)
-sin(0)
0
sin(0) 0
cos(0) 0
0 In-2
is a Lie group homomorphism of S1 into SO(n).
Example 5.38. The map 17(1, H) -> SU(2) of Proposition 5.30 is a Lie
group isomorphism.
Example 5.39. The conjugation map Cg : G —> G given by χ н* дхд~1
is a Lie group automorphism. Note that Cg = Lg о Rg-i.
Proposition 5.40. Let f\:G-*H and /2 : G —> Η be Lie group homo-
morphisms that agree in a neighborhood of the identity. If G is connected,
then jΐ = /2.
Proof. By Theorem 5.6, any g € G is a product of elements in the set on
which /1 and /2 agree, so the homomorphism property forces /1 = /2. □
Exercise 5.41. Show that the multiplication map μ : G χ G -> G has
tangent map at (e,e) € G χ G given as T(e^(v,w) — ν + w. Recall that
we identify Г(в|в)(С x G) with TeG χ ГеС.
Exercise 5.42. GL(n, R) is an open subset of the vector space of all η χ η
matrices Mnxn(R). Using the natural identification of reGL(n,R) with
Mnxn(R), show that
TeCg(x) = gxg~l,
where g e GL(n,R) and ж € MnXn(R).
Example 5.43. The map t И» elt is a Lie group homomorphism from R to
Definition 5.44. A Lie group homomorphism from the additive group R
into a Lie group is called a one-parameter subgroup. (Note that despite
the use of the word "subgroup", a one-parameter subgroup is actually a
map.)
5.3. Lie Group Homomorphisxtis
203
Example 5.45. We have seen that the torus S1 x S1 is a Lie group under
multiplication given by (е'т^ег01)(егТ2,ег*2) - (e*(n+r2)jei(0i+02)). Every
homomorphism of R into Sl χ S1y that is, every one-parameter subgroup of
S1 x S\ is of the form t \-> (et<M, etbt) for some pair of real numbers a, b Ε R.
Example 5.46. The map R : R -> SO(3) given by
cosi — sini 0
t ь* | sin ί cos £ 0
0 0 1
is a one-parameter subgroup. Also, the map
cos t — sin t 0
t н> | sini cosi 0
0 0 e*
is a one-parameter subgroup of GL(3).
Recall that an nxn complex matrix A is called Hermitian (resp. skew-
Hermitian) if A* — A (resp* A* — —-A). Let su(2) denote the vector space
of skew-Hermitian matrices with zero trace. We will later identify su(2) as
the "Lie algebra" ofSU(2).
Example 5.47. Given g G SU(2), we define the map Adp : 5u(2) -> su(2)
by Adg : χ ι-» gxg~l. The skew-Hermitian matrices of zero trace can be
identified with R3 by using the following matrices as a basis:
These are just —г times the Pauli matrices σχ,σ2,σ3, and so the
correspondence su(2) —> R3 is given by — χχσ\ — yiai — iza% н> (х, у, ζ). Under
this correspondence, the inner product on R3 becomes the inner product
{A,B) = itrace(AB') = -±trace(AB). But then
(AdgA,AdgB) = --traceigAg^gBg-1)
= -- trace (AB) - (A,B).
So, Adp can be thought of as an element of 0(3). More is true; Adp acts
as an element of SO(3), and the map g н> Adp is then a homomorphism
from SU(2) to SO(su(2)) ^ SO(3). This is a special case of the adjoint map
studied later. (This example is related to the notion of "spin". For more,
see the online supplement.)
204
5. Lie Groups
Definition 5.48. If a Lie group homomorphism ρ : G —> G is also a covering
map then we say that G is a covering group and ρ is a covering
homomorphism. If G is simply connected, then G (resp. p) is called the universal
covering group (resp. universal covering homomorphism) of G.
Exercise 5.49. Show that if ρ : Μ —> G is a smooth covering map and
G is a Lie group, then Μ can be given a unique Lie group structure such
that ρ becomes a covering homomorphism. (You may assume that Μ is
paracompact.)
Example 5.50. The group Mob of Mobius transformations of the complex
plane given by TA : ζ н> §*±} for A = (*bd) Ε SL(2,C) can be given the
structure of a Lie group. The map ρ : SL(2, C) —> Mob given by ρ : А м> Та
is onto but not injective. In fact, it is a (two fold) covering homomorphism.
When do two elements of SL(2, C) map to the same element of Mob?
5.4. Lie Algebras and Exponential Maps
Definition 5.51. A vector field Χ Ε X(G) is called left invariant if and
only if {Lg)*X = X for all g Ε G. A vector field X Ε X{G) is called right
invariant if and only if (J?5)*X = X for all g Ε G. The set of left invariant
(resp. right invariant) vector fields is denoted XL(G) (resp. XR(G)).
Recall that by definition (Lg)*X = TLg ο Χ ο L~l, and so left invariance
means that TLgoXoL~l = X or that given any χ Ε G we have TxLg-X(x) =
X{gx) for all g Ε G. Thus Χ Ε X(G) is left invariant if and only if the
following diagram commutes for every jEG:
TLg
TG *TG
G—-+G
There is a similar diagram for right invariance.
Lemma 5.52. XL(G) is closed under the Lie bracket operation.
Proof. Suppose that Χ,Υ Ε XL(G). Then by Proposition 2.84 we have
{L9)*[X% Y] = [L9*X} L9*Y] = [X, Y]. D
Given a vector ν Ε TeGr we can define a smooth left (resp. right)
invariant vector field Lv (resp. Rv) such that Lv(e) = ν (resp. R°(e) = v)
by the simple prescription
Lv{g) =TLgv (resp. Rv{g) = TRg · v).
5.4. Lie Algebras and Exponential Maps
205
A bit more precisely, Lv(g) = Te (Lg) - v. The proof that this prescription
gives smooth invariant vector fields is left to the reader (see Problem 7).
Given a vector in TeG there are various notations for denoting the
corresponding left (or right) invariant vector field, and we shall have occasion
to use some different notation later on. We will also write L(v) for Lv and
R(v) for Rv. The map ν ь-> L(v) (resp. ν н> R{v)) is a linear isomorphism
from TeG onto XL{G) (resp. XR(G)):
Exercise 5.53. Show that ν »-» Lv gives a linear isomorphism TeG — XL(G).
Similarly, TeG ^ XR{G) by ν н> R(v).
We now restrict attention to the left invariant fields but keep in mind
that essentially all of what we say for this case has analogies in the right
invariant case. We will discover a conduit (the adjoint map) between the
two cases. The linear isomorphism TeG = XL(G) just discovered shows that
XL(G) is, in fact, a vector space of finite dimension equal to the dimension
of G. Prom this and Lemma 5.52 we immediately obtain the following:
Proposition 5.54. If G is α Lie group of dimension n, then XL(G) is on
η-dimensional Lie algebra under the bracket of vector fields (see Definition
2.75J.
Using the isomorphism TeG = XL(G), we can transfer the Lie algebra
structure to TeG. This is the content of the following:
Definition 5.55. For a Lie group G, define the bracket of any two elements
ti,weTeG by
hici]:-[^L»](e).
With this bracket, the vector space TeG becomes a Lie algebra (see
Definition 2.78), and so we now have two Lie algebras, XL(G) and TeG,
which are isomorphic by construction. The abstract Lie algebra isomorphic
to either/both of them is often referred to as the Lie algebra of the Lie
group G and denoted variously by £(G) or g. Of course, we are implying
that £(#) is denoted f) and £{K) by Ϊ, etc. In some computations we will
have to use a specific realization of g. Our default convention will be that
β = £(G) := TeG with the bracket defined above.
Definition 5.56. Given a Lie algebra g, we can associate to every basis
vi?..., vn for g, the structure constants c^ which are defined by
\vuVj] = Σc*3Vk for l - {Л>к < n'
к
206
5. Lie Groups
It follows from the skew symmetry of the Lie bracket and the Jacobi identity
that the structure constants satisfy
0 cij = -^
(5.1)
u) Efe 4^i + <&4r + 44* ~ °-
The structure constants characterize the Lie algebra, and structure constants
axe sometimes used to actually define a Lie algebra once a basis is chosen.
We will meet the structure constants again later.
Let α and b be Lie algebras. For (ai,bi) and (a2,&2) elements of the
vector space α χ b, define
[(αϊ, 6ι), (α2, b2)] := ([ab a2], [bu Ь2])·
With this bracket, α χ b is a Lie algebra called the Lie algebra product
of α and b. Recall the definition of an ideal in a Lie algebra (Definition
2,79). The subspaces α x {0} and {0} χ b are ideals in α xb that are clearly
isomorphic to α and b respectively. We often identify α with α χ {0} and b
with {0} χ b.
Exercise 5.57. Show that if G and Η are Lie groups, then the Lie algebra
g χ ij is (up to identifications) the Lie algebra of G χ #.
Definition 5.58. Given two Lie algebras over a field F, say (a, [,]a) and
(b, [,]ь), an F-linear map σ is called a Lie algebra homomorphism if and
only if
a([v,w]a) = [av}aw]b
for all и, ги G a. A Lie algebra isomorphism is defined in the obvious way. A
Lie algebra isomorphism q —> q is called an automorphism of 0.
It is not hard to show that the set of all automorphisms of g, denoted
Aut(fl), forms a Lie group (actually a Lie subgroup of GL(g)).
Let V be a finite-dimensional real vector space. Then GL(Y) is an open
subset of the linear space L(V, V), and we identify the tangent bundle of
GL(V) with GL(V) χ L(V,V) (recall Definition 2.58). The tangent space
at A e GL(V) is then {A} x L(V3V). Now the Lie algebra is r/GL(V),
and it has a Lie algebra structure derived from the Lie algebra structure on
XL(GL(V)). The natural isomorphism of T/GL(V) with L(V, V) puts a Lie
algebra structure on L(V, V). We now show that the resulting bracket on
L(V, V) is just the commutator bracket given by [A, B] := А о В — Β ο Α.
In the following discussion we let X denote the left invariant vector field
corresponding to X £ Z,(V, V) ^ T/(GL(V)). By definition we have [A, B] =
[A, B]. We will need some simple results from the following easy exercises.
5.4. Lie Algebras and Exponential Maps
207
Exercise 5.59. For a fixed A G GL(V), the map LA : GL{V) -¥ GL(V)
given by A i-> AoB has tangent map given by (A, X) »->· (AoB, ΑοΧ) where
(A,X) G GL(V) χ L(V, V) ^ T{GL(V)). Show that if X denotes the left
invariant vector field corresponding to X G L(V, V), then X(A) = (A, AX).
We are going to consider functions on GL(V) that are restrictions of
linear functionals on the vector space L(V,V). We will not notationally
distinguish the functional from its restriction. If / is such a linear function
and X G L(V, V), let ftX be given by ftx{A) := f(A ο Χ). It is easy to
check that ftx is also a linear functional, and so for Υ G L(V,V) we also
have (/,x) y. It is clear that (/,χ) y = /,ΥοΧ (notice the reversal of order).
Exercise 5.60. Show that the map L(V, V) -► (L(V,V))* given ЬуХи
/д is linear over R.
Exercise 5.61. Let X be the left invariant field corresponding to X G
L(V, V) as above. If / is the restriction to GL(V) of a linear functional on
L(V, V) as above, then Xf - fx. Solution/Hint: (Xf)(A) = d/|A (Хл)
f(AoX).
Recall that if one picks a basis for V, then we obtain a global chart on
GL(V) which is given by coordinate functions x%3 defined by letting хг3(А) be
the ij-th entry of the matrix of A with respect to the chosen basis. These
coordinate functions are restrictions of linear functions on L(V, V). Using
this we see that if /,χ = fy for all linear /, then f(X) = f,x{I) = /,yC0 ""
f(Y) for all linear / and this in turn implies X = Y.
Proposition 5.62. The Lie algebra bracket on L(V, V) induced by the
isomorphisms Xl{GL(V)) ~ TjGL(V) = L(V, V) is the commutator bracket
[X,Y]:=XoY-YoX.
Proof. Let X,Y G L(V,V) and let [X,Y] denote the bracket induced on
L(V, V). Then for any linear / we have
/,[x,y] = [ВД/ = [X,Y]f = XYf -YXf = X (M - Υ (/,x)
= (MtX ~~ (/^),y = f,*°Y " /,^οΑ- = /,(ΧοΥ-ΚοΧ).
Since / was an arbitrary linear functional, we conclude that [-Х",У] =
ХоУ-YoX. D
A choice of basis gives the algebra isomorphism L(V, V) = MnX7l(R)
where η = dim(V). This isomorphism preserves brackets and restricts to
a group isomorphism GL(V) = GL(n, R). It is now easy to arrive at the
following corollary.
208
5. Lie Groups
Corollary 5.63. The linear isomorphism o/gl(n,R) = TiGL(n, R) with
Mnxn(M) induces a Lie algebra structure on Mnxn(M) such that the bracket
is given by [A, B] := AB — В A.
Prom now on we follow the practice of identifying the Lie algebra fll(V) of
GL(V) with the commutator Lie algebra L(V, V). Similarly for the matrix
group, the Lie algebra of GL(n, R) is taken to be Mnxn(R) with commutator
bracket.
If G С GL(n) is some matrix group, then TjG may be identified with
a linear subspace of Mnxn. This linear subspace will be closed under the
commutator bracket, and so we actually have an identification of Lie alge*
bras: g is identified with a subspace of Mnxn. It is often the case that G
is defined by some matrix equation or equations. By differentiating these
equations we find the defining equations for g (as a subspace of Mnxn), We
first prove a general result for Lie algebras of closed subgroups, and then we
apply this to some matrix groups. Recall that if N is a submanifold of Μ
and t : N «-» Μ is the inclusion map, then we identify TpN with Tpl(TpN)
and Tpl is an inclusion. If Я is a closed Lie subgroup of a Lie group G, and
ν € f) = Te#, then ν corresponds to a left invariant vector field on G which
is obtained by using the left translation in G. But ν also corresponds to a
left invariant vector field on Η obtained from left translations in H. The
notation we have been using so far is not sensitive to this distinction, so let
us introduce an alternative notation.
Notation 5.64. For a Lie group G, we have the alternative notation v°
for the left invariant vector field whose value at e is v. If Η is a closed Lie
subgroup of G and υ € Tetf, then vG e XL(G) while vH G XL(H).
Proposition 5.65. Let Η be a closed Lie subgroup of a Lie group G. Let
XL{H) := {X e XL(G) : X(e) e TeH}.
Then the restrictions of elements of XL(H) to the submanifold Η are the
elements ofXL(H). This induces an isomorphism of Lie algebras of vector
fields ЩН) ~ XL(H). For v,w e \), we have
and so ij is a Lie subalgebra of q; the bracket on ij is the same as that
inherited from fl.
Proof. The Lie bracket on ij = TeH is given by [v, w] := [vH', и?я](е). Notice
that if Я is a closed Lie subgroup of a Lie group G, then forheH we have
left translation by h as a map G —l· G and also as a map Η —> H. The latter
is the restriction of the former. To avoid notational clutter, let us denote
5.4. Lie Algebras and Exponential Maps
209
Lh\ff by lh- If l : Η '-¥ G is the inclusion, then we have t о lh = Lh ° t, and
so Tl о Tlh = TLh о Tl. If vH € XL(H), then we have
Tht (vH(h)) = Thc(Telh{v)) = Τι ο Tlh(v)
= TLh о Tl(v) = TeLh (Tei{v))
= TeLh(v) = vG(h) = (vGol)(h),
so that vH and vG are ^-related for any ν G TeH с TeG. Thus forv,w € TeH
we have
the formula we wanted. Next, notice that if we take Γμ as an inclusion
so that Tht (vH{h)) = vH(h) for all /i, then we have really shown that if
t; Ε ГеЯ, then vH is the restriction of v° to Я. Also it is easy to see that
&(H) = {vG:ve ГеЯ},
and so the restrictions of elements oi3tL(H) are none other than the elements
of XL(H). Prom what we have shown, the restriction map XL(H) —>> XL(H)
is given by vG ι—> vH and is a surjective Lie algebra homomorphism. It
also has kernel zero since if vH is the zero vector field, then ν = 0, which
implies that vG is the zero vector field. D
Because [v, w]^ = [v, w]fl, the inclusion f) ^ q is a Lie algebra
homomorphism. Examining the details of the previous proof we see that we have a
commutative diagram
XL{H) = *· XL{H)
of Lie algebra homomorphisms where the top horizontal map is a restriction
to Я, the left diagonal map is ν ι—> vG and the right diagonal map is
ν ι—> vH. In practice, what this last proposition shows is that in order
to find the Lie algebra of a closed subgroup Я С G, we only need to find
the subspace 1) = TeH, since the bracket on f) is just the restriction of the
bracket on g. The following is also easily seen to be a commutative diagram
of Lie algebra homomorphisms:
&(H) c > XL(G)
Ь
where both vertical maps are ν \—> vG.
210
5. Lie Groups
The Lie algebra Lie group correspondence works in the other direction
too:
Theorem 5.66. Let G be a Lie group with Lie algebra g. If ij is a Lie
subalgebra o/g, then there is a unique connected Lie subgroup HofG whose
Lie algebra is f).
The above theorem uses the Probenius integrability theorem, so we defer
the proof until we have that theorem in hand.
Since the Lie algebra of GL(n) is the set of all η χ η matrices with
the commutator bracket, and since we have just shown that the bracket
for the Lie algebra of a subgroup is just the restriction of the bracket on
the containing group, we see that the bracket on the Lie algebra of matrix
subgroups can also be taken to be the commutator bracket if that Lie algebra
is represented by the appropriate space of matrices. We record this as a
proposition:
Proposition 5.67. If G С GL(V) is a linear Lie group, then the Lie algebra
of G may be identified with a subalgebra of End(V) with the commutator
bracket. A similar statement holds for matrix Lie algebras.
Example 5.68. Consider the orthogonal group O(n) С GL(n). Given a
curve of orthogonal matrices Q(t) with Q(0) = / and ^|i=0Q(0) = A1 we
compute by differentiating the defining equation / = QlQ\
d
QlQ
t
= (IL*)W)+<?'(0)(IL·)
so that the space of skew-symmetric matrices is contained in the tangent
space Τ/0(η). But both T/0(n) and the space of skew-symmetric
matrices have dimension n(n — l)/2, so they are equal. This means that we can
identify the Lie algebra o(n) = £(0(n)) with the space of skew-symmetric
matrices with the commutator bracket. One can easily check that the
commutator bracket of two such matrices is skew-symmetric, as expected.
We have considered matrix groups as subgroups of GL(n), but it is
often more convenient to consider subgroups of GL(n,C). Since GL(n,C)
can be identified with a subgroup of GL(2n), this is only a slight change
in viewpoint. The essential parts of our discussion go through for GL(n, C)
without any significant change.
Example 5.69. Consider the unitary group U(n) С GL(n,C). Given a
curve of unitary matrices Q(t) with Q(0) = / and ^|i==0Q(0) = A, we
5.4. Lie Algebras and Exponential Maps
211
compute by differentiating the defining equation / = QtQ. We have
d
°=dt
Q'Q
t-0
Examining dimensions as before, we see that we can identify u(n) with the
space of skew-Hermitian matrices (A* = —A) under the commutator bracket.
Along the Unes of the above examples, each of the familiar matrix Lie
groups has a Lie algebra presented as a matrix Lie algebra with commutator
bracket. This subalgebra is defined in terms of simple conditions as in
the examples above, and the chart below lists some of the more common
examples.
Group
SL(n,R)
O(n)
SO(n)
U(n)
5U(n)
Sp(2n,F)
Lie algebra
sI(n,R)
o(n)
so(n)
u(n)
su(n)
sp(2n,F)
Conditions defining
the Lie algebra
Trace(A) = 0
At = -A
A* = -A
At = -A
A1 = -A, Trace(A) = 0
J A* J = A
In the chart above the matrix J is given by
НУЛ
Exercise 5.70. Find the defining conditions for the Lie algebra of the
semiorthogonal group 0(p,q).
We would like to relate Lie group homomorphisms to Lie algebra homo-
morphisms.
Proposition 5.71. Let f : G\ —> G<i be α Lie group homomorphism. The
тпар Tef : Qi —> Q2 is a Lie algebra homomorphism called the Lie
differential, which is denoted in this context by df : gi —> 32-
Proof. For ν G 0i and χ e G, we have
Txf-Lv{x)=Txf.(TeLx-v)
= Te{foLx)-v = Te(Lnx)of).v
= TeLf{x) (Tef · v) = TeLf{x)(Tef -1;)
= L*W(/(x)),
212
5. Lie Groups
so Lv ^f L^^v\ Thus by Proposition 2.84 we have that for any v, w G gi,
In other words, [£*(">, !*(»)] о / = Г/ о I>'w\ which at e gives
[df(v),df(w)] = [v,w]- Q
Theorem 5.72. Invariant vector fields are complete. The integral curves
through the identity element are the one-parameter subgroups.
Proof. We prove the left invariant case since the right invariant case is
similar. Let X be a left invariant vector field and с : (α, b) —> G be an
integral curve of X with c(0) = X(p). Let a < t\ < £2 < b and choose
an element g G G such that gc{t\) = 0(^2)· Let Δί = <2 — ii, and define
с : (α + Δί, b + At) -> G by c(i) = £c(i - Δί). Then we have
eft
c(i) = TLg ■ c(i - Δί) = TLg ■ X(c(i - Δί))
= Χ(5ο(ί-Δί)) = Χ(ο(ί)),
and so с is also an integral curve of X. On the intersection (α + Δί,δ) of
their domains, с and с are equal since they are both integral curves of the
same field and since c(i2) = #c(ii) = c(i2). Thus we can concatenate the
curves to get a new integral curve defined on the larger domain (a, b + At).
Since this extension can be done again for the same fixed Δί, we see that с
can be extended to (a, 00). A similar argument shows that we can extend
in the negative direction to get the needed extension of с to (—00,00).
Next assume that с is the integral curve with c(0) = e. The proof that
c(s + i) = c(s)c(t) proceeds by considering 7(i) = c(s)~1c(s + i). Then
7(0) = e and also
7(i) = TLc(s)-i · c(s + t)= TLC{S) 1 ■ X(c(s +1))
= X(c(s)-1c(s + t)) = XMt)).
By the uniqueness of integral curves we must have c(s)^1c(s + t)= c(i),
which implies the result. Conversely, suppose с : R —> G is a one-parameter
subgroup, and let Xe — c(0). There is a left invariant vector field X such that
X(e) — Xe, namely X = Lx&. We must show that the integral curve through
e of the field X is exactly c. But for this we only need that c(i) — X{c(t))
for all t. We have c(i + s) = c(t)c(s) or c(i + s) = Lc^c(s). Thus
*>-s
C(t + S) = (rc(t)L).c(0)=A-(c(f)).
5Λ. Lie Algebras and Exponential Maps
213
Proposition 5.73. Let ν £ g = TeG. We have the corresponding left
invariant field Lv and flow ψ\%'. Then} with (pv(t) := ψ\χ'(e), we have that
(5.2) ^(rt) = ^(t).
A similar statement holds with Rv replacing Lv.
Proof. Let и = st We have that a|t!=0 ¥>"(**) = £|υ=0^Η^(°) = sv
and so by uniqueness φν{βί) = φ8ν{ί). D
Theorem 5.74. Let G be a Lie group. For a smooth curve с : R —У G with
c(0) = e and c(0) = v, t/ie following are all equivalent:
(i) с is a one-parameter subgroup with c(0) = v.
(ii) c(t) = рПе)/or oil t.
(iii) c(i) = y?f * (e) /or <Ш i.
(iv) <pfv = jRc(t) /or aU i.
(ν) φ{? = Lc(t) /or all t.
Proof. Prom Theorem 5.72 we already know that (ii) is equivalent to (i).
The proof that (i) is equivalent to (iii) is analogous. Also, (iv) implies (i)
since then φ% (e) = Rc^(e) = c(t). Now assuming (ii) we show (iv). We
have c(t) = Lv(e) = ν and
dt
gc(t) =
t=o
dt
t=o
Lg(c(t))
= TLgv = Lv{g)ioraxiyg.
In other words,
dt
t=o
Rcit)9 = Lv(g)
for any 3. But also, Rc(o)9 = eg = 3 and <£q (з) = 3 so by uniqueness we
have i?c(^)p = <Pt°{g) for all p. We leave the remainder to the reader. D
It follows from (iv) and (v) of the previous theorem that the bracket of
any left invariant vector field with any right invariant vector field is zero
since their flows commute (see Theorem 2.109).
Definition 5.75 (Exponential map). For any ν G fl = TeG, we have the
corresponding left invariant field Lv which has an integral curve t H> φν(ί) :=
ψΐ (e) through e. The map exp : 0 —> G defined by exp : ν Η- φν{1) is
referred to as the exponential map.
214
5. Lie Groups
Thus for 5, t Ε R and vGgwe have
exp((s + t)v) = exp(sv) exp(ti>),
exp(—tv) = (ехр(£г>))~ .
Note that we usually use the same symbol "exp" for the exponential map of
any Lie group, but we may also write expG to indicate that the group is G.
By Proposition 5.73 we have
Thus by Theorem 5.74 we obtain the following:
Proposition 5.76. For ν Ε fl, the map R —> G given by t>-+ exp(tv) is the
one-parameter subgroup that is the integral curve of Lv.
Lemma 5.77. The map exp : g —> G is smooth.
Proof. Consider the map KxGxg^Gxg given by
(t,g,v)>4> {g-exp{tv),v).
This map is easily seen to be the flow on G x g of the vector field X ;
(9>v) ^ (Lv(g),0) and so is smooth. The restriction of this smooth flow to
the submanifold {1} χ {e} χ g is (l,e,v) »-> (ехр(г>),г;) and is also smooth.
This clearly implies that exp is smooth also. D
Note that exp(O) = e. In the following theorem, we use the
canonical identification of the tangent space of TeG at the zero element (that
is, T0(TeG)) with TeG itself.
Theorem 5.78. The tangent map of the exponential map exp : g—l· G is the
identity at 0 G TeG = 0, and exp is a diffeomorphism of some neighborhood
of the origin onto its image in G}
reexp = id:TeG-^TeG.
Proof. By Lemma 5.77, we know that exp : g —> G is a smooth map. Also,
2j|0 exp(iv) — v, which means that the tangent map is ν ь-> v. If the reader
thinks through the definitions carefully, he or she will discover that we have
here used the identification of g with Tog. D
Prom the definitions and Theorem 5.74 we have
¥>Г(Р) =pe*ptv,
VtV(p) = (exptv)p
for all ν G g, alii G Ж and all ρ G G.
5Λ. Lie Algebras and Exponential Maps
215
Proposition 5· 79. For α (Lie group) homomorphism f : G\
following diagram commutes:
Cr2, the
df
01 ^02
exp"
Gl
Gx
exp
σ2
G2
Proof. For υ in the Lie algebra of G\, the curve t ■-> /(expGl (tv)) is clearly
a one-parameter subgroup. Also,
£
dt
f(exVG4tv)) = df(v),
and so by uniqueness of integral curves /(expGl(ft;)) = expG2(tdf(v)). D
The Lie algebra of a Lie group and the group itself are closely related
in many ways, and the exponential map is often the key to understanding
the connection. One simple observation is that by Theorem 5.6, if G is a
connected Lie group, then for any open neighborhood V С 0 of 0 the group
generated by exp(V) is all of G.
If Я is a Lie subgroup of G, then the inclusion ι: Η «->■ G is an injective
homomorphism and Proposition 5.79 tells us that the exponential map on
Ϊ) С 0 is the restriction of the exponential map on 0. Thus, to understand the
exponential map for linear Lie groups, we must understand the exponential
map for the general linear group. Let V be a finite-dimensional vector space.
It will be convenient to pick an inner product (·, ·) on V and define the norm
of υ Ε V by ||v|| := y/{v,v). In case V is a complex vector space, we use a
Hermitian inner product. We put a norm on the set of linear transformations
b(V,V) by
ИИ = sup
\\Av\\
v\\*o IMI '
We have ||Ao B|| < p|| ||S||, which implies that \\Ak\\ < \\A\\k. If we use
the identification of fll(V) with L(V, V) (or equivalently the identification of
g[(n, R) with the vector space of η χ η matrices MnXn), then the exponential
map is given by a power series
А^ехР(А) = 1£^Ак.
fc=0
216
5. Lie Groups
which can be seen from the following argument: The sequence of partial
sums sn := ^2k 0 ^Ak is a Cauchy sequence in the normed space fll(V),
k=o k=o II \\k=M+i
< Σ й»л|1
к=М+1
From this we see that
lim
M,JV-too
Ν Μ
Ση*-Σ π*
k=0
к О
= 0,
and so {sjv} is a Cauchy sequence. Since 0l(V) together with the given
norm is known to be complete, we see that Σ£10 Ь.^ converees· For a
fixed A € fli(V), the function a : t ь-> a(t) = exp(L4) is the unique solution
of the initial value problem
a'(t) = Aa(t), a{0) = A.
This can be seen by differentiating term by term:
к 0 k=l v /
= ΛΣ Jkh)\tk lAk~l = AeMtA).
Under our identifications, this says that α is the integral curve corresponding
to the left invariant vector field determined by A. Thus we have a concrete
realization of the exponential map for 0l(V) and, by restriction, each Lie
subgroup of fli(V). Applying what we know about exponential maps in
the abstract setting of a general Lie group we have in this concrete case
exp((s +1) A) — exp(sA) exp(L4) and exp(-tA) = (exp(tA))^1. Let А, В €
flt(V). Then
exp(A)exp(B)= i£i^j фъ*вЫ)
OO OO -
j 0 fc=0
j\k\
5.4. Lie Algebras and Exponential Maps
217
On the other hand, suppose that AoB = BoA. Then we have
oo - oo 1 / oo * \
m-0 m=0 \j+k=mJ }
oo oo 1
j=0*=0e/'
Thus in the case where A commutes with Б, we have
exp(A + В) = exp(A) ехр(Б).
Since most Lie groups of interest in practice are linear Lie groups, it will
pay to understand the exponential map a bit better in this case. Let V be
a finite-dimensional vector space equipped with an inner product as before,
and take the induced norm on fll(V). By Problem 18 we can define a map
log : U -> fll(V), where
C/ = {BeGL(V):||5||<l},
by using the power series:
1оёЯ:=£Ц>-(В-/)*.
fe=i ft
If we compute formally, then for A E fll(V),
+ 1(λ+Ιλ·)' + ...
-*+(а*-И + (ял'-5А,+И+·-
We will argue that the above makes sense if ||A|| < log 2 and that there
must be cancellations in the last line so that log(expA) = A. In fact,
ЦехрА — I\\ < е11дИ — 1, and so the double series on the first line for
log(expA) must converge absolutely if e^A^ — 1 < 1 or if ||j4|| < log 2.
This means that we may freely rearrange terms and expect the same
cancellations as we find for the analogous calculation of log(exp z) for complex
ζ with \z\ < log 2. But since log(expz) = ζ for such z, we have the desired
conclusion. Similarly one may argue that
exp(logB)=Sif \\B-I\\ < 1.
218
5. Lie Groups
Next, we prove a remarkable theorem that shows how an algebraic
assumption can have implications in the differentiable category. First we need
some notation.
Notation 5.80. If S is any subset of a Lie group G, then we define
S"1 = {s-1 : s € 5},
and for any χ Ε G we define
xS = {xs : s Ε S}.
Theorem 5.81. An abstract subgroup Η of a Lie group G is a (regular)
submanifold and hence a closed Lie subgroup if and only if Η is a closed set
in G.
Proof. First suppose that Я is a (regular) submanifold. Then Я is locally
closed. That is, every point χ Ε Я has an open neighborhood U such that
U Π Я is a relatively closed set in Я. Let U be such a neighborhood of the
identity element e. We seek to show that Я is closed in G. Let у Ε Я and
χ Ε yU~l Π Я. Thus χ Ε Η and у Ε xU. This means that у Ε Я Π xU, and
hence x~ly eHr\U = HnU. So j/ G Я, and we have shown that Η is
closed.
Conversely, suppose that Я is a closed abstract subgroup of G. Since we
can always use left/right translation to translate any point to the identity,
it suffices to find a single-slice chart at e. This will show that Я is a regular
submanifold. The strategy is to first find £(Я) = f) and then to exponentiate
a neighborhood of 0 Ε f).
First choose any inner product on TeG so we may take norms of vectors
in TeG. Choose a small neighborhood U of 0 G TeG = gon which exp is a
diffeomorphism, say exp :U -+U, and denote the inverse by logu :U -+U.
Define the set Я in U by Я = logoff Π17).
Claim. If hn is a sequence in Я converging to zero and such that
^n = hn/ \\hn\\ converges to ν G fl, then exp(ti;) Ε Я for alii G R.
Proof of the claim: Note that thn/ ||/in|| -* tv while ||hn|| converges to
zero. But since \\hn\\ -)· 0, we must be able to find a sequence k(ri) Ε Ζ such
that fe(n) \\hn\\ -)· i. From this we have exp(k(n)hn) = exp(fc(n) \\hn\\ τπ^ττ)
-* exp(iv). But by the properties of exp proved previously, we have that
exp(k(n)hn) = (exp(hn))k(n\ But also exp(hn) Ε Я П f/ С Я and so
(exp(hn))fc(n) Ε Я. Since Я is closed, we have
exp(ta) = Urn (exp(/in))fc(n) Ε Я,
η—юо
Claim. If W is the set of all sv where sGK and ν can be obtained as
a limit hnf ||hn|| —» υ with /ιη Ε Я and /ιη —>■ 0, then W' is a vector space.
5.4. Lie Algebras and Exponential Maps
219
Proof of the claim: We just need to show that if hn/ \\hn\\ —> ν and
^n/II^nll ""► w w*th h'n,hn G Я, then there is a sequence of elements Λ£
from Я with h'n —> 0 such that
n/ ч n" ||« + ш||
Using the previous claim, observe that
h(t) :=log[/(exp(ii;)exp(iii;)) = (log^o/z) (exp(tv),exp(tw)).
Here μ is the group multiplication map. But, by the first claim, exp(tv) and
exp(ftu) are in Η for all t, and so h(t) is in Η for small i. By Exercise 5.41
and the fact that Te log = id, we have that
Thus,
l\mh(t)/t = ti(0) = v + w.
h(t) h{t)/t v + w
\\h(t)\\ \\h(t)/t\\ \\v + wW
Now just let tn I 0 and let h'^ := h(tn). Notice that by the first claim,
ехр(И0 С Я.
Claim. Let W be the set from the last claim. Then exp(W) contains
an open neighborhood of e in Я.
Proof of the claim: Let W1- be the orthogonal complement of W with
respect to the inner product chosen above. Then we have TeG = W±®W.
It is not difficult to show that the map Σ:^θ^4(? defined by
w + χ н> exp(tu) exp(x)
is a diffeomorphism in a neighborhood of the origin in TeG. Denote this
diffeomorphism by φ. Now suppose that exp(W) does not contain an open
neighborhood of e in Я. Then we may choose a sequence hn —> e such
that hn is in Я but not in exp(W). But this means that we can choose
a corresponding sequence (wnixn) G W θ W1- with (wn,xn) -* 0 and
exp(wn) ехр(жп) G Я and yet, xn φ 0. The space W1- is closed and the
unit sphere in W1- is compact. After passing to a subsequence, we may
assume that xn/ \\xn\\ 4i6 W1, and of course ||z|| = 1. Now exp(iun) G Я
since exp(W) С Я and Я is at least an algebraic subgroup, so we see that
ехр(гуп)ехр(жп) G Я. Thus it must be that ехр(жп) G Я also and so
xn G Я. But xn -> 0 and xn/ \\xn\\ —> 0» and so, since we now know that
xn G Я, we have that χ G W by definition. This contradicts the fact that
| ж И = 1 and χ G W1-. Thus exp(W) must contain a neighborhood of e in
Я.
Finally, we let О С exp(W) be a neighborhood of e in Я. The set О
must be of the form О = Я П V for some open У in TeG containing 0. By
220
5. Lie Groups
shrinking V further we obtain a diffeomorphism ψ\ν. The inverse of this
diffeomorphism, φ : U —»■ V, has the required properties by construction:
φ(Η Π U) = О П W. We have actually constructed a chart with values in
TeG, but this is clearly good enough since we can choose a basis of TeG
adapted to W. D
5.5. The Adjoint Representation of a Lie Group
Definition 5.82. Fix an element g € G. The map Cg : G —l· G defined by
Cg(x) = gxg~l is a Lie group automorphism called the conjugation map,
and the tangent map TeCg : q —> g, denoted Ads, is called the adjoint map.
Proposition 5.83. The map Cg : G —> G is a Lie group homomorphism.
The map С : jh Cg is a Lie group homomorphism G —* Aut(G).
Proof. See Problem 4. D
Using Proposition 5.71, we get the following
Corollary 5.84. The map Ad5 : g —> g is a Lie algebra homomorphism.
Lemma 5.85. Let f\MxN—*Nbea smooth map and define the partial
map at χ Ε Μ by fx(y) = /(x, j/). Suppose that for every χ Ε Μ the point
Уо is fixed by fx:
fxiyti^yoforallx.
Then the map Ayo : χ н+ Tyofx is a smooth map from Μ to GL(TyoN).
Proof. It suffices to show that Ayo composed with an arbitrary coordinate
function from an atlas of charts on GL(TyoN) is smooth. But GL(TyoN) has
an atlas consisting of a single chart. Namely, choose a basis v\, V2,..., vn
of TyoN and let ν1, г>2,..., vn be the dual basis of T*QN. Then χ) : A H>
v%(Avj) is a typical coordinate function. Now we compose:
χ) ο Ayo{x) = v%{Ayo{x)vj) = v\TyJx · Vj).
It is enough to show that Tyofx · Vj is smooth in x. But this is just the
composition of the smooth maps Μ -* TM χ ΤΝ ^ Γ(Μ χ Ν) -+ TN
given by
Ж I-+ (0X) Vj) \-> {dif) (Ж, Уо) · Οχ + (<%/) (X, Уо) · Vj = Гуо/χ . Vj.
(Recall the discussion leading up to Lemma 2.28.) Π
Proposition 5.86. The map Ad : g ь-> Adp is a Lie group homomorphism
G —¥ GL(g), which is called the adjoint representation ofG.
5.5. The Adjoint Representation of a Lie Group
221
Proof. We have
Ad(5l52) = TeC9192 = Te(C91 о С92)
= TeC91 о TeC92 = Adffl о Adff2,
which shows that Ad is a group homomorphism. The smoothness follows
from the previous lemma applied to the map С : (#, χ) н-> С9(х). D
Recall that for ν 6 g we have the associated left invariant vector field
Lv as well as the right invariant field Ft?. Using this notation, we have
Lemma 5.87. Let ν e fl. Then Lv{x) = RAd*v.
Proof. We calculate as follows:
Lv{x) = Te{Lx) · υ = T(RX)T{RX i)Te{Lx). ν
= T{RX)T(RX ioLx)-v = RAd№. D
We now go one step further and take the differential of Ad.
Definition 5.88. For a Lie group G with Lie algebra g, define the adjoint
representation of g as the map ad : g -> gi(g) given by
ad = reAd = d(Ad).
Proposition 5.89. ad(v)w = [v,w] for all v,w € g.
Proof. Let v1,..., vn be a basis for g so that Ad(x)w = Σ α>ΐ{%)ν% f°r some
functions щ. Let с be a curve with c(0) = v. Then we have
a,d(v)w = —
Ad(c(t))ta = 2i
to ac
аг(с(<))г/
On the other hand, by Lemma 5.87 we have
Lw(x) = flAd(*)« = R (^α,(χχ)
= ^а,(х)^г(х).
Prom the fact that the bracket of any left invariant vector field with any
right invariant vector field is zero, we have
222
5. Lie Groups
Finally, using the equation for ad(v)w derived above, we have
[v,w] = [Lv,L~\(e)
= £ь>;)(е)Д»*(е) = £Ь»К)(еУ
= y~](vai)vl = ad(u)iu. D
The map ad : g —^βί(β) — End(TeG) is given as the tangent map at
the identity of the map Ad which is a Lie group homomorphism. Thus by
Proposition 5.71 we obtain the following:
Proposition 5.90. ad : g —>β!(β) is a Lie algebra homomorphism.
Since ad is defined as the Lie differential of Ad, Proposition 5.79 tells
us that the following diagram commutes for any Lie group G:
ad
0
exp
flKfl)
G
Ad
exp
GL(0)
On the other hand, for any j G G, the map Cg : χ ь* gxg^1 is also a
homomorphism, and so Proposition 5.79 applies again, giving the following
commutative diagram:
exp
exp
In other words,
exp (t Adg v) = g ехр{Ьь)д~г
for any g Ε G, ν Ε Q and t Ε R.
In the case of linear Lie groups G С GL(V), we have identified q with a
subspace of 0l(V), which is in turn identified with L(V, V). In this case, the
exponential map is given by the power series as explained above. It is easy
to show from the power series that В о exp(iA) ο β-1 = exp(£B о А о В~х)
for any A e fll(V) and В Ε GL(V). In this special set of circumstances, we
have
AdBA = BoAoB~l.
This is seen as follows:
AdBA =
=
d
dt
d
dt
В о exp(i.A) ο β"1
i=0
= — ехр(ШоЛоБ~1) = 5о^оБ
?-ι
ί=0
5.5. The Adjoint Representation of α Lie Group
223
Earlier we noted that for a general Lie group we always have Adoexp =
expo ad. In the current context of linear Lie groups, this can be written as
exp(A) о В о exp(-A) = £ -(ad(A))*B
κι
fc=0
for any A G fl[(V) and any В G GL(V).
We end this section with a statement of the useful Campbell-Baker-
Hausdorff (CBH) formula. A proof may be found in [Helg] or [Mich].
First notice that if G is a Lie group with Lie algebra g, then for each X G g
we have ad X G L(g, g). We choose a norm on g, and then we have a natural
operator norm and consequent notion of convergence in L(g, g). The analytic
function jj*M[ is defined by the power series
n=0
which converges in a small ball about ζ = 1. We may then use the
corresponding power series to make sense of (In A) (A-I)"1 for AG L(g,g)
sufficiently close to the identity map / = id0.
Theorem 5.91 (CBH Formula). Let G be a Lie group and g its Lie algebra.
Let f(z) = j^ be defined by a power series as above. Then for sufficiently
small x,y G g,
exp(x) exp(y) = exp(C(x,y));
where
C(x,y) = y+ f /(eiad*etad*0.xdt
./o
n=l ^ J0 \fc,l>0,fc+J>l /
OO
= x + y +
It follows from the above that
OO
71=1
where C\{x,y) = χ + у, С2(ж,у) = \[х,у] and
Сз(х,у) = γ2 Ш^У]>У] + [fa/i *]>*])·
In particular,
exp(te) exp(ty) = exp(t(x + y) + 0(t2))
for any х, у and sufficiently small t. One can show, using the above results,
that G is abelian if and only if g is abelian (i.e. [ж, у] = 0 for all x, у G g). The
224
5. Lie Groups
CBH formula shows that the Lie algebra structure on g locally determines
the multiplicative structure on G.
5.6. The Maurer-Cartan Form
Let G be a Lie group, and for each g G G, define u)o{g) ' TgG —> g and
b#*fo):T,G->gby
uG(g)(Xg)=TLgl.Xg,
and
u«g*(g)(Xg) = TRg-1.Xg.
The maps ωο ' g *-> uc{g) and cj^g : ρ Μ· cj^g (<?) are g-valued 1-forms
called the left Maurer-Cartan form and right Maurer-Cartan form
respectively. We can view dg and cj^s * as maps TG —> g. For example,
u>G(Xg):=u>G(g)(Xg).
As we have seen, GL(n) is an open set in a vector space, and so its
tangent bundle is trivial, TGL(n) — GL(n) χ Μηχη (recall Definition 2.58).
A general abstract Lie group G is not an open subset of a vector space, but
we are still able to show that TG is trivial. There are two such trivializations
obtained from the Maurer-Cartan forms. These are triv^ : TG —¥ G χ g and
trivia '· TG -> G χ g defined by
tnvL(vg) (g,DG{vg)),
triyR(v9) - (g^h\vg))
for Vg Ε TgG. Observe that triv^^t;) = Lv(g) and triv^1^,^) = Я?(д).
It is easy to check that trrvx and triv# are trivializations in the sense of
Definition 2.58. Thus we have the following:
Proposition 5.92. The tangent bundle of a Lie group is trivial: TG =
Gxg.
We will refer to trivi, and triv# as the (left and right) Maurer-Cartan
trivializations. How do these two trivializations compare? There is no
special reason to prefer left multiplication. We could have used right
invariant vector fields as our means of producing the Lie algebra, and the whole
theory would work "on the other side", so to speak. The bridge between
left and right is the adjoint map:
Lemma 5.93 (Left-right lemma). For any ν Ε g, and g Ε G we have
trivfl о triv^p, v) = (g,Adg(v)).
5.6. The Maurer-Cartan Form
225
Proof. We compute:
trivflotriv^1^)
= (g,TRg iTLgv) = (g,T{Rg-iLg)*v)
= (g,TC9.v) = (g,Adg(v)). Π
It is often convenient to identify the tangent bundle TG of a Lie group
G with Gxj. Of course we must specify which of the two trivializations
described above is being invoked. Unless indicated otherwise we shall use
the "left version" described above: vg ι->· (д,и;о(уд)) = (g,TLjl(vg)).
Warning: It must be realized that we now have three natural ways to
trivialize the tangent bundle of the general linear group. In fact, the usual
one which we introduced earlier is actually the restriction to TGL(n) of the
Maurer-Cartan trivialization of the abelian Lie group (Mnxn,+).
In order to use the (left) Maurer-Cartan trivialization as an identification
effectively, we need to find out how a few basic operations look when this
identification is imposed.
The picture obtained from using the trivialization produced by
the left Maurer-Cartan form:
(1) The tangent map of the left translation TLg : TG -¥ TG takes
the form uTLg"
: (x,v)
commutes:
TG
Gxg
t-> (gx,v). Indeed, th
^ TG
—> Gxq
where elementwise we have
υ
\
(x,Tl
x\
'
t,v)
TLg
>- TLg · VX
У
, (gx,TL-
= (9
f
xTLgvx)
x,v)
(2) The tangent map of multiplication: This time we will invoke two
identifications. First, group multiplication is a map μ : G x G -> G,
and so on the tangent level we have a map T(G χ G) —> TG. Recall
that we have a natural isomorphism T(G xG)~ TG χ TG given by
Γπι x Γπ2 : (v(Xf„)) »-> (Γπι · υ^ν),Τπ2 ■ v^y)). If we also identify
TG with Gxj, then TG xTG ~ [G χ g) χ [G χ β), and we end
226
5. Lie Groups
up with the following "version" of Τ μ:
"Τ/χ" : [G χ g) χ (G χ 0) -> G χ β,
"7>" : ((χ, г;), (у,υ;)) н> (ху,TRyV + ГЗД
(see Exercise 5.94).
(3) The (left) Maurer-Cartan form is a map ωο : TG -> TeG = 0, and
so there must be a "version", 'W, that uses the identification
TG = Gxj. In fact, the map we seek is just projection:
"ωοη : {x,v) ь-> v.
(4) The right Maurer-Cartan form is a little more complicated since
we are currently using the isomorphism TG = Gxg obtained from
the left Maurer-Cartan form. Prom Lemma 5.93 we obtain:
"w£ght" :{x,v)^Adg{v).
The adjoint map is nearly the same thing as the right Maurer-
Cartan form if we decide to use the left trivialization TG = Gxj
as an identification.
(5) A vector field Χ Ε X(G) should correspond to a section of the
projection G x Q —* G which must have the form ¥ :x^(x,Fx{x))
for some smooth g-valued function Fx G C°°(G]q)> It is an
easy consequence of the definitions that Fx{x) — ωο(Χ{χ)) =
TL~l · X(x). Under this identification, a left invariant vector field
becomes a constant section of G χ β. For example, if X is left
invariant, then the corresponding constant section is χ \-ϊ (x, X(e))>
Exercise 5.94· Refer to 2 above. Show that the map "Τμ" defined so that
the diagram below commutes is ((ж, г>), (у, w)) »-► (xy, TRyV + TLxw).
T{GxG) — *TG
τ
(G x g) x (G χ g) >-Gxg
We have already seen that using available identifications the Lie algebra
of a matrix groups and associated formulas take a concrete form. We now
consider the Maurer-Cartan form in the case of matrix groups. Suppose
that G is a Lie subgroup of GL(n) and consider the coordinate functions x%
on GL(n) defined by xlj(A) = aj where A = [aj]. We have the associated 1-
forms dxly Both the functions xj and the forms dxlj restrict to G. Denoting
these restrictions by the same symbols, the Maurer-Cartan form can be
expressed as
5.6. The Maurer-Cartan Form
227
One sometimes sees the shorthand, g~ldg, which is a bit cryptic. Our goal
is to understand this expression better. First, from a practical point of view
we think of it as follows: An element vg of the tangent space TgG is also an
element of TgGL(n) and so can be expressed in terms of the vectors d/dx%j\gy
say
Σ г Я !
}dx\
Then,
*с{уд) = щ- !мк) = и ->}],
where g = [p*-] and the matrix [g^\ [v*-] is interpreted as an element of the
Lie algebra g. For instance, if G = SO(2), then the Maurer-Cartan form is
given by
~Xn
-Хл
dx\
J L dx{
dx\
dxl
ХсуСьХл
XnCuX-t XtyObXry XryCLXty
'—X-yCLX-t ~\~ X-tdX-i —X-tCLXn ι X-tCLXo I
XidX-t — XndX-t X-^QiXn Х<2&Х<£
Xndx-ι H-Xidx-\ XtjdXo ι x-tdx<y
since on SO(2) we have x\ = x\, x\ = —x\ and x\x\ — xlxi = 1· ^u^ this
can be further simplified. If we let vg G TPS0(2), then u>g (vg) is in so(2)
and so must be antisymmetric. We conclude that
U X-tCLXtj XcydXty
,~X-idX'2 ι XndXy U
wSO(2) =
Let us try to understand things a bit more thoroughly. If G is a subgroup
of GL(n), then consider the inclusion map j : G <-> MnXn, where Mnxn is
the set of η χ η matrices. Then we have the differential dj : TG —> Mnxn
and the two left multiplications Lg : G —> G and Cg : Mnxn —> Mnxn for
g£G. We have the following commutative diagrams:
Mnxn
Mnxn
Mnxn
G
+ G
ThG
ThLg
Mnxn
к
<ti\gh
TghG
for any h e G> We have used the fact that since Cg is linear, DCg(A) = Cg
for all A e MnXn. Then for vg G Г5(? we have
dj\e {ug(v9)) = dj\e {TLg^ivg) = £g-i dj\ [vg)
.-1
= 9 * dj\g (vg) = (j ο π(^)) dj {vg),
where π : TG —> G is the tangent bundle projection. Notice that the effect
of dj\e is simply to interpret elements of TeG as matrices, and so can be
228
5. Lie Groups
suppressed from the notation. Taking this into account and applying a
reasonable abbreviation for (j ο π(ν9))~ dj (vg), we arrive at
But notice that for a matrix group, j is none other than the map [xl3] : А H·
[Xj-(A)] = [aj], so that we are returned to the expression ωο = [#j]-1[d#j] =
"g~ldg". This tells us the meaning of g in the expression g~xdg.
5.7. Lie Group Actions
The basic definitions for group actions were given earlier in Definitions 1.99
and 1.6.2. As before we give most of our definitions and results for left
actions and ask the reader to notice that analogous statements can be made
for right actions.
Definition 5.95. Let I : G χ Μ -> Μ be a left action, where G is a Lie
group and Μ is a smooth manifold. If I is a smooth map, then we say that
I is a (smooth) Lie group action.
As before, we also use any of the notations gp, g-p or lg{p) for l{g,p)> We
will need this notational flexibility. Recall that for ρ G M, the orbit of ρ is
denoted Gp or G«p, and the action is transitive if Gp = M. Recall also that
an action is effective if lg(p) = ρ for all ρ only if g = e. For right actions
r : Μ xG —> Af, similar definitions apply and we write pg = rg(p) = r(p,g).
A right action corresponds to a left action by the rule gp := рд~г.
Definition 5.96. Let I be a Lie group action as above. For a fixed ρ Ε Μ,
the isotropy group of ρ is defined to be
Gp:={geG;gp = p}.
The isotropy group of ρ is also called the stabilizer of p.
Exercise 5.97. Show that Gp is a closed subset and an abstract subgroup
of G. This means that Gp is a closed Lie subgroup.
Recalling the definition of a free action (Definition 1,100), it is easy to
see that an action is free if and only if the isotropy subgroup of every point
is the trivial subgroup consisting of the identity element alone.
Definition 5.98. Suppose that we have a Lie group action of G on M. If
AT is a subset of Μ and Gx С N for all χ G iV, then we say that N is an
invariant subset. If N is also a submanifold, then it is called an invariant
submanifold.
In this definition, we include the possibility that N is an open
submanifold. If N is an invariant subset of M, then it is easy to see that gN = N,
5.7* Lie Group Actions
229
where gN = lg(N) for any p. Furthermore, if N is a submanifold, then the
action restricts to a Lie group action G χ Ν —> ΛΓ.
If G is zero-dimensional, then by definition it is just a group with discrete
topology, and we recover the definition of a discrete group action. We have
already seen several examples of discrete group actions, and now we list a
few examples of more general Lie group actions.
Example 5.99. The maps G xG -4 G given by (p, x) н· Lgx and (3, x) ь+
Rgx are Lie group actions.
Example 5.100. In case Μ = Rn, the Lie group GL(n,R) a^ts on W1 by
matrix multiplication. Similarly, GL(n,C) acts on Cn. More abstractly,
GL(V) acts on the vector space V. This action is smooth since Ax depends
smoothly (polynomially) on the components of A and on the components of
xew1.
Example 5.101. Any Lie subgroup of GL(n,R) acts on Rn also by matrix
multiplication. For example, 0(n,R) acts on Rn. For every χ Ε Rn, the
orbit of χ is the sphere of radius | ж||. This is trivially true if χ — 0. In
general, if ||ж|| φ 0, then \\gx\\ = ||x|| for any g Ε 0(η, R). On the other
hand* if я, у Ε Rn and \\x | = \\y\\ — r, then let χ := x/r and у := у jr. Let
χ :— ei and у = fi and then extend to orthonormal bases (ei,..., en) and
(/1 >. - - 5 /n)· Then there exists an orthogonal matrix 5 such that Set = ft
for г =з 1,..., η. In particular, Sx — y, and so Sx — y.
Exercise 5.102. From the last example we can restrict the action of 0(n, R)
to a transitive action on Sn~l* Now SO(n,R) also acts on Sn~l by
restriction. Show that this action on Sn~l is transitive as long as η > 1.
Example 5.103. If Я is a Lie subgroup of a Lie group G, then we can
consider L^ for any h Ε Η and thereby obtain a Lie group action of Η
onG.
Recall that a subgroup Я of a group G is called a normal subgroup
tigkg^1 Ε Κ for any к Ε Я and all g Ε G. In other words, Я is normal if
gHg'1 С Я for all g Ε С?, and it is easy to see that in this case we always
have gHg~~l = Я.
Example 5.104. If Я is a normal Lie subgroup of G, then G acts on Я by
conjugation:
g · h := Cgh = ghg^1.
Notice that the notation g · h cannot reasonably be abbreviated to gh in this
example.
Suppose that a Lie group G acts on smooth manifolds Μ and N. For
simplicity, we take both actions to be left actions, which we denote by I and
230
5. Lie Groups
λ, respectively. A map / : Μ —>· JV such that / о lg = \g о f for all 5 Ε G,
is said to be an equivariant map (equivariant with respect to the given
actions). This means that for all g the following diagram commutes:
(5.3)
If / is also a diffeomorphism, then / is an equivalence of Lie group actions.
Example 5.105. If φ : G —> Η is a Lie group homomorphism, then we can
define an action of G on Η by λ(ρ, h) — Xg(h) — Ьф^Ъ,. We leave it to the
reader to verify that this is indeed a Lie group action. In this situation, φ
is equivariant with respect to the actions λ and L (left translation).
Example 5.106. Let T71 = S1 x ■ · ■ x Sl be the η-torus, where we identify
S1 with the complex numbers of unit modulus. Fix к = (fci,..., kn) G Rn,
Then R acts on Rn by rk(t, x) = t-x:=x + tk. On the other hand, R acts
on T1 by t · (z\ ..., zn) = (eiiklzl,..., eitknzn). The map Rn -> Tn given by
(a;1,...,^71) и- (ег|
, егх ) is equivariant with respect to these actions.
Theorem 5.107 (Equivariant rank theorem). Suppose that f : Μ —> N is
smooth and that a Lie group G acts on both Μ and N with the action on Μ
being transitive. If f is equivariant, then it has constant rank. In particular,
each level set of f is a closed regular submanifold.
Proof. Let the actions on Μ and N be denoted by / and λ respectively
as before. Pick any two points pi,P2 € M. Since G acts transitively on
My there is a g with lgpi = p^. By hypothesis, we have / о lg = \g 0 /,
which corresponds to the commutative diagram (5.3). Upon application of
the tangent functor we have the commutative diagram
TP1M
Tpxlg
Τρ,Μ
Tnf
Tp2f
lf(pi)
N
Tf(Pl)X9
ЧЫ
N
Since the maps Tpxlg and Tj^Xg are linear isomorphisms, we see that TPlf
must have the same rank as TP2f. Since pi and P2 were arbitrary, we see
that the rank of / is constant on M. Apply Theorem C.5. D
There are several corollaries of this nice theorem. For example, we know
that 0(n,R) is the level set /_1(/), where / : GL(n,R) -> flI(n,R) =MnXn
is given by f(A) = ATA. The group 0(n,R) acts on itself via left
translation, and we also let 0(n,R) act on gt(n, R) by Q · A := QTAQ (adjoint
5.7. Lie Group Actions
231
action). One checks easily that / is equivariant with respect to these
actions, and since the first action (left translation) is certainly transitive, we
see that 0(n, R) is a closed regular submanifold of GL(n, R). It follows from
Proposition 5.9 that 0(n,R) is a closed Lie subgroup of GL(n,R). Similar
arguments apply for U(n, С) С GL(n,C) and other linear Lie groups. In
fact, we have the following general corollary to Theorem 5.107 above.
Corollary 5.108. If φ : G —► Η is a Lie group homomorphism, then the
kernel Ker(h) is a closed Lie subgroup of G.
Proof. Let G act on itself and on if as in Example 5.105. Then φ is
equivariant, and 0"1(e) = Ker(h) is a closed Lie subgroup by Theorem
5.107 and Proposition 5.9. D
Corollary 5.109. Let I : G χ Μ —» Μ be a Lie group action, and let Gp
be the isotropy subgroup of some ρ € M. Then Gp is a closed Lie subgroup
ofG.
Proof. The orbit map θρ : G -+ Μ given by 9p(g) = gp is equivariant with
respect to left translation on G and the given action on M. Thus by the
equivariant rank theorem, Gp is a regular submanifold of G, and then by
Proposition 5.9 it is a closed Lie subgroup. D
Proper Actions and Quotients. At several points in this section,
such as the proof of Proposition 5.111 below, we follow [Lee, John].
Definition 5.110. Let I: G χ Μ —► Μ be a smooth (or merely continuous)
group action. If the map P:GxM-^MxM given by (ρ,ρ) *-> (lgp,p) is
proper, we say that the action is a proper action.
It is important to notice that a proper action is not defined to be an
action such that the defining map I : G x Μ —> Μ is proper. We now give
a useful characterization of a proper action. For any subset К с М, let
g-K:={gx:xeK}.
Proposition 5.111. Let I: GxM —> Μ be a smooth (or merely continuous)
group action. Then I is a proper action if and only if the set
GK:={geG:(g-K)r\K^<b}
и compact whenever К is compact
Proof. Suppose that I is proper so that the map Ρ is a proper map. Let
tg be the first factor projection G χ Μ —^ G. Then
Gk = {g : there exists an χ G К such that gx e K}
= {g : there exists anxGM such that P(g, x) G Κ χ Κ}
= πα(ρ-1(ΚχΚ)),
and so Gk is compact.
232
5. Lie Groups
Next we assume that Gk is compact for all compact K. If С С Μ χ Μ
is compact, then letting К = 7Ti(C) U К2{С), where πι and π2 are the first
and second factor projections Μ χ Μ —>· Μ respectively, we have
Ρ λ{0) С Ρ 1{КхК)с {(д,х) : дх G К and χ е К}
cGKxK.
Since P~r(C) is a closed subset of the compact set Gk χ Κ, it is compact.
This means that Ρ is proper since С was an arbitrary compact subset of
MxM. 0
Using this proposition, one can show that Definition 1.106 for discrete
actions is consistent with Definition 5.110 above.
Proposition 5.112. IfGis compact, then any smooth action I: G χ Μ -»
Μ is proper.
Proof. Let В С Μ χ Μ be compact. We find a compact subset К С М
such that В С К х К as in the proof of Proposition 5.111.
Claim: P~l(B) is compact. Indeed,
Р-НВ) С P-\K xK)= UkeKP'HK χ {fc})
= UeK-ite.P) = (ЯР,Р) € Κ χ {k}}
= VkzK{(9>k)--9P£K}
cUm(GxW) = GxX
Thus P~1(B) is a closed subset of the compact set G χ Κ and hence is
compact. D
Exercise 5.113. Prove the following:
(i) If I : G χ Μ 4 Μ is a proper action and Η С G is a closed
subgroup, then the restricted action Η χ Μ —> Μ is proper.
(ii) If N is an invariant submanifold for a proper action I: G χ Μ -> Μ,
then the restricted action G χ Ν —>> N is also proper.
Let us consider a Lie group action i:GxM-)M that is both proper
and free. The orbit map at ρ is the map θρ : G -> Μ given by 0р(д) = g -p.
It is easily seen to be smooth, and its image is obviously G - p. In fact, since
the action is free, each orbit map is injective. Also, θρ is equivariant with
respect to the left action of G on itself and the action I:
θρ(9χ) ~ (fls) ·Ρ = 9-{χ·Ρ)
= 9 · 0p(*)
for all χ, ρ Ε G. It follows from Theorem 5.107 (the equivariant rank
theorem) that θρ has constant rank, and since it is injective, it must be an
5.7. Lie Group Actions
233
Figure 5.1. Action-adapted chart
immersion. Not only that, but it is a proper map. Indeed, for any compact
К С Μ the set θρι(Κ) is a closed subset of the set G#u{p}> an(l s*nce the
latter set is compact by Theorem 5.111, θ~λ(Κ) is compact. By Exercise
3.9, θρ is an embedding, so each orbit is a regular submanifold of M.
It will be very convenient to have charts on Μ which fit the action of G
in a nice way. See Figure 5.1.
Definition 5.114. Let Μ be an η-manifold and G a Lie group of dimension
fc. If Ζ: G χ Μ —> Μ is a Lie group action, then an action-adapted chart
on Μ is a chart (J7, x) such that
(i) x(U) is a product open set V\ x V2 С Rk x Rn~* = Kn;
(ii) if an orbit has nonempty intersection with f/, then that intersection
has the form
{xk+1=c\...,xn = cn-k}
for some constants c1,..., cn~k.
Theorem 5.115. Ifl:GxM-+Misa free and proper Lie group action,
then for every ρ Ε Μ there is an action-adapted chart centered at p.
Proof. Let ρ Ε Μ be given. Since G · ρ is a regular submanifold, we may
choose a regular submanifold chart (W, y) centered at ρ so that (G · ρ) Π W
is exactly given by yk+1 = · ■ · = yn = 0 in W\ Let S be the complementary
slice in W given by y1 = «· · = yk = 0. Note that 5 is a regular submanifold.
The tangent space TPM decomposes as
TpM = Tp{G-p)®TpS.
Let φ : G χ S -* Μ be the restriction of the action Ζ to the set Gx S. Also,
let ip : G -> G x 5 be the insertion map g ь* (g,p) and let je : S -> G? χ 5
be the insertion map s \-> (e, s). (See Figure 5.2.) These insertion maps
234
5. Lie Groups
are embeddings, and we have θρ — φ ο ip and also φ ο je = 6, where t is the
inclusion S «-► M. Now Te0p(TeG) = Tp(G-p) since 0p is an embedding. On
the other hand, Τθρ — Τφ ο Tip, and so the image of Τ^ρ^ψ must contain
Гр(С? ·ρ). Similarly, from the composition ψ ο je = t we see that the image
of T(ejP)V? must contain TPS. It follows that Τ^φ : T^iP)(G x £) -► ΓΡΜ
is surjective, and since T(6iP) (G x S) and TpM have the same dimension^ it
is also injective.
By the inverse mapping theorem, there is a neighborhood О of (e,p)
such that φ\α is a diffeomorphism. By shrinking О further if necessary we
may assume that φ{0) С W. We may also arrange that О has the form
of a product О = Α χ В for A open in G and В open in S. In fact, we
can assume that there are diffeomorphisms a : Ik -» A and /J : /n^fe —» B,
where Ik and /n~fe are the open cubes in Шк and Rn~k given respectively
by Ik (—l5l)fe and 7n~fc = (-l,l)n-fe and where a^(e) = 0 Ε lfe
and β~ι(ρ) = 0 e Mn~*. Let Ϊ7 := v>(^ * 5). The map ν? ο (α χ /3) :
Ik χ In k —>> 17 is a diffeomorphism, and so its inverse is a chart. We now
show that В can be chosen small enough that the intersection of each orbit
with В is either empty or a single point. If this were not true, then there
would be a sequence of open sets {Д·} with compact closure and Bi+ι С В%
(and with corresponding diffeomorphisms β% : Ik —> B{ as above), such that
for every г there is a pair of distinct points pup[ £ Д with gipi = p[ for
some sequence {gi} С G. (We have used the fact that manifolds are first
countable and normal.) This forces both ръ and p[ = giPi to converge to p.
Prom this we see that the set К = {(giPi,Pi), (p,p)}cMxMis compact.
Recall that by definition, the map Ρ : (ρ, χ) н· (дх,х) is proper. Since
(ЯиРг) — Ρ l(9iPi->Pi), we see that {(#*>№)} is a subset of the compact set
P~l(K). Thus after passing to a subsequence, we have that {guPi) converges
to (g,p) for some g and hence g%-+ g and gipi —> gp. But this means that
we have
gp = lim gip{ = lim p- = p,
ι—too г—too
and since the action is free, we conclude that g = e. However, this is
impossible since it would mean that for large enough г we would have gi Ε A)
and this in turn would imply that
¥>(ft,Pi) = l9l{Pi) = P[ = Je(pi) = ¥>(e,p·)
contradicting the injectivity of <p on Л χ Б. Thus after shrinking В we may
assume that the intersection of each orbit with В is either empty or a single
point. One may now check that with χ := (φ ο (α χ β))"1 : U -> Ikxln-k с
Rn, we obtain a chart (17, x) with the desired properties. Write χ = (χ, у),
where у takes values in J71""* С Rn~k and я takes values in Ik cMfe, Each
у = с slice is of the form <^(-A χ {g}) С Gq for q Ε Β and so is contained
in a single orbit. We see that the intersection of an orbit with U must be
5.7. Lie Group Actions
235
Figure 5.2. Construction of action-adapted charts
a union of such slices. But each orbit can only intersect В in at most one
point, and so it is clear that each orbit intersects U in one slice or does not
intersect U at all. D
Notice that we have actually constructed action-adapted charts that
have image the cube Jn. For the next lemma, we continue with the
convention that I is the interval (—1,1).
Lemma 5.116. Let χ := {φ ο (α χ β))*1 : U -* Ik x In'k = In С Шп
be an action-adapted chart map obtained as in the proof of Theorem 5.115
above. Then given any pi Ε С/, there exists a diffeomorphism ψ : In —> In
such that ф ox is an action-adapted chart centered at p\ and with image
F', Furthermore ψ can be decomposed as (a, b) и· {i>i{Q>),ip2{b)), where
1: Ik -» Ik and Ψ2 : In~k —> In k are diffeomorphisms.
Proof. We need to show that for any a G /n, there is a diffeomorphism
ψ ι Ιη -* In such that ψ (a) = 0. Let a1 be the i-th component of a. Let
: / -b I be defined by
Ψί #-= Φ'1 ° *-*(α0 ° ώ
where t-c(x) := χ — с and ф : (—1,1) —> Ш is the diffeomorphism ф : χ t-l·
tan(|x). The diffeomorphism we want is ψ(χ) = (^ι(a:1),... ,*ψ\{χη)). The
last part is obvious from the construction. D
We now discuss quotients. If I : G χ Μ -> Μ is a Lie group action,
then there is a natural equivalence relation on Μ whereby the equivalence
classes are exactly the orbits of the action. The quotient space (or orbit
space) is denoted G\M, and we have the quotient map π : Μ —> G\M. We
put the quotient topology on G\M so that А С G\M is open if and only if
236
5. Lie Groups
π~ι (A) is open in M. The quotient map is also open. Indeed, let U С М
be open. We want to show that π(ϋ) is open, and for this it suffices to show
that π^1 (π([7)) is open. But π"*1 {π(ΙΙ)) is the union \Jglg(U), and this is
open since each lg{U) is open.
Proposition 5.117. Let G act smoothly on M. Then G\M is a Hausdorff
space if the set Γ := {{gp>p) : g Ε G% ρ € Μ} is о dosed subset of Μ χ Μ.
// Μ is second countable then G\M is also.
Proof. Let p,?E G\M with π(ρ) = ρ and 7r(g) = g. If ρ φ q, then ρ and
q are not in the same orbit. This means that (p, q) £ Γ, and so there must
be a product open set U x V such that (p, q) Ε Ϊ7 x V and ί7 χ У is disjoint
from Г. This means that π (U) and π (V) are disjoint neighborhoods of ρ
and g respectively. Finally, if {Ui} is a countable basis for the topology on
M, then {π (ΪΛ)} is a countable basis for the topology on G\M. О
Proposition 5.118. If I : G χ Μ —ϊ Μ is a free and proper action, then
G\M is Hausdorff.
Proof. To show that G\M is Hausdorff, we use the previous lemma. We
must show that Γ is closed. By Problem 2, proper continuous maps are
closed. Thus Γ = P(G χ Μ) is closed since Ρ is proper. Π
We will shortly show that if the action is free and proper, then G\M
has a smooth structure which makes the quotient map π : Μ —> G\M b
submersion. Before coming to this let us note that if such a smooth structure
exists, then it is unique and is determined by the smooth structure on M.
Indeed, if (G\M)a is G\M with a smooth structure given by a maximal
atlas Λ and similarly for (G\M)& for another atlas By then we have the
following commutative diagram:
(G\M)A ^ > (G\M)B
Since π is a surjective submersion, Proposition 3.26 appUes to show that
(G\M)a -^ (G\M)b is smooth as is its inverse. This means that Λ = В.
Theorem 5.119. Ifl:GxM-±Misa free and proper Lie group action,
then there is a unique smooth structure on the quotient G\M such that
(i) the induced topology is the quotient topology, and G\M is a smooth
manifold;
5.7. Lie Group Actions
237
(ii) the projection π : Μ -> G\M is a submersion;
(iii) dim(G\M) - dim(M) - dim(G).
Proof· Let dim(M) = η and dim(G) — k. We have already shown that
G Μ is a Hausdorff space. All that is left is to exhibit an atlas such that
the charts are homeomorphisms with respect to this quotient topology. Let
q € G\M and choose ρ with π(ρ) = q. Let (£/, x) be an action-adapted chart
centered at ρ and constructed exactly as in Theorem 5.115. Let π(!7) = V С
G\M and let В be the slice x1 = ■ · · = xk 0. By construction π\Β : В ^V
is a bijection. In fact, it is easy to check that π\Β is a homeomorphism,
and σ :— (тг|в)~ *s ^he corresponding local section. Consider the map
у π2 ο χ ο σ} where π2 is the second factor projection П2 : Шк х Шп~к —>
Жп к. This is a homeomorphism since (π2 ο χ)|β is a homeomorphism and
7Г2 ο χ ο σ = (тг2 ο χ)|Β ο σ. We now have a chart (V, y).
Given two such charts (V,y) and (V,y), we must show that yoy J
is smooth. The (V,y) and (V,y) are constructed from associated action-
adapted charts (17, x) and (f7, x) on M. Let q Ε V Π V. As in the proof of
Lemma 5.116, we may find diffeomorphisms φ and φ such that (ϋ,ψ ο χ)
and (ϋ,φοχ) are action-adapted charts centered at points p\ Ε тг_1(д) and
ί>2 € π 1(g) respectively. Correspondingly, the charts (V,y) and (V,y) are
modified to charts (V, y^) and (V, y^) centered at q, where
Уф :== 7Γ2 ° X^ ° 0* >
y^ := 7Γ2οχψο σ',
with x^ := φ ο χ and similarly for x^,. Also, the sections σ' and σ' are
constructed to map into the zero slices of χψ and %ψ. However, it is not
hard to see that y^ and у ψ are unchanged if we replace σ' and σι by the
sections σ and σ corresponding to the zero slices of χ and y. Now, recall
that φ was chosen to have the form (a, b) η* {φι{α),Ф2{Ь)). Using the above
observations, one checks that y^ oy*""1 = ф2 and similarly for y^ о у l. Erom
this it follows that the overlap map у"1 о y^ will be smooth if and only if
у""1 о у-1 is smooth. Thus we have reduced to the case where (17, x) and
U9x) are centered at p\ Ε n~x(q) and P2 Ε 7r~1(q) respectively. This entails
that both (V, y) and (V, y) are centered at q Ε V Π V- If we choose a g EG
such that lg(pi) = P2> then by composing with the diffeomorphism lg we
can reduce further to the case where p\ = p2- Here we use the fact that
lg takes the set of orbits to the set of orbits in a bijective manner and the
special nature of our action-adapted charts with respect to these orbits. In
this case, the overlap map χ ox"1 must have the form (a, 6) ну (/(a, 6), g(b))
for some smooth functions / and g. It follows that yoy1 has the form
Ь»д(Ь).
238
5. Lie Groups
Finally, we give an argument that G\M is paracompact. Since G\M is
Hausdorff and locally Euclidean, we will be done once we show that every
connected component of G\M is second countable (see Proposition B.5).
Let Qo be any such connected component of G\M. Let Xo be a connected
component of π"1((?ο)· Then Xo is a second countable manifold and open
in M. We now argue that π(Χο) = Qo- To this end we show that the
connected set π(Χο) is open and closed in Qo· It is open since π is an open
map. Let χ be in the closure of π(Χο) and choose χ Ε Μ with π(χ) — χ.
Let U be the domain of an action-adapted chart centered at χ and let S
be the slice of U that maps diffeomorphically onto an open neighborhood
О of x. Now we may find у Ε Ο Π π(Χο) and a corresponding у G S such
that n(y) = у. Since у is in the image of Xq, there is a y* G Xo such that
π(ί/) = У· But then there exists g EG such that gy = y'. The open set gU is
diffeomorphic to U under lg and hence is path connected. It contains y1 and
gx. Since Xo is a path component, gU С Xo and so gx G Xo- But ж = n(gx)
so χ Ε тг(Хо). We conclude that π(Χο) is closed (and open) and connected.
Hence 7г(Хо) = Qo· Now since Xo is a second countable manifold, we argue
as in Proposition 5.117 that Qo is second countable. Conclusion: G\M is
paracompact. D
Similar results hold for right actions. Some of the most important
examples of proper actions are usually presented as right actions. In fact,
principal bundle actions studied in Chapter 6 are usually presented as right
actions. We shall also encounter situations where there are both a right and
a left action in play.
Example 5.120. Consider S2n~l as the subset of Cn given by S2n_1 =
{ξ Ε Cn : \ξ\ = 1}. Here ξ = (г1,,..,*") and \ζ\ = £zV. Let S1 act on
£2n-i ^у ^ ξ^ ^ αξ = (azl,..., azn). This action is free and proper. The
quotient is the complex projective space €Ρη~χ,
g2n-l
I
These maps (one for each n) are called the Hopf maps. In this context, 51
is usually denoted by U(l).
In what follows, we will consider the similar right action Sn χ U(l) -> Sn.
In this case, we think of Cn+1 as a set of column vectors, and the action is
given by (ξ, α) η+ ξα. Of course, since U(l) is abelian, this makes essentially
no difference, but in the next example we consider the quaternionic analogue
where keeping track of order is important.
5.7. Lie Group Actions
239
The quaternionic projective space ШРп α is defined by analogy with
CPn~4 The elements of ШРп г are 1-dimensional subspaces of the right
H-vector space Hn. Let lis refer to these as Η-lines. Each of these are
of real dimension 4. Each element of Hn\{0} determines an Η-line and
the Η-line determined by (£x,... ,fn)* will be the same as that determined
by (i1, · -. ,ξη)1 ^ and only if there is a nonzero element α € И such that
(?, - ■, £nY = (f \ · · · ι ξηΥα (£4 · · ·, ξηα)Κ This defines an equivalence
relation ~ on Hn\{0}, and thus we may also think of ELPn l as (Ηη\{0}) /~.
The element of ШРп l determined by (ξ1,...,ξ")* is denoted by [ξ1,..., ξη].
Notice that the subset {ξ e Шп : \ξ\ = 1} is S4""1. Just as for the
complex projective spaces, we observe that all such Η-lines contain points of
54n_1, and two points f, ζ Ε S4n λ determine the same Η-line if and only if
ξ ζα for some α with \a\ = 1. Thus we can think of ELF1"*1 as a quotient
of 54n l. When viewed in this way, we also denote the equivalence class of
С (£Xi · ■ · >Ο1 € 54η_1 by [ξ] = [ξ1,...,ξη]. The equivalence classes are
clearly the orbits of an action as described in the following example.
Example 5.121. Consider 54n x regarded as the subset of Шп given by
S*n~l {ξΕΜη: \ξ\ = 1}. Here ξ = (ξ\ ... ,ξη)* and \ξ\ = Ef С ■ Now
we define a right action of 17(1, H) on S471"1 by (ξ, a) ■-► ξα (ξια,..., ξηα)\
This action is free and proper. The quotient is the quaternionic projective
space ELPn_1, and we have the quotient map denoted by p,
c4n—1
Ρ
τ
Hpn-1
This map is also referred to as a Hopf map. Recall that Z2 = {1, — 1}
acts on Sn~l = Rn on the right (or left) by multiplication, and the action is
a (discrete) proper and free action with quotient RPn_1, and the examples
above generalize this.
For completeness, we describe an atlas for HP71-1. View HFn 1 as the
quotient S*n~l/~ described above. Let
and define φΗ : Uk -> H""1 Si R4""1 by
vfc(K]) = (iiffc1.--»i,...,ene*1)»
where as before, the caret symbol л indicates that we have omitted the 1 in
the fc-th slot to obtain an element of ЕР-1. Notice that we insist that the
ξΖ1 in this expression multiply from the right. The general pattern for the
240
5. Lie Groups
overlap maps become clear from the example φ% ο ψ2 . Here we have
Ψ3 о v?2 ЧУЫЙ! -'-,Уп) = </>з([Уь1,2/3,.·., 2/n])
= {viVs \ yj\ У4!^\ · · ·, УпУ^1) ·
In the case π = 1, we have an atlas of just two charts {(ί7ι,¥?ι), (t/2,^2)}.
In close analogy with the complex case we have U\ Π Ό<ι = H\{0} and
vi ° vj x(y) ~ у-1 = V2 о ^гх(у) for у e и\{о}.
Exercise 5.122. Show that by identifying Η with R4 and modifying the
stereographic charts on S3 С Μ4 we can obtain an atlas for 53 with overlap
maps of the same form as for HP1 given above. Use this to show that
HP1 ^ S3.
Combining the last exercise with previous results we have
MP1 ~ S1, CP1 ^ S2, HP1 ^ S3.
5.8. Homogeneous Spaces
Let Я be a closed Lie subgroup of a Lie group G. Then we have a right
action of Η on G given by right multiplication r : G χ Η —¥ G. The orbits
of this right action are exactly the left cosets of the quotient G/H. The
action is clearly free, and we would like to show that it is also proper. Since
we are now talking about a right action, and G is the manifold on which we
are acting, we need to show that the map Phght : G χ Η -l· G χ G given
by (p, h) h-> (piph) is a proper map. The characterization of proper action
becomes the condition that
HK:={heH:(K-h)riKJ: 0}
is a compact subset of Η whenever К is compact in G. Let К be any
compact subset of G. It will suffice to show that Ηχ is sequentially com*
pact. To this end, let {Κ)τ^Ν be a sequence in Ηκ- Then there must be
sequences (аг) and (6;) in К such that ath% = b{> Since К is compact
and hence sequentially compact, we can pass to subsequences (a^)) -
and (&i(j)) -eN such that lim^oo a^ = a and Hindoo b^ — 6. Here
i i-4 i(j) is a monotonic map on positive integers; N —► N. This means
that Hindoo h^j) = lim^oo <^/jA(j) = a~1b. Thus the original sequence
{hi} is shown to have a convergent subsequence. Since by Theorem 5.81, Я
is an embedded submanifold, this sequence converges in the topology of #.
We conclude that the right action is proper. Using Theorem 5.119 (or its
analogue for right actions), we obtain the following Proposition.
Proposition 5.123. Let Η be a closed Lie subgroup of a Lie group G.
Then,
5.8. Homogeneous Spaces
241
(i) the right action G χ Η —> G is free and proper;
(ii) the orbit space is the left coset space G/H, and this has a unique
smooth manifold structure such that the quotient map π : G —> G/H
is a surjection. Furthermore, dim(G/H) = dim(G) — dim(if).
If AT is a normal Lie subgroup of G, then the quotient is a group with
multiplication defined by [pi] [^2] = {g\K){92K) = 9\9гК. In this case, we
may ask whether G/K is a Lie group. If К is closed, then we know from the
considerations above that G/K is a smooth manifold and that the quotient
map is smooth. In fact, we have the following:
Proposition 5.124 (Quotient Lie groups). If К is a closed normal subgroup
of a Lie group G, then G/K is a Lie group and the quotient map G —^ G/K
is a Lie group homomorphism. Furthermore, if f : G -> Η is a surjective
Lie group homomorphism, then Ker(/) is a closed normal subgroup, and the
induced map f : G/Ker(f) —l· Η is a Lie group isomorphism.
Proof. We have already observed that G/K is a smooth manifold and that
the quotient map is smooth. After taking into account what we know from
standard group theory, the only thing we need to prove for the first part,
is that the multiplication and inversion in the quotient are smooth. It is
an easy exercise using Corollary 3.27 to show that both of these maps are
smooth.
Consider a Lie group homomorphism / as in the hypothesis of the
proposition. It is standard that Ker(/) is a normal subgroup and it is clearly
closed* It is also easy to verify fact that the induced / map is an
isomorphism. One can then use Corollary 3.27 to show that the induced map / is
smooth. D
If a Lie group G acts smoothly and transitively on Μ (on the right or
left), then Μ is called a homogeneous space with respect to that action.
Of course it is possible that a single group G may act on Μ in more than
one way and so Μ may be a homogeneous space in more than one way. We
will give a few concrete examples shortly, but we already have an abstract
example on hand.
Theorem 5.125. If Η is a closed Lie subgroup of a Lie group G, then the
map G χ G/H —> G/H, given by I : (g ,31 ii) —> gg\H, is a transitive Lie
group action. Thus G/H is a homogeneous space with respect to this action.
Proof. The fact that I is well-defined follows since if g\H = giH, then
$2 gi 6 H, and so ддгН = ggig^gxH = gg\H. We already know that
G/H is a smooth manifold and π : G —> G/H is a surjective submersion.
242
5. Lie Groups
We can form another submersion idc χπ : G x G
following diagram commute:
GxG *G
G x G/H making the
Υ ^
G χ G/H * G/H
Here the upper horizontal map is group multiplication and the lower
horizontal map is the action 1. Since the diagonal map is smooth, it follows from
Proposition 3.26 that I is smooth. We see that I is transitive by observing
that if <7i#, g2H e G/H, then ^ ι {gxH) = g2H. D
It turns out that up to appropriate equivalence, the examples of the
above type account for all homogeneous spaces. Before proving this let us
look at some concrete examples.
Example 5.126. Let Μ = Rn and let G = Euc(n,R) be the group of
Euclidean motions. We realize Euc(n, R) as a matrix group
0
Q
Euc(n,R)
-{[;
: υ e Rn and Q € O(n)
}
x = Qx + v,
The action of Euc(n,R) on Rn is given by the rule
1 0
ν Q
where χ is written as a column vector. Notice that this action is not given
by a matrix multiplication, but one can use the trick of representing the
points χ of Rn by the (n +1) χ 1 column vectors [*], and then we have
[Iq] [χ] = [qsVv]· The action is easily seen to be transitive.
Example 5.127. As in the previous example we take Μ = Rn, but this
time, the group acting is the affine group Aff(n,R) realized as a matrix
group:
Aff(n,R) = j
The action is
1
ν
0
A
ν e Rn and A G
GL(n,R)i
1 0
ν A
and this is again a transitive action
■ χ = Ax + υ.
Comparing these first two examples, we see that we have made Rn into
a homogeneous space in two different ways. It is sometimes desirable to give
different names and/or notations for Rn to distinguish how we are acting on
the space. In the first example we might write En (Euclidean space), and
5.8, Homogeneous Spaces
243
in the second case we write An and refer to it as affine space. Note that,
roughly speaking, the action by Euc(n,M) preserves all metric properties of
figures such as curves defined in En. On the other hand, Aff(n,R) always
sends lines to lines, planes to planes, etc.
Example 5.128. Let Μ = Η := {ζ G С : Im* > 0}. This is the upper
half-plane. The group acting on Η will be SL(2,R), and the action is given
by
ία b \ _az + b
\ с d ) ~ cz + d'
This action is transitive.
Example 5.129. We have already seen in Example 5.102 that both O(n)
and SO(n) act transitively on the sphere Sn~l С R71, so Sn~l is a
homogeneous space in at least two (slightly) different ways. Also, both SU(n) and
U(n) act transitively on 52n_1 С С71.
Example 5.130. Let V^k denote the set of all fc-frames for Rn, where by
a Jfc-frame we mean an ordered set of к linearly independent vectors. Thus
an n-frame is just an ordered basis for Rn. This set can easily be given
a smooth manifold structure. This manifold is called the (real) Stiefel
manifold of fc-frames. The Lie group GL(n,R) acts (smoothly) onl^fc
by д · (ei,..., e&) = (ge\,..., ge^). To see that this action is transitive,
let (ei,..., efc) and (/i,..., Д) be two it-frames. Extend each to n-frames
ei,..., e&) - - -1 en) and (/i,..., Д,..., fn). Since we consider elements of
ln as column vectors, these two η-frames can be viewed as invertible nxn
matrices Ε and F. If we let g := EF~l, then gE = F, or g · (ei,..., e^)
flei,...,^efc) = (/ι,..-,Λ).
Example 5.131. Let Vnjk denote the set of all orthonormal fc-frames for Rn,
where by an orthonormal /c-frame we mean an ordered set of к orthonormal
vectors. Thus an orthonormal η-frame is just an orthonormal basis for Mn.
This set can easily be given a smooth manifold structure and is called the
Stiefel manifold of orthonormal fc-frames. The group 0(n,R) acts
t ansitively on Vn>k f°r reasons similar to those given in the last example.
Theorem 5.132. Let Μ be a homogeneous space via the transitive action
I : G χ Μ —l· M, and let Gp be the isotropy subgroup of a point ρ G M.
Recall that G acts on G/Gp. If G/Gp is second countable (in particular
f G is second countable), then there is an equivariant diffeomorphism φ :
G Gp-> Μ such that ф{дСр) = g - p.
Proof. We want to define φ by the rule ф{дСр) — д-р, but we must show
that this is well-defined. This is a standard group theory argument; if
giGp = g2Gp, then gx lg2 G Gp, so that (дг lg2) - ρ = ρ or gi - ρ = g2 · p.
244
5. Lie Groups
This map is surjective by the transitivity of the action /. It is also injective
since if <f>(giGp) = 0(52^), then gi · ρ = g2 · ρ or (fl^1^) · Ρ = Ρ, which
by definition means that #f x<72 € Gp and g\Gp = Р2^р. Notice that the
following diagram commutes:
G/Gp -^ Μ
Prom Corollary 3.27 we see that φ is smooth.
To show that φ is a diffeomorphism, it suffices to show that the rank
of φ is equal to dim Μ or in other words that φ is a submersion. Since
Φ{99ι@ρ) = (99i) ' Ρ = 9Φ{9ι@ρ)> the map φ is equivariant and so has
constant rank. By Lemma 3.28, φ is a submersion and hence in the present
case a diffeomorphism. D
Without the technical assumption on second countability, the proof
shows that we still have that φ : G/Gp —> Μ is a smooth equivariant bijec-
tion.
Exercise 5.133. Show that if instead of the hypothesis of second count-
ability in the last theorem we assume that θρ has full rank at the identity,
then φ : G/Gp —> Μ is a diffeomorphism.
Let I : G χ Μ -> Μ be a left Lie group action and fix po € M. Denote
the projection onto cosets by π and also write iP():j4 gpo as before. Then
we have the following equivalence of maps:
G = G
π \ΘΡ0
Υ У
G/Gp - Μ
Exercise 5.134. Let G act on Μ as above. Show that if P2 = fiPi for some
g G G and pi,p2 £ M, then is a natural Lie group isomorphism GP1 = Gn
and a natural equivariant diffeomorphism G/GPl = G/GP2.
We now look again at some of our examples of homogeneous spaces and
apply the above theorem.
Example 5.135. Consider again Example 5.126. The isotropy group of the
origin in MP is the subgroup consisting of matrices of the form
(0 q)>
5.8. Homogeneous Spaces
245
where Q eO(n). This group is clearly isomorphic to 0(n,R), and so by the
above theorem we have an equivariant diffeomorphism
En~Euc(n,R)
0(n,R)
Example 5.136. Consider again Example 5.127. The isotropy group of the
origin in Rn is the subgroup consisting of matrices of the form
U a)*
where A E GL(n,R). This group is clearly isomorphic to GL(n,R), and so
by the above theorem we have an equivariant diffeomorphism
Aff(n,R)
GL(n,R)"
It is important to realize that there is an implied action on Rn, which is
different from that in the previous example.
Example 5.137. Consider the action of SL(2, R) on the complex upper half-
plane Η = C+ as in Example 5.128. We determine the isotropy subgroup
for the point г = у/^Л . A matrix A = (ac \) is in this subgroup if and only
if
ai + b
——; —*·
сг + а
This is true exactly if be — ad = 1 and bd + ac = 0, and so the isotropy
subgroup is SO(2,R) (= S1 = E7(1,C)). Thus we have an equivariant
diffeomorphism
„ SL(2,R)
+ ~SO(2,R)·
Example 5.138. Prom Example 5.129 we obtain
0(n) cn-x^ SO(n)
qn-l ~ ^\'Ч qn-1 ~
~0{n-lY ~SO(n-l)'
S2n 1 ^ U(n) g2n 1 s SU(n)
U(n-l)' SU(n-l)'
Example 5.139. Let (ei,... ,en) be the standard basis for Rn. Under the
action of GL(n,R) on V^k given in Example 5.130, the isotropy group of
the fc-frame (e^+i,..., en) is the subgroup of GL(n, R) of the form
(i!)
for AeGL(n-k,M).
We identify this group with GL(n — k, R) and then we obtain
, „ GL(n,R)
n·* GL(n-fc,R)"
246
5. Lie Groups
Example 5.140. A similar analysis leads to an equivariant diffeomorphism
„ Q(n,R)
n* 0{п-кЛУ
where Vn,k is the Stiefel manifold of orthonormal fc-planes of Example 5.131.
Notice that taking к = 1 we recover Example 5.135.
Exercise 5.141. Show that if к < η, then we have Vn^ — so(n k Ш) *
Next we introduce a couple of standard results concerning connectivity.
Proposition 5.142. Let G be a Lie group acting smoothly on M. Let the
action be a left (resp. right) action. If both G and M\G (resp. M/G) are
connected, then Μ is connected.
Proof. Assume for concreteness that the action is a left action and that G
and M\G are connected. Suppose by way of contradiction that Μ is not
connected. Then there are disjoint open sets U and V whose union is M.
Each orbit G -pis the image of the connected space G under the orbit map
9 *-> 9P and so is connected. This means that each orbit must be contained
in one and only one of U and V* Since the quotient map π is an open map,
7r(i7) and ir(V) are open, and from what we have just observed they must
be disjoint and ir(U) U7r(Vr) = M\G. This contradicts the assumption that
M\G is connected. D
Corollary 5.143. Let Η be a closed Lie subgroup ofG. If both Η and G/H
are connected, then G is connected.
Corollary 5.144. For each η > 1, the groups SO(n), SU(n) and U(rc) are
connected while the group O(n) has exactly two components: SO(n) and the
subset o/0(n) consisting of elements with determinant — 1.
Proof. The groups SO(l) and SU(1) are both connected since they each
contain only one element. The group U(l) is the circle, and so it too is
connected. We use induction. Suppose that SO (A:), SU(fc) are connected for
1 < к < η — 1. We show that this implies that SO(n), SU(n) and U(n) are
connected. Prom Example 5.138 we know that Sn~l = SO(n)/SO(n- 1).
Since Sn г and SO(n — 1) are connected (the second by the induction
hypothesis), we see that SO(n) is connected. The same argument works for
SU(n) and U(n).
Every element of O(n) has determinant either 1 or —1. The subset
SO(n) С O(n) is closed since it is exactly {g G O(n) : detg = 1}. Fix an
element ao with det ao = — 1. It is easy to show that aoSO(n) is exactly the set
of elements of 0(?г) with determinant —1 so that SO(rc) U aoSO(n) — O(n)
and SO(n) П aoSO(n) = 0. Indeed, by the multiplicative property of
determinants, each element of aoSO(n) has determinant —1. But aoSO(n)
5.8. Homogeneous Spaces
247
also contains every element of determinant —1 since for any such д we have
д — αο {%l9) ancl ao19 ^ SO(n). Since SO(n) and aoSO(n) are
complements of each other, they are also both open. Both sets are connected, since
9 H· aoff is a diffeomorphism which maps the first to the second. Thus we
see that SO(ra) and aoSO(n) are the connected components of O(n). D
We close this chapter by relating the notion of a Lie group action with
that of a Lie group representation. We give just a few basic definitions, some
of which will be used in the next chapter.
Definition 5.145. A linear action of a Lie group G on a finite-dimensional
vector space V is a left Lie group action A:GxV->V such that for each
д £ G the map X9 : ν ь-» \(д, ν) is linear.
The map G —l· GL(V) given by д ι-> \(g) :== \g is a Lie group homo-
morphism and will be denoted by the same letter λ as the action so that
\(g)v := X(g,v). A Lie group homomorphism λ : G —>· GL(V) is called
a representation of G. Given such a representation, we obtain a linear
action by letting \{g,v) := X(g)v. Thus a linear action of a Lie group is
basically the same as a Lie group representation. The kernel of the action
is the kernel of the associated homomorphism (representation). An effective
linear action is one such that the associated homomorphism has trivial
kernel, which, in turn, is the same as saying that the representation is faithful.
Two representations λ : G ->■ GL(V) and λ': G -> GL(V') are equivalent
if there exists a linear isomorphism Τ : V -> V such that Τ о Хд = \'g ο Τ
for all g.
Exercise 5 Л 46. Show that if λ : G χ V -* V is a map such that \g :v \-l·
λ g}v) is linear for all g, then λ is smooth if and only if Xg : G -> GL(V) is
smooth for every g G G. (Assume that V is finite-dimensional as usual.)
We have already seen one important example of a Lie group
representation, namely, the adjoint representation. The adjoint representation came
from first considering the action of G on itself given by conjugation which
leaves the identity element fixed. The idea can be generalized:
Theorem 5.147. Let I: G χ Μ —>· Μ be α (left) Lie group action. Suppose
thatpo G Μ is a fixed point of the action (lg(po) = Po for all g ). The map
№ : G-+GL(TPQM)
given by
$ й Lie group representation.
248
5. Lie Groups
Proof. Since
1ы(дт) = ЗДШ) = ΓΛ(ϊβ1 о гя)
= ΤΡ0ίί71οΓΡ0Ζ92 = /(«')(^1)^(ρ2))
we see that /to) is a homomorphism. We must show that №°) is smooth. It
will be enough to show that g н-> a(TPolg-v) is smooth for any ν Ε T^M and
any a Ε ГД M. This will follow if we can show that for fixed vo Ε Τ^Μ, the
map G -> ΓΜ given by g ь-> Тро^ · vq is smooth. This map is a composition
G-+TGxTM^T(GxM)™ TM,
where the first map is g \-> (05,г>о), which is clearly smooth. By Exercise
5.146 this implies that the map G xTPoM -* TP0M given by (3, v) н· TPQlg-v
is smooth. D
Definition 5.148. For a Lie group action I :G χ Μ —► Μ with fixed point
po, the representation i^°) from the last theorem is called the isotropy
representation for the fixed point.
Now let us consider a transitive Lie group action I : G χ Μ —► Μ
and a point po. For notational convenience, denote the isotropy subgroup
Gpo by Я. Then Я acts on Μ by restriction. We denote this action by
A : Я χ Μ -> Μ,
A : (Λ, ρ) ■-> ftp for h Ε Я = Gpo.
Notice that po is a fixed point of this action, and so we have an isotropy
representation \&°) ; Я х T^M -* TpoM. On the other hand, we have
another action С : Я χ G ->· G, where C^ : C? -* G is given by g i-> hph *
for h Ε H. The Lie differential of C^ is the adjoint map Ad^ : g —» g. The
map Сл fixes Я, and Ad^ fixes fj. Thus the map Ad^ : g —> g descends
to a map Ad^ : g/ϊ) -> g/f). We are going to show that there is a natural
isomorphism TpoM = g/f) such that for each h Ε Я the following diagram
commutes:
(5-4) ф-^+Ф
Υ д(Р0) Υ
ТроМ^^ГроМ
One way to state the meaning of this result is to say that h \-> Adh is
a representation of Я on the vector space g/f), which is equivalent to the
linear isotropy representation. The isomorphism T^M = g/f) is given in the
5.9. Combining Representations
249
r«*«>=*
following very natural way: Let ξ Ε ij and consider Τβπ(ξ) Ε ΤΡ0Μ. We
have
7г(ехр££) = О
Ιί=0
since βχρξί G f) for all t> Thus f) С Ker(Te7r). On the other hand,
dimf) = dimuT = dim(Ker(Te7r)), so in fact I) = Ker(Te7r) and we obtain an
isomorphism g/f) = TP0M induced from Te7r. Let us see why the diagram
(5.4) commutes. First, L^ is well-defined as a map from G/H to itself and
the following diagram clearly commutes:
G
ch
+ G
G/H -^ G/H
Using our equivariant difFeomorphism φ : G/H
lent commutative diagram
G—^G
M, we obtain an equiva-
UP0
'PO
Ah
M- *M
Applying these maps to exp££ for ξ Ε g, we have
exp£f h
ch
/i(expi£)/i
-1
λ„
"po
(exp t£)po ι s- /i(exp ίξ)ρο
Applying the tangent functor (looking at the differential), we get the
commutative diagram
Adh
0
^►fl
\ λ(Ρο) \
TnM-^T^M
and, taking quotients, this gives the desired commutative diagram (5.4).
5.9. Combining Representations
We close this chapter with a bit about constructing new linear
representations from old ones. Suppose that V is an F-vector space and let В =
(v\y..., vn) be a basis for V. Then denoting the matrix representative of
\9 with respect to В by [Хд]в we obtain a homomorphism G —У GL(n,F)
250
5. Lie Groups
given by g \-+ [\д]в· In general, a Lie group homomorphism of a Lie group
G into GL(n,F) is called a matrix representation of G. We have already
seen that any Lie subgroup of GL(IRn) acts on Mn by matrix multiplication,
and the corresponding homomorphism is the inclusion map G M- GL(Rn).
More generally, a Lie subgroup G of GL(V) acts on V in the obvious way
simply by employing the definition of GL(V) as a set of linear
transformations of V. We call this the standard action of the linear Lie subgroup
of GL(V) on V, and the corresponding homomorphism is just the inclusion
map G «-► GL(V). Choosing a basis, the subgroup corresponds to a matrix
group, and the standard action becomes matrix multiplication on the left of
Fn, where the latter is viewed as a space of column vectors. This action of
a matrix group on column vectors is also referred to as a standard action.
Given a representation A of G in a vector space V, we have a dual
representation λ* of G in the dual space V* by defining A* (5) := (X{g *))*:
V* -> V*. Recall that if L : V -> V is linear, then L* : V* -* V* is
defined by L*(a)(v) = a(Lv) for a e V* and ν e V. This dual representation
is also sometimes called the contragredient representation (especially
whenF = R).
Now let λν and Aw be representations of a lie group G in F-vector
spaces V and W respectively. We can then form the direct sum
representation λν Θ Aw by (λν Θ Xw)g :- Aj Θ Xf for 5 <E G, where we have
W<B\T){v,w) (A^,Aww).
We will not pursue a serious study of Lie group representations but
simply note that a major goal in the subject is the identification and
classification of irreducible representations. A representation A : G —► GL(V) is
said to be irreducible if there is no nonzero proper subspace W of V such
that Afl(W) С W for all g. A large class of Lie groups known as semisimple
Lie groups have the property that their representations break into direct
sums of irreducible representations.
Example 5.149. A homogeneous polynomial of degree d on C2 is a linear
combination of monomials of total degree d. Let H3 denote the vector space
of homogeneous polynomials of degree 2j, where j is a nonnegative "half-
integer" (j = к/2 for some nonnegative integer k). Define Xj : SU(2) ->
GL(tfj) by (Xj(g)f)(*) -'= f(9~1*) for ζ = (zuz2) e C2. Then X3 is
an irreducible representation called the spin-j' representation of SU(2 .
The spin-1/2 representation turns out to be equivalent to the standard
representation of SU(2) in C2. These spin representations play an important
role in quantum physics.
One can also form the tensor product of representations. The definitions
and basic facts about tensor products are given in the more general context of
5.9. Combining Representations
251
module theory in Appendix D. Here we give a quick recounting of the notion
of a tensor product of vector spaces, and then we define tensor products
of representations. Given two vector spaces V and W over some field F,
consider the class Cyxw consisting of all bilinear maps VxW^X, where
X varies over all F-vector spaces, but V and W are fixed. We take members
of CvxW as the objects of a category (see Appendix A). A morphism from,
say, μι; V χ W —> X to μ<ι: V χ W —> Υ is defined to be a linear map I: X
-+ Υ such that the diagram
VxW
commutes.
There exists a vector space Tv,w together with a bilinear map : V x
W -> Ty,w that has the following universal property: For every bilinear
map μ : V χ W —> X, there is a unique linear map μ : Ty,w ~* X such that
the following diagram commutes:
® /^
TV,w
If such a pair (Tv,w?®) exists with this property, then it is unique up to
isomorphism in Cvxw· In other words, if (8) : V χ W —> Ty,w also has this
universal property, then there is a linear isomorphism Ty,w — Ty,w such
that the following diagram commutes:
TV,w
VxW
7V,w
We refer to such universal object as a tensor product of V and W. We
will indicate the construction of a specific tensor product that we denote by
V W with corresponding bilinear map ®:VxW->V<g)W. The idea is
252
5. Lie Groups
simple: We let V® W be the set of all linear combinations of symbols of the
form ν ® w for ν G V and w € W, subject to the relations
(v\ + V2) ® w = v\ ® w + V2 ® w,
V ® (W\ + ^2) = V ® Ιϋχ + V ® 1^2,
г (г? ® w) = то ® it; = г; ® гг^, for r Ε F.
The map ® is then simply ® : (v7 w) —> г;®w. Somewhat more pedantically,
let F(V x W) denote the free vector space generated by the set V χ W (the
elements of V χ W are treated as a basis for the space, and so the free space
has dimension equal to the cardinality of the set V x W). Next consider the
subspace Д of F(V χ W) generated by the set of all elements of the form
(av,w) — α(ν,ΐϋ),
(ν,aw) — a(v,w),
(vi + v2,w) - {vi,w) + (v2,tu),
(v, tui + w2) - (ν,ιοί) + (v, ги2),
for vi, V2, υ Ε V, wi, гУ2, w G W, and α Ε F. Then we let V ® W be defined
as the quotient vector space F(V χ W)/i2, and we have a corresponding
quotient map F(V χ W) -* V®W. The set V χ W is contained in F(V x W),
and the map ®:VxW—»V®Wis then defined to be the restriction of
the quotient map to V χ W. The image of (v, w) under the quotient map is
denoted by ν ® w.
Tensor products of several vector spaces at a time are constructed
similarly to be a universal space in a category of multilinear maps (Definition
D.13). We may also form the tensor products two at a time and then use the
easily proved fact that (V ® W) ® U = V ® (W ® U), which is then denoted
by V ® W ® U. Again the reader is referred to Appendix D for more about
tensor products.
Elements of the form ν ® w generate V ® W, and in fact, if (ei,.,., er)
is a basis for V and (/1,. · ·, fs) is a basis for W, then
{ei ® fj : 1 < i < r, 1 < j < s}
is a basis for V ® W, which therefore has dimension rs = dim V dim W.
One more observation: If A : V —► X and В : W —>► Υ are linear maps,
then we can define a linear map A®£:V®W—»X®Y. First note that
the map {v,w) ь->- Αν ® Bw is bilinear. Thus, by the universal property,
there is a unique map A ® В such that
(A ®B)(v®w) = Av® Вю,
for all ν e V, w € W.
Problems
253
Notice that if A and В are invertible, then A ® В is invertible with
(A ® £)_1(v ® w) = A~lv ® В lw. Pick bases for V and W as above and
bases {e'l5..., e'r} and {f[,..., f3} for X and Υ respectively. For τ € V® W,
we can write r = rlJei (8) f3 using the Einstein summation convention. We
have
A ® В(т) = А (8) B(ry e, (8) /,·)
= т'Мец ® Б/j
= т*4*в} (βί ® //),
so that the matrix of A <8> В is given by (^4 (8> 5)fj = A^Bly
Let λν and Aw be representations of a Lie group G in F-vector spaces
V and W, respectively. We can form a representation of G in the tensor
product space V<8>W by letting (λν ® Aw) :- λ^®λ^ for all g e G. There
is a variation on the tensor product that is useful when we have two groups
involved. If Av is a representation of a Lie group G\ in the F-vector space
V and Aw is a representation of a Lie group G<i in the F-vector space W,
then we can form a representation of the Lie group G\ χ G2, als° called the
tensor product representation and denoted λν ® Aw as before. In this case,
the definition is (λν ® Aw), . := λ^ ® A^. Of course if it happens that
G\ = G2, then we have an ambiguity since AV®AW could be a representation
of G or of G χ £?. One can usually determine which version is meant from
the context. Alternatively, one can use pairs to denote actions so that an
action A : G χ V —> V is denoted (G, A). Then the two tensor product
representations would be (G χ G, Av ® Aw) and (G, λν ® Aw), respectively.
Problems
(1) Verify that each of the groups described in Section 5.2 is (isomorphic to)
a Lie subgroup of an appropriate Lie group of linear automorphisms.
(2) Show that proper continuous maps are closed maps.
(3) Show that SL(2, C) is simply connected and that ρ : SL(2, C) -> Mob is
a universal covering homomorphism. See Example 5.50.
(4) Prove Proposition 5.83.
(5) Let 0 be a Lie algebra of a Lie group G. Show that the set of all
automorphisms of g, denoted Aut(g), forms a Lie group (actually a Lie
subgroup of GL(g)).
254
5. Lie Groups
(6) Show that if we consider SL(2,R) as a subset of SL(2, C) in the obvious
way, then SL(2,R) is a Lie subgroup of SL(2,C) and p(SL(2,R)) is a
Lie subgroup of Mob. Show that if Τ Ε p(SL(2,R)), then Τ maps the
upper half-plane of С onto itself (bijectively).
(7) Show that for ν Ε TeG, the field defined byp^ Lv(g) := TLg . ν is
automatically smooth.
(8) Determine explicitly the map Γ/inv : T7GL(n,R) -►TjGL(n,R), where
inv : GL(n,R) ->GL(n,R) is defined by inv(A) = A"1.
(9) Let Η be the set of real 3x3 matrices of the form
Л =
1 a b
0 1c
0 0 1
Find a global chart for Η and show that this and the usual matrix
multiplication gives Η the structure of a Lie group.
(10) If G is a connected Lie group and h : G -» Η is a Lie group homomor-
phism with discrete kernel K, then К С Z(G), where Z(G) = {χ Ε G:
xg = gx for all g G G} is the center of G.
(11) Show that for a Lie group G, the conjugation map Cg : G —l· G defined
Ьужн gxg"1 is a Lie group isomorphism. Show that the map С : g ->
Diff (G) is a group homomorphism. Note that we have not defined any
Lie group structure on Diff (G).
(12) Consider the map TeCg : TeG -> TeG. Show that g н^ ТеСд is a Lie
group homomorphism from G into GL(TeG).
(13) Show that SO(3) is the connected component of the identity in 0(3).
Show that the special Lorentz group SO(1,3) is not connected. Show
that the first entry of elements of SO(1,3) must have absolute value
greater than or equal to 1. Define SO(3,1)^ as the subset of SO(l,3)
consisting of matrices with positive first entry (which must be greater
than 1). Show that SO(3,1)^ is connected (and hence the connected
component of the identity in 0(1,3)).
(14) Let A E $1{V) = L(V,V) for some finite-dimensional vector space V.
Show that if A has eigenvalues {Аг}г-1т_)П, then ad(A) has eigenvalues
{λ; — Хк}з,к=1,...,п' Hint: Choose a basis for V such that A is represented
by an upper triangular matrix. Show that this induces a basis for gt(V)
such that with the appropriate ordering, ad (A) is upper triangular.
(15) Fix a nonzero vector w Ε R3 with length θ = \ w\\. Let Lw : R3 -> R3
denote the linear transformation ν ь-> wxv, where χ is the vector cross
product. Show that for any right handed orthonormal basis {е1,б2,ез}
Problems
255
with ез parallel to w we have
ехр(1^)е1 = cos θ e\ + sin θ e2,
exp(Lw)e2 = — sin θ e\ + cos θ β2,
exp(Lw)e3 = e3.
16) Let L-u, be as in the previous problem. Show that
r T sin θ _ 1 — cos 0 _ о
exp I,™ = I + —q-Lw + —^ Lw>
where ^^ and * ffi8* are defined in the obvious way using power series.
17) Let Α, Β Ε fll(V), where V is a finite-dimensional vector space over the
field F = R or C, and show that the following statements are equivalent:
(a) [A,B] = 0.
(b) expsA and exptB commute for all s,t Ε F.
(c) exp (sA + tB) = exp(sA)exp (IB) for all s,iEF.
18) Let V be a finite-dimensional normed space over the field R (resp. C).
Show that if Σ^° ο αηΧη is an absolutely convergent real (resp. complex)
power series with radius of convergence i?, then
η Ο
converges (absolutely) in the normed space fli(V) for \\A\\ < R.
19) Show that if G is a connected Lie group, then K\{G) is abelian.
20) Let G be a Lie group and denote by μ : G x G —> G the multiplication
map.
(a) Identify T(GxG) with TG χ TG in the usual way. Show that the
tangent map Τ μ : TG χ TG —> TG defines a Lie group structure
on TG and show that if (vg,Wh) G TgG χ T^G, then
Τμ(ν9, wh) = ТД^^ + TLgWh,
where i?^ and Z^ are right and left multiplications respectively.
(b) Show that under the isomorphism Gxj with TG, the Lie group
multiplication takes the form
(g,A)-(h,B) = (gh,Adh ιΑ + Β).
21) Let Μ be a smooth manifold, let G be a Lie group, and let r : Μ χ G —>
Μ be a right Lie group action. Recall from the previous problem that
TG is naturally a Lie group.
(a) Show that Tr : TM x TG ->· TM defines a right Lie group action.
256
5. Lie Groups
(b) For each A E Q = TeG, define a vector field σ(Α) on Μ by
σ(Α)(ρ) := -jfi\t=zQpexp(Ai). Show that for А,Б G g, we have
σ([ΑΒ]) = ΚΛ),σ(Β)].
(c) Show that if AL denote the left invariant vector field generated by
A, then for vp e TpM,
Tr(vPi AL(g)) = TR9 (tip) + a(A)(pg).
Chapter 6
Fiber Bundles
The notion of a bundle is basic in both topology and geometry. The reader
need not master everything in this chapter before going on to later chapters
and should skip forward rather than become too bogged down. The
definition and basic examples of a vector bundle are most important. In this
chapter we also introduce the more advanced notion of a structure group
for a bundle. There is more than one approach to structure groups. We
start out with an approach that takes the notion of a G-atlas as basic. This
is essentially the approach of Steenrod [St]. One may also approach
unbundle structures by first introducing the notion of a principal G-bundle
(see [Hus]). We discuss principal bundles near the end of this chapter.
6.1. General Fiber Bundles
Definition 6.1. Let F, M, and Ε be Cr manifolds and let π : Ε -ϊ Μ be
a Cr map. The quadruple (£?, π, Μ, F) is called a (locally trivial) Cr fiber
bundle if for each point ρ € Μ there is an open set U containing ρ and a
Cr diffeomorphism φ : π~ι(υ) —^ U χ F such that the following diagram
commutes:
п~г{и) ■ >- UxF
In differential geometry, attention is usually focused on C°° fiber bundles
smooth fiber bundles), but the continuous case is also of interest. We will
restrict ourselves to the smooth case, but the reader should keep in mind
that most of the definitions and theorems have analogous C° versions where
257
258
6. Fiber Bundles
1Ш
I
* Μ
Ρ
Figure 6.1. Schematic for fiber bundle
the spaces are assumed merely to be sufficiently nice topological spaces and
the maps are only assumed to be continuous. In this chapter, all maps and
spaces will be smooth unless otherwise indicated.
Definition 6.2. If (Ε, π, Μ, F) is a smooth fiber bundle, then Ε is called the
total space, π is called the bundle projection, Μ is called the base space
and F is called the typical fiber. For each ρ e Μ, the set Ep := π^1^) is
called the fiber over p.
Because the quadruple notation is cumbersome, it is common to
denote a fiber bundle by a single symbol. For example, we could write ξ
(Ε, π, Μ, F). In the literature, it is common to see Ε refer both to the total
space and to the fiber bundle itself (an abuse of notation). The map π is
also a common way to reference the fiber bundle.
Example 6.3. For smooth manifolds Μ and F, we have the projection
prx : Μ χ F -> M. Then, (Μ χ i^pr^M,^) is a fiber bundle called a
product bundle (or trivial bundle).
Exercise 6.4. Show that if ξ — (Ε, π, Μ, F) is a (smooth) fiber bundle, then
π : Ε —> Μ is a submersion and each fiber π^1 (ρ) is a regular submanifold
which is difFeomorphic to F. Show that if both F and Μ are connected,
then Ε is connected.
There are various categories of bundles with corresponding notions of
morphism. We give two very general definitions and modify them as needed.
Definition 6.5 (Bundle morphism (type I)). Let ξι = (ΕΊ,πι,Μ, Fi) and
ξ2 = (£^2,π2, M,i<2) be smooth fiber bundles with the same base space M.
A (type I) bundle morphism over Μ from ξι to £2 is a smooth map
6.1. General Fiber Bundles
259
h : E\ —У E2 such that the following diagram commutes:
Ex *E2
Μ
This type of morphism is also called an M-morphism or a morphism over
M. If h is also a diffeomorphism, then h is called a bundle isomorphism
over Μ and in this case the bundles are said to be isomorphic (over M)
or equivalent. A bundle isomorphism from a bundle to itself is called a
bundle automorphism.
Definition 6.6 (Bundle morphism (type II)). Let ξι = (£?ι,πι, Afi,Fi)
and £2 = №2>Я2,М2, F2) be smooth fiber bundles. A (type II) bundle
morphism from ξι to £2 is a pair of smooth maps / : Ει —> E2 and
/ : M\ —> M2 such that the following diagram commutes:
/
Ει >■ E2
πι
π2
Μι^-^Μ2
We write (/, /) : ξι —> ξ2 and say that / is a bundle morphism along /. If
both / and / are diffeomorphisms, then we call (/, /) a bundle isomorphism.
In this case, we say that the bundles are isomorphic over /.
Note that as a fiber preserving map, / determines / and so it is also
proper to refer to / as the bundle morphism and we sometimes say that /
is a bundle morphism along (or over) /.
Warning: The definitions of bundle morphism above are quite relaxed.
There are a variety of definitions in the literature that require more than
the definitions above, especially when structure groups (discussed below)
are emphasized.
Definition 6.7. A (global) smooth section of a fiber bundle ξ = (£7,πτ
M,F) is a smooth map σ : Μ —> Ε such that π ο σ = \ам (i-e-, σ(ρ) G Ep).
A local smooth section over an open set U is a smooth map σ : U —» Ε
such that π ο σ = idt/. The set of smooth sections of ξ is denoted by Γ (ξ)
or sometimes by Γ (Ε) or Γ (π).
A very important point is that a fiber bundle may not have any global
smooth sections. If two bundles are equivalent, via a bundle isomorphism h
(of type I), then there is a natural bijection between the spaces of sections
given by σ Η /ι ο σ. This means that one quick way to conclude that two
260
6. Fiber Bundles
bundles are not equivalent is by showing that one bundle has global sections
while the other does not.
A bundle chart essentially gives a local type I bundle isomorphism, but
it is sometimes more natural to consider bundle charts which are local type
II isomorphisms. We will call these type II bundle charts. These are of
the form (0,x), where φ : π_1ϊ7 —» V x F and χ : U —> V are smooth
diffeomorphisms such that the following diagram commutes:
π-1!/ Λ VxF
υ Α ν
Usually, the pair (U, x) is a chart on the base manifold. The two types of
bundle charts are equivalent since one may always compose a type II chart
with (x_1, idjr) to obtain a type I bundle chart.
More restricted notions of bundle morphism can be obtained by making
requirements such as that the induced maps on fibers /|π-ΐ/ \ : π^ίρ) -►
7Г21{р) are C°° diffeomorphisms.
The maps φ : 7Γ—1(ί7) -> U χ F occurring in the definition of a fiber
bundle are said to be local trivializations of the bundle. It is easy to see
that such a local trivialization must be a map of the form φ = (π\π-ια;) > $)
where Φ : ir~l(U) —> F is a smooth map with the property that Φ\Ε :
Ep —>· F is a diffeomorphism. We will just write (π, Φ) instead of the more
pedantic (Tr^-imx , Φ). Thus
Ф{у) = (тг,Ф)(у) := (π(ι/),Φ(ί/)).
The second component map Φ is called the principal part of the local
trivialization. A pair (17,0), where φ is a local trivialization over U С М, is
called a bundle chart. (Clearly, a local trivialization and a bundle chart
are essentially the same thing.) A family {(иа,фа)}аеА of bundle charts
such that {Ua}aeA is a cover of Μ is said to be a bundle atlas. Given two
such bundle charts (Ua> φα) and (Щ, φβ), we have
Φα = Κ Φα) : ΤΓ-^ϋα) -> Ua Χ F
and similarly for φ β — (π, Φ^). If Ua Π Up is not empty, then π_1(ί/α) Π
n~l(Up) = π l(Ua Π Uβ) is not empty and we have overlap maps
Φα ο φβ1 : {Ua n^)xF-> {Ua Π υβ) x F.
Since Φα|я is a diffeomorphism for each ρ Ε Ua, the map Φα|# ° Φ^Ι^1:
F —> F is a diffeomorphism for all ρ Ε Ua Π i7^. We then obtain a map
Φα/з '. UanUfi-+ Diff (F) defined by
риФ^(р) = Фа|£,оФ41.
6.1. General Fiber Bundles
261
It follows that
0α ° Φβ1^ ν) = (P> $afl(p){v))-
The functions Φαβ : Ua Π ΐ/β —> Diff (F) are called transition maps or
transition functions. Given a bundle atlas, the corresponding transition
functions clearly satisfy the following "cocycle conditions":
Φαα(ρ) =β forpE Ua,
Φαβ{ρ) = (Φβα(ρ))~1 for peUaH Up,
Φαβ(ρ) ο ΦβΊ(ρ) ο ΦΊα{ρ) = id for ρ^υαηυβΠ Ι77>
for all α,/3,7.
Notation 6.8. We will often denote Φαβ(ρ) (у) by Φαβ\ν (j/)> which is, in
many contexts, more transparent.
Diff (F) is a group, and we have a group action DiflF(F) χ F —> F given
by (V>>2/) *-> Ф(у)- However, Diff(F) is too big for many purposes, and we
have certainly not attempted to give Diff (F) a Lie group structure. Even if
we were to somehow extend the notion of Lie group sufficiently to include
Diff(F), it would be infinite-dimensional and thereby take us out of the circle
of ideas we have been developing. Because of this, the transition functions
Φαρ above, which could be called "raw transition functions", might not
be appropriate for our needs. We remedy this below by bringing Lie groups
into the picture. First we give a simple example of a nontrivial bundle.
Example 6.9. The circle Sl can be considered as a quotient R/~ where
χ is equivalent to у if and only if χ — у is an integer multiple of 2π. For
this example, we put an equivalence relation on R χ (—1,1) according to
the prescription (x,t) ~ (χ + 2πη, (—l)nt) for any integer n. The quotient
(Ex (-1,1)) /~ can easily be seen to be a smooth manifold and is none
other than the familiar Mobius band which we denote by MB. Define a
map π: MB ->· R/~ = S1 by π([α:,ί]) = [ж]. We show that this is a fiber
bundle by exhibiting an atlas consisting of three bundle charts. We call it
the Mobius band bundle. We use three bundle charts instead of two in
order that the overlaps be connected sets. Let U\ — {[χ] Ε R/^ : —2π/3 <
χ < 2тг/3} and U2 = {[x] € R/~ : 0 < χ < 4π/3} and [/3 = {[χ] 6 R/~ :
2π/3 < χ < 2π}. Then U\ U U2 U U3 = R/~ = S1. For г = 2,3 define
^••^{Uij^UixS1 by
&(МН(И,«)»
where (ζ,ί) is the unique representative of [x,i] in the set (0,2π) χ (—1,1).
For 0i: 7γ"^1(Ζ7ι) —> U\ χ 51, we define φι{[χ, i\) = ([a?], t), where (ж, t) is the
unique representative of [x,t] in the set (—2π/3,2π/3) χ (—1,1). One can
check that φ2 ο φ%1 = фз о ф^1 = id on the overlap /κ~1{υ2) Π π"1^). Now
consider the overlap π_1(ί7ι) Ππ~ι(υ2). If [x}t] G π_1(ϋ7ι) Ππ-1^)* then
262
6. Fiber Bundles
Figure 6.2. Mobius band
[x, t] is uniquely represented by some (x, i) Ε (0,2π/3) χ (—1,1) and in view
of the definitions we see that <f>2°<t>3l = id also. Finally we consider φχοφ^1.
If [χ,ί] G π_1(ί/ι) Π 7г_1({7з), then it has a unique representative (χ,ί) in
(4π/3,2π) χ (—1,1) and then φ^1 ([χ], t) = [χ, t]. For <£i, we need to represent
[x, t] properly. We use the fact that [x, t] = [x — 27Г, —t] and (x — 2π, — ί) €
(-2π/3,0) χ (-1,1) so that 0i([x - 2ττ, -ί]) = ([χ - 2ττ], -ί) = ([χ], -ί). In
short, we have φι ο ^1([χ,ί]) = ([χ],—£). From these considerations and
the fact that in general φα ο φ β1 (ρ, у) = (ρ, Фа/з (ρ) (y)) we see that
Ф12(р) = id(_U) e Diff(-1,1) for peUif] U2,
Фгз(р) = id(-ifi) e Diff(-1,1) for ρ e ?72 Π C/3,
*1з(р) = - id(_ifi) € Diff (-1,1) for ρ e Ux Π ?73.
A 'twist" occurs on the overlap π~χ(ί7ι) Π n^l(U^). There is no way to
construct an atlas for this bundle without having such a twist on at least
one of the overlaps.
Notice that if we define an action λ of Z2 = {1,-1} on the interval
(—1,1) by λ(<7, χ) н-> дх, then we can describe the transition functions in the
last example by
Φαβ(ρ)(χ) = λ(ββ^(ρ)>χ)>
where g^ : Ua П ϋβ -► Z2 is given by
012 = 1 on Ε/Ί П ?72,
#23 = 1 on [72 Π t/з,
013 = -1 on U± П £/з,
and in this case the g^ satisfy a cocycle condition like the Φαβ. This is
convenient since we understand Z2 very well. It is a zero-dimensional Lie
group. Inspired by this, we seek to put Lie groups into the formalism. This
6.1. General Fiber Bundles
263
will alleviate our concerns about the group Diff (F) mentioned above. We
are also led to the theory of G-bundles that involves the group in subtle
ways.
Definition 6.10. Let {Ua} be an indexed open cover of a smooth manifold
Μ and let G be a Lie group. A G-cocycle on {Ua} is the assignment of a
smooth map gap : Ua Π ΐΐβ —¥ G to every nonempty intersection Ua Π Ιίβ
such that the cocycle conditions hold:
9aa{p) = e forpG Ua,
9αβ(ρ) = (Ρ^α(ρ))"1 for ρΕΐ/αΠί/β
9αβ{ρ)9βΊ(ρ)9Ία{ρ) = e for ρ G Ua Π Щ П С/7,
where e is the identity in G. The family of maps {gap} forms a cocycle.
The idea that we wish to pursue is that of representing the action of
the raw transition maps by using Lie group actions. There is a subtle point
here that the reader should not miss. Consider the following fact: If λ :
G x F —> F is a group action, then by letting К = {g : Xg(p) = ρ for all
ρ Ε F} (the kernel of the action) we obtain an effective action of G/K on
F. Things are not so simple on the global level of bundles as becomes clear
when dealing with the notion of spin structure (See [L-M]). The best way
to explain what is at stake is by the use of the notion of a principal bundle
which we introduce later. Even before we get to that point, we will mention
some things that will provide some idea as to why we need to be careful
about ineffective actions.
We start out assuming that the action is effective (see Definition
1.100):
Definition 6.11. Let ξ = (Ε, π, Μ, F) be a fiber bundle and G a Lie group.
Suppose that we have an effective left action λ : GxF —> F. Let {(0a,Ua)}
be a bundle atlas for ξ. Suppose that for every nonempty intersection ΙΙαΠΐ/β
there exists a smooth map g^ : Ι/αΠί/β -l· G such that А(ра/з(р),у) =
ΦαβΙρ {у) f°r all Ρ ^υαΓ\υβ and у Ε F. Then the atlas {(<£<*, Ua)} is called
a (G, A)-bundle atlas. If the action λ is understood or standard in some
way, one also speaks of a G-bundle atlas.
Because the action λ in the above definition is assumed effective, it
follows that the family {g^} satisfies the cocycle conditions of Definition
6.10. Notice that if we had not assume the action to be effective, then
the maps g^ would not be unique and may not satisfy a cocycle condition
(although they would do so modulo the kernel of the action). Thus if we do
not assume effectiveness, then we have to make the cocycle {g^} part of
the definition. We return to this below.
264
6. Fiber Bundles
The basic definition in the case of an effective action can be formulated
as follows:
Definition 6Л2. Let ξ = (Ε, π, Μ, F) be a fiber bundle and G a Lie group.
Suppose that we have an effective left action λ : G χ F —>· F. Two (G, A)-
bundle atlases for f, say {(φα, Ua)} and {($*, t^)}> are strictly equivalent
if the union of the atlases is also a (G, A)-bundle atlas. A strict equivalence
class of atlases is referred to as an effective (G, A)-bundle structure on ξ,
and we say that ξ together with this (G, A)-bundle structure is an effective
(G, A)-bundle. Again, if the action is standard or understood, then it is
common to speak of a G-bundle structure and refer to ξ as a G-bundle.
Actually, there is a tiny point to be made. To keep things neat we should
always arrange that the indexing map a \-¥ (φα, Ua) for any atlas is
infective. Thus when taking the union of two atlases per the definition of strict
equivalence, one may need to reindex so that the g^ are notationally
unambiguous. For example, if we take the union of an atlas {(0i, C/i), (Φ2, U2)}
with an atlas {(V>1, ^i)> (^2)^2)}, then how should the transition maps for
the bigger atlas be denoted? What would g\\ mean? Sometimes the trar
ditional indexing scheme is wisely dropped. Instead one uses the set of
trivializing maps itself as the index set so that a chart is written as (ф, Щ)
or (ψ,υψ), and then one denotes transitions maps by дфф etc. The notion
of maximal (G, A)-bundle atlas is defined in the obvious way by direct
analogy with the notion of maximal atlas for a smooth structure.
Our main emphasis will be on the effective (G, A)-bundles, but as
mentioned above, if we wish to allow ineffective actions, then the notion of atlas
should include the cocycle as part of the data. But even then we have
to be careful. Indeed, for an ineffective action it is conceivable that there
could be a different cocycle {g^} such that A(p^(p),y) = Φαβ\ρ (у) for all
у Ε F. Then {(фа, Ua), (gafi) > λ} and {{φα, Ua), (p^), A} would be
different (but possibly equivalent) (G, A)-bundle atlases. For {(0a, Ua,(g^), X}
and {(</>j, Щ, {g[j), A} to define the same (G, A)-bundle it must be the case
that both of the cocycles are contained in a larger cocycle that gives the
transitions for the atlas obtained as the union of the collection of charts
from {{фа,иа)у (5Q/s),A} and {{ф'^Щ), (g'tj),X}. We handle ineffective
actions this way so as to keep aligned with the notion of associated bundle
introduced later.
If there is no chance of confusion, we will drop the adjective effective.
The reader is warned that some standard expositions on fiber bundles allow
ineffective actions right from the start, but in some cases assertions are made
that would only be true in the effective case! It is interesting to note that
in his famous book on the subject [St], Norman Steenrod restricts himself
6.1. General Fiber Bundles
265
to effective actions, although he announces this restriction in one easily
overlooked sentence early in the book.
Notice that an alternative way to say that A(pa^(p),y) = Φ<χβ\ρ(ν) is
ΦαΟφβ1^) = (Ρ, λ(9αβ(ρ)*ν)) ОТ
Φα ο Φβ1(ρ,υ) = (ρ, 9αβ(ρ) ' У)·
(Actually, we shall at first avoid the notation g · у for the action of a group
element g on у since we do not want the beginner to forget the role of the
choice of action.) The maps gap are also called transition functions for
the (G, A)-bundle atlas. If the G is literally a subgroup of Diff (F) and the
action is simply (Ф, у) н-» Ф(у)> then the transition maps gap are simply the
maps Φαβ.
When dealing with (G, A)-bundles there is a more specific notion of mor-
phism and equivalence:
Definition 6.13. Let £l = (£i,7ri,Mi,F) be a (G,A)-bundle with its
(G, A)-bundle structure determined by the strict equivalence class of the
(G, A)-atlas {{φα, Ua)}a€A. Let ξ2 = (#2, π2, Мь F) be a (G, A)-bundle with
its (G, A)-bundle structure determined by the strict equivalence class of the
(G, A)-atlas {(ij>0, Vp)}a€B. Then a type II bundle morphism (h, h) : ξι -> ξ2
is called a (G, A)-bundle morphism along h if
(i) h carries each fiber of Εχ diffeomorphically onto the corresponding
fiber of £2;
(ii) whenever Ua Л h~l (Vfi) is not empty, there is a smooth map Ηαβ :
Ua Π hr1 (Vfi)-+G such that for each ρ 6 Ua Π /ι"1 (Τ^) we have
Γψ/, о ho (φβΙ^χ^)) Ί Ы = ΚΚβ{ρ),υ) for all у e F>
where as usual φα = (πι, Φα) and ψβ = (π2, Φ^).
If Μι — M2 and h = id^ >then we call h a (G, A)-bundle equivalence
over M- (In this case, h is a diffeomorphism.) Condition (ii) simply says that
h must be given by the action on each fiber when viewed in (G, A)-charts. For
this definition to be good it must be shown to be well-defined. That is, one
must show that condition (ii) is independent of the choice of representatives
{{4>aiUa)}aeA and {(v?ijVi)iGj} of the strict equivalence classes of atlases
that define the (G, A)-bundle structures. We leave this as an exercise. Later
we will discover another, perhaps better, way to talk about equivalence of
(G, A)-bundles.
The product bundle prx : Μ χ F -» Μ has a trivial (G,A)-bundle
structure for any A acting on F. Indeed, we just take the structure given
266
6. Fiber Bundles
by the single bundle chart (idjMxF, M) where we have the resulting cocycle
{#11}, and where gn (x) := e for all x. In fact, if {Ua}aeA is any open cover of
MY then {(idt/aXjF, Ua)}aeA is a (G, A)-atlas strictly equivalent to the atlas
{{i&MxF,M)} and so defining the same trivial (G, A)-bundle. By trivial
(G, A)-bundle over Μ we will mean either this product (G, A)-bundle or
one that is (G, A)-equivalent to it.
Remark 6.14. The special case of a (G, A)-bundle equivalence in the case
that Ει = E2 and πι = π2 is interesting but easily misunderstood. Suppose
that {(ipa,Ua)}aeA determines a (G, A)-bundle structure on (E,7r,M,F)
and suppose that {(ψα, Ua)}a€A determines another (G, A)-bundle structure
on (#,π, M, F). Then these two (G, A)-bundle structures might be (G,A)-
bundle equivalent without being the very same. We would have (G, A)-
bundle equivalence in case the two atlases give "cohomologous" cocycles.
This is not the same as strict equivalence despite what one occasionally
finds stated in the literature. Strict equivalence is the notion that defines the
(G, A)-bundle structure and is analogous to equivalence of manifold atlases
which defines the notion of differentiable structure on a manifold. The other,
weaker, notion of equivalence above is analogous to diffeomorphism and we
know that two atlases on a manifold may define different smooth structures,
which may or may not be diffeomorphic. For a more detailed discussion of
these issues see the online supplement [Lee, Jeff].
We have defined fiber bundles, (G, A)-bundles and various notions of
morphism and equivalence. The question of classifying fiber bundles
generally, or (G, A)-bundles more specifically, is a huge part of topology which
we do not have the space to discuss. The interested reader should consult
[Hus], [Span] and [St].
G-bundle structures vs. G-structures. We have deliberately opted
to use the term "G-bundle structure" rather than simply "G-structure",
which could reasonably be taken to mean the same thing. Perhaps the
reader is aware that there is a theory of G-structures on a smooth manifold
(see [Stern]). One may rightly ask whether a G-structure on Μ is nothing
more than a G-bundle structure on the tangent bundle TM where G is a
Lie subgroup of GL(n) acting in the standard way. The answer is both yes
and no. First, one could indeed say that a G-structure on Μ is a kind of
G-bundle structure on TM even though the theory is usually fleshed out in
terms of the frame bundle of Μ (defined below). However, the notion of
equivalence of two G-structures on Μ is different from what we have given
above. Roughly, G-structures on Μ are equivalent in this sense if there is a
diffeomorphism φ such that (Γ0, φ) is a type II bundle isomorphism that is
also a G-bundle morphism along φ.
6.1. General Fiber Bundles
267
Let ξ = (Ε, π, Μ, F) have a (G, A)-bundle structure given by a
maximal (G, A)-bundle atlas {(</><*> ΪΛχ)}α€л· Suppose that there is a subatlas
{(^j,{77)}j€j, J С -A, such that the transition maps for the subatlas take
values in a Lie subgroup Η of G. Then clearly ξ has an (Я, A|H)-bundle
structure, and this is called a reduction of the structure group. We say
that the structure group is reducible.
The following theorem shows how we can build a fiber bundle using a
cocycle and a choice of action.
Theorem 6.15 (Fiber bundle construction theorem). Let Μ and F be
smooth manifolds and let G be a Lie group. Let {Ua}aeA be a cover of Μ
and {Οαβ} a G-cocycle for the cover. For every action λ :G χ F —> F, there
exists a fiber bundle with bundle-atlas {(Ua, фа)} satisfying φα ° φ~β (ρ, у) =
(p>M<7a0(p),y)) on nonempty overlaps Ua Π ϋβ. Thus the resulting bundle
has a (G, λ) -bundle structure.
Proof. On the union Σ := ЦДа} *Ua x F define an equivalence relation
such that (a,p, y) G {a} x Ua x F is equivalent to (0,ρ', yl) G {β} x^xi1
if and only if ρ = ρ* and у = £а/з(р) · у'. Notice that ρ — ρ' is possible
only in the case Ua Π ϋβ ^ 0. The first member of the triple is only needed
to make the union above disjoint. The cocycle conditions ensure that the
equivalence relation is well-defined.
The total space of our bundle is then Ε := Σ/~. The set Σ is essentially
the disjoint union of the product spaces Ua x F and so has an obvious
topology. We then give Ε := Σ/~ the quotient topology. The bundle
projection π is induced by (α,ρ, ν) и· p. То get our trivializations, we
define
фа(е) := (ρ, у) for e G π_1(ϊ7α),
where (p, y) is the unique member of Ua χF such that (a, p, y) G e (recall that
e is an equivalence class). The point here is that (α,ρι, yi) ~ (a,P2, У2) only
^ iPh 2/i) = (P2> У2)- Now suppose U^ Π ϋβ Φ 0. Then for ρ G Ua Π ΌβΛ the
element 071 (p, y) is in π"
[{ft ft I/)]· But since ф^1(р,у) is in 7Γ_1(ί7α), it must be equal to [(ев, q, j/a)]
for some 2/2· This means that p = q and 2/2 = 9αβ(ρ)·ν = λ(<7αβ(ρ), у). Thus
tf/(ft У) = [<*>ft 0α/?(ρ) ■ У] and so
0α о <^(ft У) = (ftfttfOO ' У)·
We leave the existence of the smooth structure ала the routine verification
of the smoothness of these maps to the reader. That the topology induced
is Hausdorff and paracompact is also easy to see. D
Example 6.16. Let S1 be the circle realized as the unit complex numbers.
We will construct a fiber bundle with typical fiber F := (—1,1) CM.
268
6. Fiber Bundles
Let Ui = {егв e S1 : 0 < θ < 2π} and U2 = {e* G S1 : -π < 0 < π}. Then
U\ Π Е/г is a disjoint union of open sets V and W where lEK and — 1 Ε W.
Now we define maps with values in the multiplicative group of two elements
{1, —1} = Z2. Let 011 (ж) 1 for all χ e Uu 922(#) = 1 for all χ € Ε/Ί, and
then let 012 and 321 be defined on U\ Π 02 by
Ч-.
, ν ix on V,
and 521 *·= gi2· Let Z2 act on the symmetric interval (—1,1) С R by
multiplication (so that —1 · χ := —χ). Using the cocycle {511,522,012,521}
and this action on M, Theorem 6.15 above gives a Z2-bundle which can be
shown to be equivalent to the Mobius band bundle described in Example
6.9.
Example 6.17. Use the same cocycle on S1 as in the last example but with
typical fiber Sl and action Z2 χ S1 —> S1 given by letting 1 act as the identity
and -1 act by a rotation of Sl by π, that is -1 · ei0 := е*^+7Г) = -eie. Then
the bundle obtained is a Z2-bundle, sometimes called the twisted torus.
It is of the upmost importance to realize that there is a fiber preserving
diffeomorphism between the twisted torus and the trivial bundle 51 χ S1 4
S1 and yet there is no Z2-bundle equivalence between the twisted torus and
the trivial Z2-bundle S1 χ S1 —ί S1. Thus the twisted torus is trivial as a
general bundle but not as a Z2-bundle (See Problem 5). This shows how
much the involvement of the group matters.
Notice that the previous theorem is true even without the assumption
that the action is effective and we still end up with a genuine (ineffective)
(G, A)-bundle since there is no problem about the cocycle existing. A
quotient by the kernel К of the action would give an effective structure group
action. Thinking of the action as a homomorphism G —> Diff (F), we have
an induced homomorphism G/K -> Diff (F) such that the following diagram
commutes:
G ^Diff(F)
G/K
Definition 6.18. Let ξ = (E,n,M,F) be a smooth fiber bundle and / :
N -4 Μ a smooth map. The pull-back bundle /*ξ = (/*£, πι, Μ, F) (or
induced bundle) is defined as follows: The total space f*E is the set
f*E:={{q,e)€NxE:f(q)=n(e)}.
Then we define πι as the restriction to f*E of the projection prx : Ν χ Ε -»
ΛΓ.
6.1. General Fiber Bundles
269
Notice that the second factor projection map ргг : Ν χ Ε —> Ε restricts
to a map / : f*E -» Ε which is a bundle morphism over the map /:
\ , \
N >M
The map / restricts to a diffeomorphism on each fiber. If φ = (π, Φ) is a
trivialization of the bundle ξ over the open set {/, then (πι, Φ) :— (πι, Φ ο /)
is a trivialization of /*£ over the open set /_1(i7). Thus a bundle atlas on
£ induces a bundle atlas on /*£. If {Φαβ} are (raw) transition maps for ξ
corresponding to a bundle atlas {(π, Фа)}аеА, then {Φαβ ° /} are transition
maps for /*£ corresponding to the atlas {(πι, Φα о /)}аед. In fact,
*-U/(f) (*'e) =ф* ° /Ц/(1) ^ = ф«к9) w.
so the inverse is
Thus *«1дхЯ/(, ° */»1дхЯ/(„ iS ^теП Ъу
У ·"► (?, *β\Ε)Μ Ш -* фаЬ/(„ ° */»1в}(„ (У)'
and so
Ф^(в)=Фа|дхВ/(д)Оф/3|Д
7(g)
Furthermore, if {(π, Фа)}аеА is a (G, A)-atlas for £, then since Φαβ\ (у) =
Κ9αβ(ρ),ν), we have Фа/з|дЫ = *ctf(/(?))(y) А(з^(/(д)),у). We see
that /*ξ has a (G, A)-bundle structure with cocycles ^o/. Note, however,
that because of the composition with /, this structure may be reducible to
a smaller group.
Definition 6.19. Let ξ = (Ή,π,Μ, F) be a smooth fiber bundle and let
/ : N -► Μ be a smooth map. A section of ξ along / is a map σ : N —> Ε
such that π ο σ = /. The set of sections along / will be denoted Γ/ (£) or
Tf(E).
If σ : TV —> Ε is a section of ξ along /, then the map σ*: N —l· f*E given
Ъу Ρ *-► (ρ, 0"(ρ)) is a section of the pull-back bundle /*£. It is not hard to
show that all sections of /*£ have this form.
Proposition 6.20. Let ξ = (Ε, π, M,F) be a smooth fiber bundle and f :
Ν -¥ Μ a smooth map. Then there is a natural bisection between Γ/(ξ) and
ГСГ0-
270
6. Fiber Bundles
Proof. Let s € Γ(/*ξ). For each ρ e N, we have *(p) € (f*E)p, which
must have the form (p, y) for some у G £/(p) · Therefore the smooth map
σ := pr2 о s : TV —> Ε has the property that π ο σ = /. Thus we obtain a
map Γ(/*£) —» Г/(£) given by 5 ι-> σ. But the inverse of this map is clearly
σ ί-+ (idjv,^). D
6.2. Vector Bundles
The tangent and cotangent bundles axe examples of a general type of fiber
bundle called a vector bundle. Roughly speaking, a vector bundle is a
parametrized family of vector spaces. We shall need both complex
vector bundles and real vector bundles, and so to facilitate definitions we let
F denote either R or С Let V be a finite-dimensional F-vector space. The
simplest examples of vector bundles over a manifold Μ are the product
vector bundles which consist of a Cartesian product Μ χ V together with
the projection onto the first factor prx : Μ χ V —> Μ. Each set of the form
{χ} χ V С Μ χ V inherits an F-vector space structure from that of V in the
obvious way: a(p, v) + b(p, w) := (p, av + bw). We think of Μ χ V as copies
of V parametrized by M.
Definition 6.21. Let V be a finite-dimensional F-vector space. A smooth
F-vector bundle with typical fiber V is a fiber bundle (E} π, Μ, V) such
that:
(i) for each χ G Μ the set Ex := π^1 (χ) has the structure of a vector
space over the field F, isomorphic to the fixed vector space V;
(ii) every ρ Ε Μ is in the domain of some bundle chart (С/, ф), such
that for each χ e U the map Φ^ : Ex —► V is a vector space
isomorphism where φ — (π, Φ).
Definition 6.22 (Terminology). We refer to a vector bundle as a complex
vector bundle (resp. real vector bundle) if F = С (resp. F = R). "Vector
bundle" shall mean either real or complex vector bundle as determined by
the context.
A bundle chart (С/, ф) of the sort described in the definition is called
a vector bundle chart (VB-chart) or a local vector bundle trivialization
(over U). In the setting of vector bundles, a local trivialization is assumed
to be linear on fibers. A family {(Uai φα)} of vector bundle charts such that
{Ua} is an open cover of Μ is called a vector bundle atlas for π : Ε -ϊ Μ.
The definition of a vector bundle guarantees that such an atlas exists. The
dimension of the typical fiber V is called the rank of the vector bundle.
Remark 6.23. By choosing a basis for V, one gets an isomorphism with
Ck or Rfc as the case may be. Composing with this isomorphism we can
6.2. Vector Bundles
271
convert the V-valued VB-charts into Ck- or Revalued VB-charts. Thus we
could have assumed from the start that we were dealing with one of these
standard vector spaces, but it is not always natural to do so since our vector
space may arise in a specific way (it could be a Lie algebra or perhaps a
space of algebraic tensors) and may not have a preferred choice of basis.
Exercise 6.24. Show that the tangent and cotangent bundles of an n-
manifold are vector bundles with typical fiber Rn (the cotangent bundle
may be viewed as having typical fiber (Rn)*).
Let F be R or С as above. The space of smooth sections Γ (ξ) of an
F-vector bundle f = (Ε, π, Μ, V) has the structure of a module over the
ring C°°{M] F); for σ, σι, σ2 6 Τ (ξ) and / € C°°{M\ F), we define
(σι + σ2) (ρ) := σι (ρ) + σ2(ρ) for all ρ Ε Μ,
/σ (ρ) := /{ρ)σ(ρ) for all ре М.
Definition 6.25. Let ξχ and £2 be F-vector bundles with respective bundle
projections πχ and π2. A bundle morphism (/, /) : ξχ —ϊ £2 is called a vector
bundle morphism if the restrictions to fibers, /|π-ΐ/ ^^{p) —> 7r2"1(/(p))>
are F-linear. If ξχ and £2 have the same base space M, then we obtain
the definition of a vector bundle morphism over Μ by specializing to the
case / = idM· We then also have the corresponding notions of vector
bundle isomorphism and automorphism (for both type I and II bundle
morphisms).
A vector bundle (2?, π, Μ, V) is said to be trivial if it is vector bundle
isomorphic to the product vector bundle prx ;MxV4 M. This happens
exactly when there is a vector bundle trivialization over the entire manifold
M, which we call a global vector bundle trivialization (a notion already
introduced for tangent bundles).
Definition 6.26. Let (Ε,π,Μ,ν) be a rank к vector bundle with typical
fiber V and fix an i-dimensional subspace V7 of V. If Er с Ε is a submanifold
with the property that for every ρ € Μ there is a VB-chart (Ε/, φ) such that
φ{π-\υ) Π Ε?) = U χ V С U х V,
then (Ef, π\Ε,, Μ, V;) is called a rank I vector subbundle of (25, π, Μ, V).
Charts with this property are said to be adapted to the subbundle.
The triple (25", π\Ε, ,Μ, V7) is a vector bundle and every adapted VB-
chart (f/, φ) on Ε gives rise to a chart on (Ef, π\Ε,, Μ, V'); namely, (С/, <£'),
where φ' is the restriction of φ to π_1(ϊ7) Π Ε' = π\^} (U). By picking a
basis for V7 and extending to a basis for V one may take V to be Rk and V'
to be R' embedded in Rk as Rl χ {0} С Шк.
272
6. Fiber Bundles
Exercise 6.27. Let Ε —► Μ be a vector bundle as above. Suppose that
a subspace E'p of Ep is given for each ρ G Μ and consider the set Ef =
UpeM-^p- Show that £" is the total space of a rank I vector subbundle
if and only if for each ρ € Μ, there is an open neighborhood U of ρ on
which smooth sections σι,..., σι are defined such that for each q £ U the
set {σι(ς),..., cri(q)} is a basis of the subspace Efq.
If h : E\ —> E<i is a vector bundle morphism over M, then
КегЛ:- UKer/lkP
is a subset of £Ί. This subset is not necessarily (the total space of) a
subbundle, at least in the ordinary sense. However, if the rank I of h\El is
independent of p, then we say that the bundle map has rank I and, in this
case, Kerh is a vector subbundle. Similarly, if h has constant rank in this
sense, then the image Im/ι is a vector subbundle of E2. Both of these facts
follow from
Proposition 6.28. Suppose that h : E\ —> E<i is a vector bundle morphism
over Μ of constant rank r and that E\ and E2 have typical fibers Vi and V2
respectively. Fix a rank r linear map A : Vi —> V2. Then for every ρ G Μ
there is a VB-chart (t/, φ) for E\ with ρ G С/ and a VB-chart ({/, φ) for E%
such that φ oho φ~χ : JJ x Vi —>■ U x V2 has the form
{p,v)\-+ (ρ,Αν).
It follows that Kerh is a vector subbundle with typical fiber KerA and Imh
is a vector subbundle of E^ with typical fiber 1mA.
Proof. Let us first make an observation. Notice that only the rank of the
linear map A : Vi —> V2 in this last proposition is important and we may
replace A by any linear map of the same rank. The reason for this is that if
В : Vi -> V2 is any other linear map with the same rank as A, then there
exist Unear isomorphisms a and β such that В = βΑοΓχ. In particular, if
one has chosen bases and identified Vi with Rfcl and V2 with Rfea, then we
may take A to be a map of the form
\tC , * ■« , X J l~~r yX , * · ■ , X , U, · t . , Uy,
so that Ker Л is a copy of Rfcl~r and Im h is a copy of W.
What we need to prove is entirely local. Thus our task is to show that
for any smooth map h : U x ¥kl —> U χ Ffca of the form (p, г;) -> (ρ, hpv),
with hp a linear map of rank r and where ρ t-> hp is smooth, we may find
maps φ and φ such that φ о ho ф^1 : U χ Ffcl —> U χ Ffc2 is given by
(p,я?1,...,xkl) 1-+ (p, a?1,..., xr> 0,...,0). Fix p$ Ε U. There exist linear
6.2. Vector Bundles
273
isomorphisms a : ¥kl -+ Ffcl and β : Ffca -> Ffc2 such that β ο hp ο α^1 is
given by a A:2 x &i matrix of the form
An(p) Ai2(p)
A2X{p) A22{p)
and where Ац{ро) is an r χ r matrix which is invertible. By shrinking C/
if needed, we may assume that Ац{р) is invertible for all ρ £U. Thus we
may as well assume from the start that hp is represented by a matrix of this
form. Now consider the map φρ : ¥kx —> ¥kl whose matrix is given by
An{p) A12(p)
0 J(*i-r)x(*i-r) JfclXfcl"
Then hp ο φ ι has a matrix of the form
L Αα(ρ)4Γί(ρ) С
fc2Xfcl
and since this matrix must have rank r, we see that (7 = 0. Let Mp
Μι{ρ)Α^Ι(ρ) and let φρ be the linear map Rfca -> R*2 with matrix
irxr
[ -Mp I(k2-r)x(k2-r)
Then ΦΡ° hp ο φρι has the form [Ir*r jj]. Now define φ{ρ,ν) — (р,фру),
h(p,x) := {p,hpx) and ψ{ρ,ν) := {ρ,ψΡν) for ρ G 17, ж Ε Ffcl, and и € F*2.
Notice that ^p, /ip and 0"1 each depend smoothly on p. The map ψ ο ho φ
has the required form.
-1
D
Proposition 6.29. Let λο : GL(V) χ Υ -^ V be the standard action of
GL(V) on the F-vector space V. A fiber bundle with typical fiber V has an
¥-vector bundle structure if and only if it admits a \$-bundle atlas (a GL(V)-
bundle atlas). Furthermore, if λ : GL(V) χ V -* V is any effective action
which acts linearly, then any fiber bundle (E} π, Μ, V) that has a X-atlas is
a vector bundle in a natural way.
Proof. That a vector bundle has a GL(V)-bundle structure follows directly
from the definition. All that remains to show is the second part of the
theorem, since this will imply the remainder of the first part. Let λ : Gx V —>
V be any effective Lie group action which acts linearly and suppose that
Ε,π,Μ,ν) has a λ-bundle structure. Let {υα,φα) and (ϋβ,φβ) be λ-
compatible bundle charts and let φα = (π, Φα) and φβ = (π,Φ^). Fix
274
6. Fiber Bundles
peUaD ΐ7β. For u,«ieV, and a, b G F, we have
Ф«/з(р)(аг; + bw) = λ(ρα/3(ρ), ατ; + бги)
= αλ(ρα^(ρ), ν) + δλ(0α0(ρ), w)
which shows that Φαβ(ρ) € GL(V) for all ρ Ε UaC\Up. We transfer the
vector space structure from V to Ep via ΦαΙ^1 and note that this is well-
defined by Proposition 2.3. With this Unear structure on the fibers it is easy
to verify that (£", π, Μ, V) is a vector bundle. D
Theorem 6.30 (Vector bundle construction theorem). Let {Ua}aeA be a
cover of Μ and let {</a/?} be a G-cocycle for a Lie group G.IfG acts linearly
on the vector space V (by say \), then there exists a vector bundle over Μ
with a VB-atlas {{Ua,<j>a)} satisfying φα ο φ^ι(ρ,ν) = (p,gap{p) ■ ν)) on
nonempty overlaps υαΠΐΙβ. In other words, there exists a vector bundle
with (G, \)-atlas.
Proof. This is essentially a special case of Theorem 6.15. One only needs
to check Hnearity of the φα on fibers. D
Perhaps some clarification is in order. In the case of a vector bundle,
the raw transition maps Φαβ take values in the general linear group GL(V),
which is a Lie group. They correspond to a λο-bundle structure where λο is
the standard Hnear action of GL(V) on V (the standard representation), and
they automatically satisfy the cocycle condition. The more general
transition maps that define a (G, A)-bundle structure (G-bundle structure) are
G-valued. It is important to note that G may be small compared to GL(V)
and certainly need not be thought of as a subset of GL(V). For
example, the tensor bundles have (possibly ineffective) GL(V)-bundle structures
coming from tensor representations, but the tensor bundles themselves
generally have rank greater than к = dim (V). Since in the vector bundle case,
the Φαβ arise directly from a VB-atlas and act by the standard action, we
will call these standard transition maps, and the corresponding GL(V)-
bundle structure will be called the standard GL(V)-bundle structure.
The standard GL(V)-bundle structure is the structure that a vector bundle
has simply by virtue of being an ¥-vector bundle with typical fiber V.
Remark 6.31. We have previously mentioned that the notion of a
representation is equivalent to that of a left linear action. When dealing with vector
bundles it is perhaps more common to use the representation terminology
and notation and this we shall do as convenient. So if λ is a left Hnear action,
then the map G -> GL(V) given by g H> X(g) := Xg is a representation
of G. Conversely, if λ is such a representation, we obtain a Unear action by
letting \(g,v) := \(g)v.
6.2. Vector Bundles
275
We already know what it means for two vector bundles over Μ to be
equivalent. Of course any two vector bundles that are equivalent in a natural
way can be thought of as the same. Since we can and often do construct our
bundles according to the above recipe, it will pay to know something about
when two vector bundles over Μ are isomorphic, based on their respective
transition functions. Notice that the standard transition functions are easily
recovered from every (G, A)-atlas by the formula λ {gafi{p)) У — Φαβ\ρ (у)·
Proposition 6.32. Two vector bundles π : Ε —> Μ and π' : Ε' -* Μ with
standard transition maps {Φαβ - Ua Π ϋβ —> GL(V)} and {Φ*αβ : Ua Π ϋβ —>
GL(V)} over the same cover {Ua} are isomorphic (over M) if and only if
there are GL(V) -valued functions fa defined on each Ua such that
(6.1) Φ'αβ{χ) = ^(χ)Φαβ{χ)^\χ) for χ e Ua П ϋβ.
Proof. (Sketch) Given a vector bundle isomorphism / : Ε —► E! over Μ,
let fa{x) := Φα ° / ° $(*\e - Check that this works. Conversely, given
functions fa satisfying equations (6.1), define fa : Ua x V —> Ua x V by
(xtv) н> (ж, fa{x)v). We define / : Ε -> Ε1 by
№ := ((Φα)~1 ° fa о фа) (б) for 6 € Ε\Ό% .
The conditions (6.1) insure that / is well-defined on the overlaps E\v Π
Ε jj = Щиапив· One eas% checks that this is a vector bundle
isomorphism. D
We can use this construction to arrive at several common vector bundles.
Example 6.33. Given an atlas {(£/"«, x«)} for a smooth manifold M, we let
9<*β{ρ) = ΤρΧαο(Γί>χ/0)~1 for allp e UanUfi. The bundle constructed
according to the recipe of Theorem 6.30 is a vector bundle which is (naturally
isomorphic to) the tangent bundle TM. If we let 9αβ(ρ) — (Τρ*.β ο (TpXa)~1)*,
then we obtain the cotangent bundle T*M.
Proposition 6.34. Let π : Ε —» Μ be an ¥-vector bundle with typical
fiber У and with VB-atlas {({/«> <£«)}· Let sa : Ua -> V be a collection of
maps such that whenever υαΓ\ΙΙβ Φ 0, we have sa{p) = Φαβ{ρ)^β(ρ) for all
ρ £ Ua Π Uβ. Then there is a global section s such that s\Ua = sa for all a.
Proof. Let φα : Ε у —> Ua x V be the trivializations that give rise to the
cocycle {Φαβ}- Let ηα{ρ) := (ρ, sa(p)) for ρ e C/"e and let з\и := φ'1 ο ηα.
276
6. Fiber Bundles
This gives a well-defined section s because for χ Ε Ua Π ϋβ we have
Φα1°Ία{ρ)=Φα1(Ρ^α(ρ))
= φ-χ(ρ,Φαβ{ρ)3β{ρ))
:=Φα1°Φα°Φβ1(Ρ^β{ρ))
= ^(Ρ, */зЫ) = Φβ1 ° 7)8 (ρ)- Π
Suppose we have two vector bundles, πι : Ε\ -ϊ Μ and 7Г2 : E2 —► M.
We give two constructions of the Whitney sum bundle πχθπ2 : Εχ®Ε2 ->
Μ. This is a globalization of the direct sum construction of vector spaces.
In fact, the first construction simply takes E\®E2 — {Jp^m^Ip®^p- Now,
we have a vector bundle atlas {(φα, Ua)} for πι and a vector bundle atlas
{(^Q, Ua)} for П2- Assume that both atlases have the same family of open
sets (we can arrange this by taking a common refinement). Now let φα®Ψα -
(vp, wp) н· (p,pr2 ο φα (νρ),pr2 ο <ψα (tup)) for all (vp, wp) 6 (Ег θ Εϊ)\Όα.
Then {(φα Θ ψα, Ua)} is a VB-atlas for πι Θ π2 : Ει Θ Ε<ι -> Μ.
Another method of constructing this bundle is to take the cocycle {gafi}
for πι and the cocycle {Ηαβ} for π2 and then let gafi θ Ηαβ : 1/α Π ϋβ -*
GL(Ffcl x Ffc2) be defined by (gafi®hafi) (x) = fttfte) Θ /^(ж) : (и,и/) к
(fla^(^)^j hafi(x)w). The maps #α£ θ Λαβ form a cocycle which determines
a bundle by the construction of Proposition 6.30, which is (isomorphic to)
πχ®π2:Ει®Ε2-+ Μ.
The pull-back of a vector bundle π : Ε —> Μ by a smooth map / :
N —> Μ is naturally a vector bundle whose linear structure on each fiber
(f*E) = {q} χ Ep is the obvious one induced from Ep. Put another way,
we give the unique linear structure to each fiber that makes the bundle map
/ ; f*E -> Ε linear on fibers. When given this vector bundle structure, we
call f*E the pull-back vector bundle.
Example 6.35. Let πι : Ει -* Μ and π2 : £?2 ->■ Μ be vector bundles and
let Δ : Μ -t Μ χ Μ be the diagonal map χ н-> (я, х). From πι and π2 one
can construct a bundle /jte1xE2 : Ει χ Ε2 -> Μ χ Μ Ъу kexxE2 (бьб2) -"=
(ττι (^l), 1Γ2 (^2))· The Whitney sum bundle Ei®E2 defined previously is
naturally isomorphic to the pull-back А*пе1хе2 '· Α* (-ΕΊ x E2) —> Μ (Problem
10).
Exercise 6.36. Recall the space Τ/(ξ) from Definition 6.19. Show that if
π : Ε —► Μ is an F-vector bundle and / : N —> Μ is a smooth map, then
both Γ/(ξ) and Г(/*0 are modules over C°°(JV;F), and that the natural
correspondence between Γ/(ξ) and Γ(/*£) is a module isomorphism.
Every vector bundle has global sections. An obvious example is the
zero section which maps each χ € Μ to the zero element 0X of the fiber
6.2. Vector Bundles
277
Ex. The image of the zero section is also referred to as the zero section
and is often identified with M. (Of course, the image of any global section
is a submanifold diffeomorphic to the base manifold.) We have the following
simple analogue of Lemma 2.65:
Lemma 6.37. Let π : Ε -± Μ be an F-vector bundle with typical fiber V.
If υ € 7г_1(р) then there exists a global section σ G Τ(ξ) such that σ(ρ) = v.
Furthermore, if s is a local section defined on U, and V is an open set with
compact closure with V С V С U, then there is a section σ € Γ (ξ) such that
σ = s onV.
Proof. Using a local trivialization one can easily get a local section a\oc
defined near ρ such that σ(ρ) = v. Now just use a cut-off function as in the
proof of Lemma 2.65. For the second part we just choose a cut-off function
β with support in U and such that β = 1 on V. Then ββ extends by zero
to the desired global section. D
If a section of a vector bundle takes the zero value in some fiber we say
that it vanishes at that point. Global smooth sections that never vanish do
not always exist; such sections are called nowhere vanishing or nonvan-
ishing. However, there is one case where it is easy to see that nonvanishing
smooth sections exist:
Proposition 6.38. Any bundle equivalent to a (trivial) product bundle must
have a nowhere vanishing smooth global section.
It is a fact that the tangent bundle of S2 does not have any such nowhere
vanishing smooth sections. In other words, all smooth (or even continuous)
vector fields on S2 must vanish at some point. This is a result from algebraic
topology called the "hairy sphere theorem" (see Theorem 10.15). If one
fancifully imagines a vector field on a sphere to be hair, then the theorem
suggests that one cannot comb the hair neatly "flat" without creating a
cowlick somewhere. More generally, the analogous result holds for S2n if
л>1.
Exercise 6.39. Modify either the construction of Example 6.9 or Example
6.16 to obtain a rank one vector bundle version of the Mobius band and give
an argument proving that every global continuous section of this bundle must
vanish somewhere.
Definition 6.40. If ξ = (ϋ7, π, Μ, V) is a vector bundle and ρ £ M, then a
vector space basis for the fiber Ep is called a frame at p.
Definition 6.41. Let π : Ε ->> Μ be a rank к vector bundle. A fc-tuple
(г = (σι, ·.., ak) of sections of Ε over an open set U is called a (local) frame
field over U if for all ρ eU, (σ"ι(ρ),..., σ*(ρ)) is a frame at p.
278
6. Fiber Bundles
If we choose a fixed basis {е*}^^...^ for the typical fiber V, then a
choice of a local frame field over an open set U С М is equivalent to a local
trivialization (a vector bundle chart). Namely, if φ is such a trivialization
over f/, then defining σι{ρ) = 0_1(р,ег), we have that σψ = (σι,.,.,σ*)
is a local frame over U. Conversely, if σψ = (σι,..., σ^) is a local frame
over U, then every ν 6 n~l(U) has the form ν = Σνι<7ι(ρ) for a unique
ρ and unique numbers vl(p). Then the map / : U χ V -> 7Γ_1(ί7) defined
by (p, v) *-» Σν1<τι(ρ) *s a diffeomorphism and its inverse <£ = /_1 is a
trivialization. Thus if there is a global frame field, then the vector bundle
is trivial. A manifold Μ is said to be parallelizable if TM -> Μ has a
global frame field; i.e. if the tangent bundle is trivial. For example, the
hairy sphere theorem mentioned above implies that S2 is not parallelizable.
On the other hand, S2 χ R is parallelizable. It is easy to show that the torus
T2 = Sl χ S1 and its higher-dimensional analogues T4 = S1 χ · · ■ χ S1 are
also parallelizable.
If G is a Lie subgroup of GL(V) and G acts in the standard way on V,
that is, if the action is λο|β, the restriction of the standard action, then
a Ao|G-bundle structure is a reduction to the group G. Put another way,
one has achieved such a reduction if one can find a cocycle of standard
transition maps {Φαβ} arising from a vector bundle atlas which take values
in G (acting in the standard way on V). By a slight extension, if λ is a
linear action on V, then an effective (G, A)-bundle structure on Ε can be
considered as a reduction of the standard GL(V)~bundle structure.
Let ai(p) = ф~х{р, щ) and σ[{ρ) = Φ~β1{ρ, щ) be frame fields
corresponding to VB-charts (ί/α, φα) and (Щ} φ β) that lie in a (G, A)-atlas and where
Ua Л ΙΙβ is nonempty. Then φα ο φΖι(ρ, ν) = (ρ, λ (gafi(p)) {ν)) for transition
functions 0αβ : Ua Π ϋβ —> G. For each g E G, there is a matrix (Х3г(д))
which represents λ9 with respect to (βχ,..., e*). Unwinding definitions, we
see that φαι is linear on the vector space {ρ} χ V. Using this we have
σ'ί(ρ) = ΦβΧ{ρ^) = Φ*1^ λ (9αβ{ρ)) (β»))
\ j / j
= ΣΛι(5α^(ρ))^(ρ).
j
Thus the smooth matrix-valued function (λ;? og^) defined on Ua Π ϋβ gives
the change of frame and embodies the transition map on иаГ\Щ. In practice,
this is often a good way to look at things. Consider the common situation
where V = Rk and where Aq is the standard representation of GL(Rfc). If
6.2. Vector Bundles
279
we identify GL(Rfc) with the matrix group GL(fc), then ((λο)^ о д^ (ρ)) is
just the matrix g^{p) itself.
Metric differential geometry begins if we have a scalar product on the
fibers of a vector bundle. We introduce the concept at this point so as to
have an example of a reduction of the structure group.
Definition 6.42. A Riemannian metric on a real vector bundle π : Ε —>
Μ is a map ρ н-> др(-, ■) which assigns to each ρ € Μ a, positive definite
scalar product gp{-, ·) on the fiber Ep that is smooth in the sense that ρ \-¥
gv(si(p),S2{p)) is smooth for all smooth sections s\ and $2. A real vector
bundle together with a Riemannian metric is referred to as a Riemannian
vector bundle.
For example, a Riemannian metric on the tangent bundle of a smooth
manifold is what one means by a Riemannian metric on the manifold, A
smooth manifold with a Riemannian metric is called a Riemannian manifold
and such will be studied later in this book. If a rank к real vector bundle
π: Ε -* Μ has a Riemannian metric, then it is convenient to assume that a
fixed inner product is chosen on the typical fiber V and that a distinguished
orthonormal basis (βχ,..., e&) has been chosen. In most applications, V is
Rfc, the inner product is the standard dot product, and the distinguished
basis is the usual standard basis. Recall that the orthogonal group O(V)
is the subgroup of GL(V) consisting of elements that preserve the inner
product. Once the inner product and distinguished orthonormal basis are
fixed, every choice of Riemannian metric on Ε corresponds to a reduction
of the standard structure group GL(V) to the subgroup O(V) as follows.
Let us first show how a metric leads to a reduction. We start with an
arbitrary VB-atlas {(ί/α, φα)}· Since we have fixed a basis for V, each chart
υα>Φα) defines a frame field (af,... ,σ£) on Ua by σ"(ρ) := <f>al(Piei) as
explained above. One can perform a Gram-Schmidt process on the basis
σί(ρ)>· ■· ισ%(ρ)) simultaneously for all ρ Ε Ua so that we have a new
orthonormal basis (ef(p),..., ea(p)) for eachp, where e?(p) = Y^A){p)(rf{p)
for all ρ € Ua and the matrix entries Α^(ρ) depend smoothly on p. Thus
ef,..., e£) is a (smooth) local frame field called an orthonormal frame
field. One then replaces the original chart (ί/α, φα) by a new chart ({7a, φα)
that is the inverse of the map (ρ, ν) ι-> Σ v%e%{p)i where ν = £ v%ei(p) in V.
In other words, фа1 : (р,г>) ι-> Σ ν'σ*(ρ) ls replaced by
C^M^ ^г/еДр).
Make this replacement for each (υα,φα) to obtain a new atlas {{Ua, Φα)}·
Any two of these orthonormal frame fields, say (ef,,.., ea) and (ef,... e^),
280
6. Fiber Bundles
are related by
for some smooth orthogonal matrix function Q%j. One now checks that the
transition maps for this new atlas (which is still a subatlas for the maximal
VB-atlas) take values in O(V). Indeed,
(ρ,Φ'αβ(ρ)(ν)) = φ'βοφ>-ΐ(ρ,υ) = ^(ДЛ'е»)
=^(Σ^?>)β?(ρ)) = (p.^lpE^frtfw)
= (p.E^w *Я„^ (ρ)) = (ρ.Σ^ω*)·
from which we see that Ф'а/3(р)(^) = Φ^ΟρΗΣ1^) = JlvJQ](p)ei* Since
(Q*(p)) is an orthogonal matrix for all p, we have Φ'αβ(ρ) € O(V) for all
p. The converse is also true. Namely, a reduction to the structure group
O(V) (acting in the standard way) is tantamount to the introduction of a
Riemannian metric. The correspondence presumes the prior choice of inner
product and distinguished orthonormal basis on V.
Exercise 6.43. Prove the converse statement referred to above.
Exercise 6.44. Let Ε be a complex vector bundle of rank k. Define by
analogy with Riemannian metric, the notion of a Hermitian metric on Ε
and show that every Hermitian metric on Ε corresponds to a reduction of
the standard GL(fc,C)-bundle structure to a U(n)-bundle structure.
Proposition 6.45. On every real vector bundle Ε there can be defined a
Riemannian metric. Similarly, on any complex vector bundle there exists a
Hermitian metric.
Proof. We prove the Riemannian case; the Hermitian case is entirely
analogous. The proof uses the fact that a strict convex combination of positive
definite scalar products is a positive definite scalar product. This allows
us to use a partition of unity argument. Endow V with an inner product.
Let {{Ua,<i><x)} be a VB-atlas and let (υαιφα) be a given VB-chart. On
the trivial bundle Ua χ V —> Ua there certainly exists a Riemannian metric
given on each fiber by {(p, v), (p, w))a — (υ, ги). We may transfer this to the
bundle π 1(Ua) —> Ua by using the map φ~ι, thus obtaining a metric ga on
this restricted bundle over Ua. We do this for every VB-chart in the atlas.
The trick is to piece these together in a smooth way. For that, we take a
smooth partition of unity (Ϊ7α, ρα) subordinate to the cover {Ua}. Let
8(ρ) = Σ^(ρ)^(ρ)"
6.2. Vector Bundles
281
The sum is finite at each ρ Ε Μ since the partition of unity is locally finite
and the functions paga axe extended to be zero outside of the corresponding
Uw The fact that pa > 0 and pa > 0 at ρ for at least one α easily gives the
result that g is positive definite at each ρ and so it is a Riemannian metric
on#. D
Example 6.46 (Tautological line bundle). Recall that RPn is the set of all
lines through the origin in Rn+1. Define the subset L(RPn) of RPn χ Rn+1
consisting of all pairs (l,v) such that ν G / (think about this). This set
together with the map π^ρη : L(RPn) —>> RPn given by (Ζ, ν) ι-» Ζ, is a rank
one vector bundle.
Example 6.47 (Tautological bundle). Let G(n, k) denote the Grassmann
manifold of fc-planes in Rn. Let ηη^ be the subset of G(n, к) х Rn consisting
of pairs (P, v) where Ρ is a fc-plane (fc-dimensional subspace) and ν is a
vector in the plane P. The projection πη>^ : 7η>£ —► G(n, к) is simply
(P,v) н> P. The result is a vector bundle (7n)fc,7rn)fc, G(n,к),Шк). We leave
it to the reader to discover an appropriate VB-atlas (see Problem 12).
These tautological vector bundles are not just trivial bundles, and in
fact their topology or twistedness (for large n) is of the utmost importance
for classifying vector bundles (see [Bo-Tu]). One may take the inclusions
Rn c Rn+i c ... c roo to constrUct inclusions G(n, к) С G(n + 1, к) С · - ■
and 7n,fc С 7n+i,fc· Given a rank к vector bundle π : Ε -> Μ, there is an η
such that π : Ε —> Μ is (isomorphic to) the pull-back of 7njfc by some map
/:Af->G(n,Jfc):
Ε = /*7n,fc ** 7n,fc
1 / l
Μ -+ G{n, к)
Exercise 6·48. To each point on a unit sphere in Rn, attach the space of
all vectors normal to the sphere at that point. Show that this normal bundle
is in fact a (smooth) vector bundle. Generalize to define the normal bundle
of a hypersurface in Rn. When is such a normal bundle trivial?
Exercise 6.49. Fix a nonnegative integer j. Let Υ = Rx(—1,1) and let
χι»3/ι) ~ (^2,2/2) if and only if χχ — X2 + jk and y\ = (—l)**ifc for some
integer к. Show that Ε := Y/~ is a vector bundle of rank 1 that is trivial
if and only if j is even. Prove or at least convince yourself that this is the
Mobius band when j is odd.
282
6. Fiber Bundles
6.3. Tensor Products of Vector Bundles
Given two vector bundles 7ri : E\ —> Μ and π2 : Ε<ι -» Μ with respective
typical fibers Vi and V2, we let
Ει ® E<i := |^J £ΊΡ ® E2p (a disjoint union).
рем
Then we have a projection map π : Ει ® £2 —> Μ given by mapping any
element in a fiber E\p ® £2? to the base point p. We show how to construct
a VB-atlas for E\ ® E2 from an atlas on each of Ει and E<i. The smooth
structure and topology can be derived from the atlas as usual in such a way
as to make all the relevant maps smooth. We leave the verification of this
to the reader. The resulting bundle is the tensor product bundle, As
usual we can assume that the atlases are based on the same open cover.
Thus suppose that {(υα,Φα)} is a VB-atlas for Εχ while {(l/a,Vv»)} is a
VB-atlas for £2. Now let Φα ® Φα : (Ει ® E2)\Ua -* Vi ® V2 be defined by
(Φα ® *α)\Εΐρ e2p - φ«\Εΐρ ® V*\E2p for ρ € Z7e. Then let
φα® Φα: {Ει ® £2)Ιαα -> ί/α χ (Vi ® V2)
be defined by φα®Φα ·= (π, Φα®Φα). To clarify, the map ΦαΙ^ ® *а1я2 *
£Ίρ ® £^2p —^ Vi ® V2 is the tensor product map of two linear maps as
described at the end of Chapter 5. To see what the transition maps look
like, we compute;
(Φβ ® ф*)ь1р№р о (Φβ ® φ,) -;p№p
= *«\e1p ® *«\E2p ο Φβ\~Ιρ ® Φβ ^
= Фа/з(р)®*а/з(р).
Thus the transition maps are given by ρ —» Φαβ{ρ) ® Фа/з(р)> which is a
map from C/a to GL(Vi ® V2). The group GL(Vi ® V2) acts on Vi ® V2
in a standard way, and this is the standard effective structure group of the
bundle as we have just seen. However, it is also true that the bundle Εχ ® Ε?
has (ineffective) structure group GL(Vi) χ GL(V2) via a tensor product
representation. Indeed, if L\ denotes the standard representation of GL(Vi)
in Vi and /,2 denotes the standard representation of GL(V2) in V2, then we
have a tensor product representation i\®i2 of GL(Vi) χ GL(V2) in Vi®V2.
This is usually not a faithful representation. Using the GL(Vi) χ GL(V2)-
valued cocycle ρ н* Ηαβ(ρ) := {Φαβ(ρ)^αβ{ρ))^ together with t\ ® ι<ι, we
see that by definition
(tl ® ΙαίΗ^φ) = {Φαβ{ρ) Ο *αβ(ρ)) (τ).
6.4. Smooth Functors
283
Furthermore, if Vi = V2 = V, then the tensor product representation is
usually defined as a representation of GL(V) rather than GL(V) χ GL(V),
and so Ει <g> Ε<χ would have a (GL(V), ι <g> £)-bundle structure where ι is the
Standard representation. In this case l®l is still not a faithful representation
since — idy is in the kernel. We can reconstruct the same vector bundle using
any of these representation-cocycle pairs via Lemma 6.30. In fact, it is quite
common that we have different representations by one group. Suppose that
we have two faithful representations Ai and X2 of a Lie group G acting on
Vi and V2 respectively. If {9αβ} is a cocycle of transition maps, then we
can use the pair {gafi, Ai} in Lemma 6.30 to form a vector bundle E\ that
has a (G, Ai)-bundle structure by construction. Similarly, we can construct
a vector bundle E2 with (G, A2)-bundle structure. If we use Αχ ® Аг and
the same cocycle {g^}·, then we obtain a bundle which, as a vector bundle,
is Εχ ® Ε2. But by construction, it has a (G, Ai ® A2)-structure (possibly
ineffective). This is the case in the following exercise:
Exercise 6.50. Suppose that Ε is a vector bundle with a (G, A)-bundle
structure given by a (G, A)-atlas with a corresponding cocycle of transition
functions. Show how one may use Theorem 6.30 to construct bundles
isomorphic to £?*, E®E and E®E* which will have a (G, A*)-bundle structure,
a (G, A®A)-bundle structure and a (G, A®A*)-bundle structure respectively.
6.4. Smooth Functors
We have seen that various new vector bundles can be constructed starting
with one or more vector bundles. Most of the operations of linear algebra
extend to the vector bundle category. We can unify our thinking on these
matters by introducing the notion of a G°° functor (or smooth functor).
With F = Μ or C, the set of all F-vector spaces together with linear maps
is a category that we denote by Lin(F). The set of morphisms from V to W
is the space of F-linear maps L(V, W) (also denoted Hom(V, W)).
Definition 6.51. A covariant G°° functor Τ of one variable on Lin(F)
consists of a map, denoted again by J", that assigns to every F-vector space
V an F-vector space TV, and a map, also denoted by T, which assigns to
every linear map A G L(V, W), a linear map ΤΑ Ε L(TV, TW) such that
(i) T: L(V, W) -► L(FV,FW) is smooth;
(ii) .F(idv) = idjry for all F-vector spaces V;
(iii) T{A oB)=TAoFB for aU A e L(U, V) and Β Ε L(V,W) and
vector spaces U, V and W.
As an example we have the C°° functor which assigns to each V the
Wold direct sum ®k V = V Θ · · ■ Θ V and to each linear map A E L(V, W)
284 6. Fiber Bundles
the map
0A:0V^0W
given by ®kA(vi,..., v*) := (-Avi,.. ·, Аг;&). Similarly there is the functor
which assigns to each V the fc-fold tensor product (g)fcV — V ® ■ ■ · ® V and
to each A G L(V, W) the map ®kA : ®kV -> ®*W given on homogeneous
elements by ((%)kA)(vi ® · ■ · ® v*) := Avi ® · ■ ■ ® Avfc.
One can also consider C°° covariant functors of several variables. For
example, we may assign to each pair of vector spaces (V,W) the tensor
product V <g> W, and to each pair (Α,Β) Ε L(V, V) x L(W, W), the map
A®B:V®W->V'®W.
There is also a similar notion of contravariant C°° functor:
Definition 6.52. A contravariant C°° functor Τ of one variable on
Lin(F) consists of a map, denoted again by T, which assigns to every F-
vector space V an F-vector space TVy and a map, also denoted by J", which
assigns to every linear map A Ε L(V,W) a linear map ТА Ε L(TW,TV)
(notice the reversal) such that
(i) Τ : L(V, W) -> L{TW,TV) is smooth;
(ii) .F(idv) = id^v for all F-vector spaces V;
(iii) JF{A о В) = ТВ о ТА for all A G L(U, V) and Β Ε L(V, W) and
vector spaces U, V and W.
The map that assigns to each vector space its dual and to each map its
dual map (transpose) is a contravariant C°° functor T. One may define the
notion of a C°° functor of several variables which may be covariant in some
variables and contravariant in others. For example, consider the functor of
two variables that assigns to each pair (V, W) the space V ® W* and to each
pair (A,B) G L(Vi, V2) xL(Wi, W2) the map A®S* : Vi®W^ -+ V2®WJ.
Theorem 6.53. Let Τ be a C°° functor of m variables on Lin(F) and
let 2?i,..., Em be ¥-vector bundles with respective typical fibers Vi, · ·., V^.
Then the set
E:=T{Eu...tEm):=\jT{El\pl...1Em\p)
ρ
together with the map π : Ε —> Μ which takes elements ofT{E\\p,..., Em\)
to ρ is naturally a vector bundle with typical fiber .F(Vi,..., Vm).
Proof. We will only prove the case of m = 2 with covariant first variable
and contravariant second variable. This should make it clear how the general
case would go while keeping the notational complexity under control.
6.5. Нот
285
Given vector bundles πι : Εχ —>> Μ and π2 : E2 -ϊ Μ, the total space
of the constructed bundle is []ρΤ(Εχ\ρ, Ε2\ρ) with the obvious projection
which we call π. Let (<j>a,Ua) be a VB-atlas for E\ and (^a,i7Q) a VB-
atlas for £2 (we have arranged that both atlases use the same cover by
going to a common refinement as usual). For each p, let Ep denote the fiber
F{Ei\ , E2\p). Fix a and for each PeUa define θα\ρ € L{EP1F(VUV2))
by
θαΙρ-^ΦαΙρ,Φαΐ;1),
where φα = (πι,Φα) and ^α = (π2,Φα). Then define θα : π~ι (Ua) —у
^(Vi, V2) by θα(ε) = θα|ρ (c) whenever б € 7"(£?i|p, £fc|p). Next define
θα = (π, θα) : π^ϋα ^^x -F(Vi, V2).
The family {(9a,Ua)} is to be a VB-atlas for £*. We check the transition
maps:
вс*(р) = вв|ровД
= ^(Φα|ροΦ^|;1ιΦ/ϊ|ροΦβ|;1)
= ЛФа/9(р),Ф^(р))-
(Remember that the functor is contravariant in the second variable.) Now
we can see from the properties of Фа^, Ф^а and the definition of C°° functor
that Γ(Φαβ(ρ),Φβα{ρ)) € GL(T(VuV2)) and the maps θαβ :U*nUfi->
GL(.F(Vi, V2)) axe smooth. D
6.5. Horn
Let ξχ := (Εχ,πχ,Μ, V) and £2 := (^,^,Μ, V) be smooth F-vector
bundles. The bundle whose fiber over pGMis L(Elp, E2p) = Rom(Eip, E2p)
is denoted by Horn (ξι, £2) °r less precisely, by referring to the total space
Rom(Ex,E2). Here Hom(£ap, E2p) denotes F-linear maps. If f : Εχ ->
E2 is vector bundle homomorphism over M, then we may obtain a
section 5 of Rom(Ei, E2) by defining 5 :рн f\El . Conversely, given s G
Τ (Rom(Ei, E2)) we define / : E\ -> E2 by requiring that
/lslp = 5(i>)·
Thus every element of Horn(£4,£2) can be identified with a vector bundle
homomorphism over M.
Exercise 6.54. Let Εχ -> Μχ and £?2 -> M2 be smooth vector
bundles. Show that the set of vector bundle homomorphisms along a smooth
286
6. Fiber Bundles
map g : M\ —> M2 is in natural bijection with the sections of the bundle
Rom(Eug*E2).
Since Τ (Εχ) and Γ (£2) are C°°(M,F) modules, we can look at the
C°°(M,F) module Нот(Г (Ει) ,Γ (E2)). Then we have
Proposition 6.55. Let Ει —> Μ and E2 —> Μ be smooth F~vector bundles.
Then Γ (Hom(i?i, £2)) and Нот(Г (Ει), Γ (£2)) are naturally isomorphic as
C°°(M,F) modules.
Proof. To each section s G Г (Hom(Ei, £2)) we assign a map <j>s : Г (Е\) -¥
Г (E2) defined by the formula
фа (σ) (ρ) = *(ρ)σ(ρ) for σ G Γ (Ει).
Then we obtain a map Φ : Г (Нот(Яь £fe)) -> Нот(Г (Ει), Γ (Ε2)) which
is defined by Φ : s ь* 0S. The smoothness of s and σ implies the
smoothness of φ8 (σ) and then the smoothness of φ3. The map φ8 is clearly in
Нот(Г (Ει) ,Γ (J52)), and it is not difficult to check that Φ is also a module
homomorphism.
Now suppose we are given a module homomorphism φ : Γ (Ει) —> Г (Ε2).
Define seT(Rom(Ei,E2)) by
s(p)(vp) := φ (σ) (ρ) for vp G EXp,
where σ is any section in Γ (Εχ) such that σ(ρ) — νρ. We need to show that
this is well-defined. It suffices to show that if σ(ρ) = 0 then φ (σ) (ρ) = О,
Let (χι,..,, Xk) be a local frame field for Εχ defined over an open set U.
Choose g G C°°(M) with support in U and g(jp) = 1. Define fields Xi := gx%
and extend by zero outside of U. Then there exist functions fl such that
к
ga = "£fXu
and since σ(ρ) = 0, we must have /г(р) = О for all i. Then we have
(φσ) (ρ) = g(p) (φ (σ)) (ρ) = (дф (σ)) (ρ)
= ф(да)(р) = фГ£ГхЛ(р)
к к
= £ (гф (Xi)) (ρ) = £ /*(ρμ (^) (ρ) = ο.
г=1 г-1
The constructed map is easily checked to be the inverse of the map Φ : 5 ь4
φ,- □
6.6. Algebra Bundles
287
If (σι,..., σ^) is a local frame field for E\ —► Μ over an open set U>
and if (0i,..., фк2) a local frame field for E<i-±M also over f/, then we may
construct a local frame field for Hom(£i, E^). The frame field is {e^} where
e*ipk) = 5fcj0i· If Hom(Ei,E2) is identified with E2®El, then e^ is φι®σ^
where (σ1,..., σ*1) is the dual frame field to (σι,..., σ^). Suppose that
over an open set V, we have frame fields (σι,..., σ^) and (0i,..., 0fc2). If
U Л V is nonempty, then
(σι,... ,σ^) = (σχ,..., σ^)0,
(01,..., фк2) = {фи · · ■> ^fe)-0
for smooth matrix-valued functions С and ί) defined on U Π V. We obtain
% = 0г (g) σ> = ΣγΛ^"1)-^· If A G Γ (Hom(Si, Я2)) and on U П V we
have
then 1} = Er>e (ГГ1), ^CJ on С/ П V.
Of special importance is End(£) := Hom(i£, E) —l· Μ for a given vector
bundle Ε -+ M. Here if (σι,..., σ&) = (σι,..., σ&)(7 is a change of frame
field, then we take φι = σι and φι = σ* so that the above rule specializes to
(6.2) % = Σ,{(Γΐ)τΑ& on UHV,
which is the signature transformation law for sections of End(E). This
bundle is important in the study of covariant derivatives and curvature.
Notice how everything is formally similar to operations in linear algebra, but
here we are dealing with fields and functions (often only locally defined).
6.6. Algebra Bundles
Let F be С or R. Recall that an F-algebra is a vector space V with a bilinear
map V χ V —> V giving a product on V. Such a bilinear map is uniquely
specified by the associated linear map V ® V —» V (see Definition D.17).
Definition 6.56. Let V be an F-algebra. An F-vector bundle ξ= (Ε, π,
M,V) is called an F-algebra bundle if each fiber Ep has an F-algebra
structure in such a way that the associated maps Ep ® Ep —> Ep combine
to give a vector bundle homomorphism Ε ® Ε —>· Ε and ξ has a VB-atlas
{(t^aj Φα)} such that for each α, $a|# · Ep —у V is an algebra isomorphism
for all ρ Ε Ua. Here φα = (π, Φα) as usual.
Example 6.57. The endomorphism bundle End(i?) —► Μ of a vector
bundle (Ε,π,Μ, V) is the bundle whose fiber at ρ is End(Ep) = Rom(EP)Ep).
The space End^p) has an F-algebra structure given by composition of linear
maps.
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6. Fiber Bundles
Example 6.58. Let ξ — (Ε,π,Μ,ν) be an F-vector bundle. The
corresponding general linear Lie algebra bundle is the bundle whose fiber at
ρ is L(Ep,Ep) but with the product being (/,p) н· [/,</] :=/oj-jo/.
When given this product, the fiber is denoted gl(Ep) and is the Lie algebra
of GL(Ep). The total space of the general linear Lie algebra bundle is then
denoted by gl(E). This provides an example of a Lie algebra bundle.
Recall that when a direct sum has an infinite number of summands, we
require that each element have finite support. In other words, ф^ Vt is the
vector space of all formal sums Σ£ι υυ where Vi e V* and all but finitely
many of the vt's are zero.
Example 6.59. The definition of tensor algebra is given as Definition D.46
of Appendix D. Let ξ = (.Ε,π,Μ,ν) be an F-vector bundle and consider
the bundle 0J51 —► Μ whose fiber at ρ is the F-tensor algebra ®Ep —
R®EP® (Ep ® Ep) Θ · ■ ■. If we consider the disjoint union
®Я = (J ®Ep,
рем
with the obvious projection 0£7 —> M, then it seems that we have a bundle
with algebra fibers. However, each fiber (g) Ep is infinite-dimensional. On
the other hand, we do at least have a nested sequence of vector bundles
... С ®-fc Ε С ®-fc+1 EC···, where
0~fe E = R®E®{E®E)®··-® ((g)^ Ε
6.7. Sheaves
Let Ε -► Μ be a vector bundle. We have seen that Γ(Μ, Ε) is a module
over the smooth functions C°°(M). It is important to realize that having a
vector bundle at hand not only provides a module, but a family of modules
{T(U, E)} parametrized by the open subsets U of M. How are these modules
related to each other?
Consider a section σ : Μ —l· E. Given any open set U С М, we may
always produce the restricted section σ\ν :U —> E. This gives us a family of
sections; one for each open set {/. Conversely, suppose that we have a family
of sections συ :U —> Ε where U varies over the open sets (or just a cover of
Μ). When is it the case that such a family is just the family of restrictions
of some section σ : Μ -> Ε? To help with these kinds of questions, and to
provide a language that will occasionally be convenient, we will introduce
another formalism. This is the formalism of sheaves and presheaves. The
formalism of sheaf theory is used for studying the interplay between the local
and the global, for example, using sheaf cohomology theory. It is especially
•
6.7. Sheaves
289
useful in complex geometry. Sheaf theory also provides a very good
framework within which to develop the foundations of supergeometry, which is an
extension of differential geometry that incorporates the important notion of
"fermionic variables". A deep understanding of sheaf theory is not necessary
for what we do here and it would be enough to acquire a basic familiarity
with the definitions since we only want the convenience of the language.
Definition 6.60. A presheaf of abelian groups (resp. rings, etc.) on
a manifold (or more generally a topological space) Μ assigns to each open
set an abelian group (resp. ring, etc.) M(U) and assigns to each nested
pair V С U of open sets a homomorphism Гу : M(U) —» M(V) of abelian
groups (resp. rings, etc.) such that
(0 rw ° rV = rw whenever W cV CU\
(ii) Гу = idy for all open V С М.
Definition 6.61. Let Μ be a presheaf and 11 a presheaf of rings over M. If
for each open U С Μ we have that M(U) is a module over the ring 7£(I7),
and if multiplication commutes with restriction, that is, if the following
diagram commutes for each nested pair V С U>
TZ(U)xM{U) >M(U)
K{V)xM(V) *M(V)
then we say that Μ is a presheaf of modules over Tl.
Definition 6.62. Let Mi and M2 be presheaves over M. A presheaf
morphism h : Mi -> M2 over Μ is a collection of morphisms, h\j :
ΛΊχ(ί7) —> M2{U), one for each open set and such that whenever V С С/,
the following diagram commutes:
Mi(U)-^M2(U)
Mi{V)-^M2{V)
Note that we have used the same notation for the restriction maps of both
presheaves.
Definition 6.63. Let Μ be a presheaf. A family {sa} with sa Ε M(Ua)
is called consistent if ГцаПи sa = ruanUesP whenever υ^ΓΧϋβφ 0.
Definition 6.64. We will call a presheaf Μ a sheaf if the following
properties hold whenever U = (Jcr ш ^« ^or some collection of open sets U.
290
6. Fiber Bundles
(i) If si, S2 € M(U) and rj^si = r^S2 for all Ua £ ZY, then si = 52.
(ii) Given a consistent family {sa : sa ΕΛΊ(Ε/"α)}, there exists 5 €
ΛΊ(Ϊ7) such that r\j s = sa.
(iii) Л4(0) is the trivial group, (resp. ring, etc.).
If we need to indicate the space Μ involved, we will write Мм instead
of Μ
Definition 6.65. A morphism of sheaves is a morphism of the underlying
presheaf.
The assignment C°°(·) : U н* C°°(U) is a sheaf of rings. This sheaf will
also be denoted by Cjg. The best and most important example of a sheaf
of modules over C°°(·) is the assignment Γ(·, Ε) : U н* Г(U, Ε) for some
vector bundle Ε —► Μ, where by definition Vy(s) = s\v for s G T(U,E). In
other words, ry is just the restriction map. Let us denote this (pre)sheaf by
TE:U^>TE(U):=T{U,E).
Exercise 6.66. For each open set U in a manifold M, let B(U) denote
the ring of bounded smooth functions defined on U. Show that UM> B{U)
defines a presheaf that is not a sheaf.
Many if not most of the constructions and operations we introduce
for sections of vector bundles are really also operations appropriate to the
(pre)sheaf category. Naturality with respect to restrictions is one of the
features that is often not even mentioned (precisely because it seems obvious).
This is the inspiration for a slight twist on our notation.
Global Local Sheaf
Functions on Μ C°°(M) C°°{U) C$
Vector fields on Μ X{M) X{U) XM
Sections of Ε Τ (Ε) T{U,E) TE
where C% : U н> C$(U) := C°°(U), XM'.U^ XM{U) := X(U) aлd so on.
Notation 6.67. For example, when we say that D : Cjg —► CjJ is a
derivation we mean that D is actually a family of algebra derivations Du :
^м(^) ~* ^м W) indexed by open sets U such that we have naturality with
respect to restrictions; i.e., diagrams of the form below for V С U commute:
cs?(v)-^cs?(v)
6.8. Principal and Associated Bundles
291
It is easy to see that all of the following examples are sheaves. In each
case, the maps ry are just the restriction maps.
Example 6.68 (Sheaf of holomorphic functions). Sheaf theory really shows
its strength in complex analysis. This example is one of the most studied.
However, we have not studied the notion of a complex manifold, and so this
example is for those readers with some exposure to complex manifolds (see
the onUne supplement). Let Μ be a complex manifold and let Om(U) be
the algebra of holomorphic functions defined on U. Here too, Ом is a sheaf
of modules over itself. Whereas the sheaf Cjg always has global sections,
the same is not true for Ом · The sheaf-theoretic approach to the study of
obstructions to the existence of global holomorphic functions has been very
successful.
For a bit more on sheaves, see the online supplement [Lee, Jeff].
6.8. Principal and Associated Bundles
Let π : Ε —> Μ be a vector bundle with typical fiber V and for every ρ Ε Μ
let GL(V, Ер) denote the set of linear isomorphisms from V to Ep. If we
choose a fixed basis (ei,..., e^) for V, then each frame (щ,..., щ) at ρ
gives an element и Е GL(V,£^) defined by
where ν = Σν1βι* We identify и with (iti,. ..,зд) and refer to it as a
frame. With this identification, notice that if σα '= <?φα is the local frame
field coming from a VB-chart (ί/α, φα) as described above, then we have
σα(ρ) = Φα\Ερ ioxpeUn.
Now let
F(E) := (J GL(V9Ep) (disjoint union).
рем
It will shortly be clear that F(E) is a smooth manifold and the total space
of a fiber bundle. Let ρ : F(E) -* Μ be the projection map defined by
p(u) — ρ for и е GL(V, Ep). Observe that GL(V) acts on the right of the set
F[E)\ the action F(E) χ GL(V) -* F(E) is given by r : (u,g) H· ug — иод.
If we pick a fixed basis for V as above, then we may view g as a matrix and
an element и Ε GL(V, Ep) as a basis (г/χ,..., u*). In this case, we have
It is easy to see that the orbit of a frame at ρ is exactly the set р~г (ρ) =
GL(V, Ep) and that the action is free. For each VB-chart (t/,0) for E,
let σψ be the associated frame field. Define /ψ : U χ GL(V) —> ρ l (U)
292
6. Fiber Bundles
by f<t>fag) = σφ(ρ)9· It is easy to check that this is a bijection. Let φ :
p~l (U) -> ?7xGL(V) be the inverse of this map. We have φ = (ρ, Φ), where
Φ is uniquely determined by φ. Starting with a VB-atlas {(?7α, φα)} for £?,
we obtain a family {φα : ρ~λ {Ua) -> Ϊ7α χ GL(V)} of trivializations which
gives a fiber bundle atlas {(υα,φα)} for F(E) —> Μ and simultaneously
induces the smooth structure.
Definition 6.69. Let π : Ε —>· Μ be a vector bundle with typical fiber
V. The fiber bundle (F(.E),p, M, GL(V)) constructed above is called the
linear frame bundle of i? and is usually denoted simply by F(E). The
frame bundle for the tangent bundle of a manifold Μ is often denoted by
F(M) rather than by F(TM).
Notation 6.70. It would be perhaps more appropriate to refer to the frame
bundle of a vector bundle ξ = (Ε, ττ, Μ, V) as F(f) since the notation F(E) is
inconsistent with the notation F(M) above. After all, Ε is itself a manifold.
Despite this we will continue with the dangerous notation.
Notice that any VB-atlas for Ε induces an atlas on F(E) according to
our considerations above. We have
Φα ο φ^1 fag) ~ φα{σβ(ρ)9) = φα($β\Ερ9) - fa®afi{p)9)-
Thus the transition functions of F(E) are given by the standard transition
functions of Ε acting by left multiplication on GL(V). The cocycle
corresponding to the bundle atlas for F(E) that we constructed from a VB-atlas
{(ί/α, φα)} for the original vector bundle JE, is the very cocycle Φαβ deriving
from this atlas on E.
We need to make one more observation concerning the right action of
GL(V) on F(E). Take a VB-chart for E7 say (17, φ), and let us look again
at the associated chart (ΙΙ,φ) for F(E). First, consider the trivial bundle
prx : U χ GL(V) -* U and define the obvious right action on the total space
4>{p~l{U)) =Ux GL(V) by (fagx),g) »-► fa gig) := fagi) · g. Then this
action is transitive on the fibers of this trivial bundle. Of course since GL(V)
acts on F(E) and preserves fibers, it also acts by restriction on p_1(C/)*
Proposition 6.71. The bundle map φ : p~l(U) —>· t/xGL(V) is equivariant
with respect to the right actions described above.
Proof. We first look at the inverse:
$~lfagi)9 = °<t>(p)gig = 4>~lfagig)·
To see this from the point of view of φ rather than its inverse, take и Е
p~l(U) С F(E) and let fagi) be the unique pair such that и = ^^1(p,ffi).
6.8. Principal ала Associated Bundles
293
Then
ф(ид) -Φ\Φ~ι($,9ΐ)9) = ФФ 1{р,д\д) = {ρ,gig) = (p,gi)-g = Ф(и)-д- □
A section of F(E) over an open set U in Μ is just a frame field over [/.
A global frame field is a global section of F(E)) and clearly a global section
exists if and only if Ε is trivial.
For a frame bundle F(E)} the following things stand out: The
typical fiber is the structure group GL(V), and we constructed an atlas which
showed that F(E) has a GL(V)-bundle structure where the action is left
multiplication. Furthermore there is a right action of GL(V) on the total
space F(E) which has the fibers as orbits. The charts constructed were of
the form (ϋ,φ), where φ is equivariant in the sense that ф{ид) = ф{и)д,
where if ф{и) = (ρ,pi), then (p,gi)g := (p,gig) by definition. These facts
motivate the concept of a principal bundle:
Definition 6.72. Let ρ : Ρ —>· Μ be a smooth fiber bundle with typical
fiber a Lie group G. The bundle (Ρ, ρ, Μ, G) is called a principal G-bundle
if there is a smooth free right action of G on Ρ such that
(i) The action preserves fibers; p{ug) = p(u) for all и £ Ρ and g £ G.
(ii) For each peM, there exists a bundle chart (C/, φ) with ρ e U and
such that if φ = (ρ, Φ), then
Ф(ид) = Ф(и)д
for all и G ρ-1 (U) and ρ Ε G-
If the group G is understood, then we may refer to (Ρ, ρ, Μ, G) simply
as a principal bundle. Define a right action on U x G by (p,gi)g =
(p,3ip). Then U χ G —► С/ is a trivial principal bundle. If 0 = (ρ, Φ), then
using this right action on U x G we have that ф(ид) = 0(г/)з ^ and only
if Ф(ид) = Ф{и)д. Charts of the form described in (ii) of the definition are
called principal bundle charts, and an atlas consisting of principal bundle
charts is called a principal bundle atlas.
Proposition 6.73. // (P, p, M, G) is α principal G-bundle, then the fibers
are exactly the orbits of the right G-action.
Proof. The definition makes it clear that each orbit is contained in some
fiber. Suppose that u\ and u<i are in the same fiber so that p{u\) — р(г/2).
We wish to find a g EG such that щ = г/25. Let g := Φ(ι/2)-1Φ(ήι). Then
Φ(ΐίι) = Ф(гб2)# and so
ФЫ) = (р(г/1),Ф(гл)) = (р{и2),Ф(и2)д)
= (р(и2д),Ф{и2д)) = Ф(и2д).
294
6. Fiber Bundles
Since φ is bijective, we see that u\ = u<ig. The conclusion can be expressed
by saying that the action is transitive on fibers. D
The definition of a principal G-bundle above does not use the notion of
a G-atlas or strict equivalence. The definition of principal bundle atlas is
given after the notion of a principal bundle is already defined. However,
once we have singled out this type of atlas, we can find out what the
transition functions are and thereby make connection with earlier developments.
Notice that if (φα, Ua) and [φβ, ϋβ) are overlapping principal bundle charts
with φα = (ρ, Φ«) and φβ = (ρ, Φ β), then
Фа{ид)Фр{ид)~1 = Фа(и)дд~гФр(и)~г = Φα(η)Φβ(η)~ι,
so that the map и н> Φα(η)Φβ(η)~ι is constant on fibers. This means that
there is a smooth function g^ :1/аГ\Щ —l· G such that
(6.3) 9αβ{ρ) = Φα{η)Φβ{η)-\
where и is any element in the fiber at p.
Lemma 6.74. Let (фа^а) and (φβ,ΙΙβ) be overlapping principal bundle
charts. For each ρ G иаГ\Щ,
φ*\ρ ΐ(ρ) ° (φ/*ΙΡ-ΐ(ρ)) is) = 9αβ{ρ)9,
where the gap are given as above.
Proof. Let (Φβ\ρ i^^ig) = u. Then g - Φβ(η) and so Φα\ρ-ΐ(ρ «
Φβ\ρ i(p\ (g) = Φα(^). On the other hand, и Ε ρ-1 (ρ) and so
9*β(ρ)9 = ia(w)$^W_1j = Φα(ν)Φβ(υ)-ιΦβ(υ)
= Фа(и) - Фа|р 1(р) о Φ/^-ι^ (д). Ώ
Prom this lemma we see that the structure group of a principal bundle
is G acting on itself by left translation. Conversely if (Ρ, ρ, M, G) is a fiber
bundle with a G-atlas with G acting by left translation, then (P, p, M, G) is
a principal bundle. To see this, we only need to exhibit the free right action.
Let и £ Ρ and choose a chart (φα, Ua) from the G-atlas. Then let ug ;—
Φαλ(Ρί Фа(и)9) where ρ = p(u). We need to show that this is well-defined,
so let (φβ, ϋβ) be another such bundle chart with ρ = p(u) £ Щ ΠΪ7α. Then
if u\ := φβ1(ρ, Φβ(u)g), we have
0a (tii) = ФаФр1(р,Фр{и)д) = (р,да0(р)Фр(и)д) = (р,Фа(и)д)
so that u\ = φ~ι (ρ, Фа(гг)<7) = ttg. It is easy to see that this action is free.
Furthermore, since
Фа^ФаМ) = ф^офа (ид) = ид := фа1(р,Фа(и)д),
6.8. Principal and Associated Bundles
295
we see that Фа(ид) = Фа(и)д as required by the definition of principal
bundle. Obviously the frame bundles of vector bundles are examples of
principal bundles.
Definition 6.75. Suppose that π : Ε —> Μ is a rank к vector bundle
with typical fiber V. Let G be a Lie subgroup of GL(V) and suppose that
7Г : Ε -> Μ has a G-bundle structure where G acts on V by the standard
action as a subgroup of GL(V). Thus we have a reduction of the structure
group to G. Let Ag — {{U<y.,<t>a)} be the maximal G-atlas that defines this
structure. For each ρ Ε Μ, let
FG(EP) := {и G GL(V,£P) : и = φ*1 {ρ, ·) for some (U, φ) Ε А?}·
Elements of Fg(Ep) are called G-frames at ρ (associated to Ag)·
The group G acts on the right on Fq{Ev) by (u, jj^wo^, and letting
Fq{E) := UpGM^b(^p) we see *ha* @ acts on ^е right on Fq{E). It is not
hard to show that Fq{E) has the structure of a principal G-bundle with this
action. It is a subprincipal bundle of the frame bundle.
Definition 6.76. The bundle Fq{E) is called the bundle of G-frames
associated to the G-bundle structure on π : Ε —> Μ.
Actually, G acts on the whole frame bundle of Ε by the same formula
and on the set of all frames F(EP) at a fixed p. Then, Fg(Ep) is an orbit of
this action on F(E). In fact, one may simply define a G~bundle structure
on a vector bundle to be a subbundle of the frame bundle such that G acts
transitively on each fiber. It is not hard to see that the notion of a bundle
of G-frames gives us another way to describe the notion of reduction of the
structure group. It is also important to notice that since there may be more
than one G-bundle structure on E, the notation Fq{E) is ambiguous. If
one must deal with two different G-bundle structures on ϋ?, then one must
resort to another notation such as Fq(E) and Fq(E).
Exercise 6.77· Show that a choice of metric on a vector bundle Ε —>■ Μ is
equivalent to a specification of a subbundle of the frame bundle such that
0(k) acts transitively on each fiber of the subbundle.
As another example of a principal bundle we also have the Hopf bundles
described in the next example and the following exercise.
Example 6.78 (Hopf bundles). Recall the Hopf map ρ : 52n"x -> CPn~l
defined in Example 5.120. The quadruple (S2n x, p, CP71"1, U(l)) is a
principal fiber bundle. We have already defined the left action of U(l) on S2n~l
in Example 5.120. Since U(l) is abelian, we may take this action to also be a
right action. Recall that in this context, we have S2n l = {ξ e Cn : \ξ\ = 1},
296
6. Fiber Bundles
where for ξ = (zl,...,zn), we have \ξ\2 ^2 ζ1 ζ*. The right action of
U(l) = Sl on S2n~l is fag) н> £д = (z1g,...,zng). It is clear that
p(&) = P(0· To finish the verification that (S2n~l, p,CPn_1,U(l)) is a
principal bundle, we exhibit appropriate principal bundle charts. For each
к = 1,2,...,η, we let Uk := {[г1,...,*"] G СРП_1 : zk φ 0} and we let
Фк ■ P~X{Uk) -> ^ik x U(l) be defined by ^fc := (p, Фк), where
Ф*(0 = Ф*(г1,...,Л:=1«*Г1«*·
We leave it to the reader to show that ф^ :— (ρ, Фд.) is a diffeomorphism.
For # e U(l), we have
**(&) = lArVs) = Ι**Ι_1(Λ) = (Ι^Γ1**)*? = Ф*(0</>
as desired. Let us compute the transition cocycle {дгз}- For ρ = [ξ] Ε Utr\U}
we have
ey(i>) = *i(0*i(0_1 = КГ1** И-1 HI e u(i).
Exercise 6.79. By analogy with the above example, show that we have
principal bundles (5n^1,p,MFn-1,Z2) and (S471"1, p,HF^1,U(l,H)).
Show that in the quaternionic case дъэ(р) = |ς*|~" ql (q*)~ \<]*\ for Ρ =
[ς1,..., qn] and that the order matters in this case.
If (C7, φ) is a principal bundle chart for a principal bundle (Ρ, ρ, Μ, G),
then for each fixed g Ε G, the map σ^ : ρ н> 0 * (ρ, g) is a smooth local
section. The canonical choice for the fixed element is the identity e. Thus
to each principal bundle chart {ϋ,φ) we associate the local section σφ :
ρ И- φ~ι(Ρι e). Conversely if σ : U —> Ρ is a smooth local section, we let
fa :U xG —l· p_1 (C/) be defined by /σ(ρ, ρ) = c(p)g. The proof of the next
proposition shows that /σ is a diffeomorphism and its inverse φ : ρ"1 (ί/) ->
UxG defines a principal bundle chart. This is a principle worth emphasizing:
Local sections of a principal bundle give rise to associated principal bundle
charts, and conversely as described above.
Proposition 6.80. If ρ : Ρ —ϊ Μ is a surjective submersion and a Lie
group G acts freely on Ρ so that for each ρ Ε Μ the orbit of ρ is exactly
P~l(p)> then (Ρ, ρ, Μ, G) is a principal bundle.
Proof. Let us assume (without loss of generality) that the action is a right
action since it can always be converted into such by group inversion if needed.
We use Proposition 3.25: For each point ρ Ε Μ, there is a local section
σ : U —> Ρ on some neighborhood U containing p. Consider the map
fa :U xG -> p_1 (U) given by /σ(ρ,#) = <т(р)д. One can check that this
map is injective and has an invertible tangent map at each point of U. Now
6.8. Principal and Associated Bundles
297
let φ := /σ *. Then we have φ = (ρ, Φ) for a uniquely determined smooth
map Φ : U —> G. If ρ — p(u), we have ф(ид) — (ρ, Φ {ид)) and so
ид = ф-1(р,Ф(ид))>
while
^(p, Ф(и)з) = /σ (Ρ, ФНз)
= σ(ρ) (Ф(гх)у) = (σ(ρ)Φ(ι*))0
= /σ(ρ,Φ(Μ))» = 0 ^ВД)^
-г13 = 0_1(р,Ф(^)).
Since </>_1 is a bijection, we have Ф(ид) = Ф{и)д. Thus the section σ gives
rise to a principal bundle chart (f/, φ), where φ = (π, Φ). D
Combining this with our results on proper free actions from Chapter 5,
we obtain the following corollary:
Corollary 6.81. If α Lie group G acts properly and freely on Μ (on the
right), then (Μ, π, M/G, G) is a principal bundle. In particular, if Η is a
closed subgroup of a Lie group G, then (G, π, G/H, H) is a principal bundle
(with structure group H).
Definition 6.82. Let (Pi, pi, Mi, G) and (P2,p2,M2,G) be two principal
G-bundles. A (type II) bundle morphism / : Pi -> P2 along a smooth map
/ : Μι —> Μ<ι is called a principal G-bundle morphism (along /) if
/(« · З) = J{u) · 3
for all g € G and и G P. If Mi = M2 and / = idj^·, then we say that / is a
principal G-bundle morphism over M.
Exercise 6.83. Show that if (Pi, pi, Mi, G) and (P2, p2, M2, G) are
principal G-bundles and f : Pi -¥ P2 is a principal G-bundle morphism along a
diffeomorphism /, then / is a diffeomorphism.
If / is a principal bundle morphism over M, then it is a diffeomorphism
and hence a bundle equivalence (or bundle isomorphism over M) with the
property /(it · g) = f{u) · g for all g Ε G and и G P. In this case, we call /
a principal G-bundle equivalence and the two bundles are equivalent
principal G-bundles over M. A principal G-bundle equivalence from a
principal bundle to itself is called a principal bundle automorphism or also
a (global) gauge transformation.
The classification problem for principal bundles (in the topological
category) is regrettably beyond the scope of this volume and can be found in
[Hus]. We can only offer the following comments: In the topological
category, the notion of a Lie group is replaced by that of a topological group,
298
6. Fiber Bundles
but the reader may continue to think of Lie groups. For every topological
group G, there is a principal bundle f(G) = ((E(G),poo,B(G),G) called
a universal bundle with the property that all the homotopy groups of
E(G) are trivial. There is then a classification theorem that states that the
equivalence classes of principal bundles with a fixed sufficiently nice1 base
space Μ are in one-to-one correspondence with the set of homotopy classes
of maps from Μ to B(G). The correspondence is given by assigning to the
homotopy class [/], the pull-back principal bundle /*£(£?).
The notion of a principal bundle morphism over Μ can be generalized
to the situation where we have two groups in play.
Definition 6.84. Let (Pi, pi, M, G\) and (P2, р2, М, G2) be principal bun*
dies and let h : G\ —> G% be a Lie group homomorphism. A bundle morphism
over Μ is called a principal bundle homomorphism with respect to h if
/(«■ g) = f(u)-h(g).
If Gi С С?2 ai*d the homomorphism h is the inclusion, then we call / a
reduction of (P2, p2, M, G2) to (Pi, pi, M, C?i).
In the most common case of a reduction, Pi is a submanifold of P2 and
/ is the inclusion Pi «->> P2. For example, this is the case when one chooses
a metric on a vector bundle and thereby obtains a bundle of orthonormal
frames Fq^(E). The inclusion PQ(n)(^) *~* F(E) is then a reduction, and
we just say that Fq^(E) is a reduction of the frame bundle F(E).
We have seen that a principal G-bundle atlas {(C/a, фа)} is associated to
a cocycle {5«/з}· From this cocycle and the left action of G on itself we may
construct a bundle which has {9αβ} as a transition cocycle. In fact, recall
that in the construction we formed the total space by putting an equivalence
relation on the set Σ :— Ц*{а) xUaxG, where (α,ρ, 5) Ε {α} χ Ua x G is
equivalent to (/3, p', g1) Ε {β} xUpxGii and only if ρ — pf and g' = gfia(p) ·£·
If we define a right action on the total space of the constructed bundle by
[αιΡ)0ι] · 9 = [a)Pi9i9]i then this is well-defined, smooth, and makes the
constructed bundle a principal G-bundle equivalent to the original principal
G-bundle.
Exercise 6.85. Prove the last assertion above.
Thus we see that G-cocycles on a smooth manifold Μ give rise to
principal G-bundles and conversely. If we start with two G-cocycles on M, then
we may ask whether the principal G-bundles constructed from these сосу-
cles are equivalent or not. First notice that the constructed bundles will
have principal bundle atlases with the respective original transition сосу-
cles. Thus we are led to the following related question: What conditions on
Μ should be a CW-complex.
6.8. Principal and Associated Bundles
299
the transition cocycles arising from principal bundle atlases on two principal
G-bundles will ensure that the bundles are equivalent principal G-bundles?
By restricting the trivializing maps to open sets of a common refinement,
we obtain new atlases and so we may as well assume from the start that the
respective principal bundle atlases are defined on the same cover of M.
Theorem 6.86, Let (Ρι,ρι,Μ, G) and (P2,p2,M, G) be principal G-
bundles with principal bundle atlases {(φα, Ua)} and {($*> Ua)} respectively.
Then (Ρι,ρι,Μ, G) is equivalent to (P2, p2,M, G) if and only if there
exists a family of (smooth) maps ra : Ua ->· G such that д'ая(р) —
(τα(ρ))~ 9αβ(ρ)τβ{ρ) for all ρ G Ua Г\Щ and for all nonempty intersections
υαΓ\ΙΙβ. (Here {gafi} is the cocycle associated to {(<£<*, Ua)} and {gfaa} is
the cocycle associated to {{<t>fa,Ua)}.)
Sketch of proof. First suppose that Pi and P2 are equivalent principal G-
bundles and let / : Pi —> P2 be an equivalence. Let ρ el/Q and choose some
и e Ρχ1{ρ), so f{u) G ρ^(ρ). Write^Q = (ρι,Φα) and ф'а - (р2,Фа)·
One can easily show that Фа(и)(Ф'а(/(и))) l is an element of G that is
independent of the choice of it G pf1 (p). For each a, define rQ : Ua -* G by
τα{ρ):=Φα(η)(Φ'α(Ηη)))-\
where и G pjf1 (p). Suppose that ρ G Ua Π Up. Then we have (τα(ρ)) l =
*tt(/(u))(*a(w))_1. Using the definitions of ραβ and g^ (see equation
(6.3)), we immediately have
9αβ(ρ) = (^(P))"19*β(ρ)τβ(ρ).
Conversely, given the maps ra :Ua —l· G satisfying
9αβ(ρ) = ОъЫГ19αβ{ρ)τβ{ρ),
we define, for each a, a map /a : p^l(Ua) —> Ρ^*(ΪΛ*) by
/«(«) == W1 (p- (Te(p))-4„(u)) .
Check that fa{u) = ίβ(ν) when pi (it) G υαΓ\ϋβ so that there is a well-
defined map / : Pi —► P2 such that /a(u) = /(и) whenever pi(it) G t/a.
Finally, check that f(u-g) = f(u)-g. D
Let ρ ; Ρ —» Μ be a principal G-bundle and suppose that we are given
a smooth left action λ : G x F —> F on some smooth manifold F. Define a
right action of G on Ρ χ F according to
(u,y)-g:= (ug,g~ly) = (ug, X(g~l,y)).
Denote the orbit space of this action by Ρ X\F (or Ρ χ с? F) and let ρ denote
the quotient map. Also denote the equivalence class of (it, у) by [it, y] so
300
6. Fiber Bundles
p(u, у) = [it, у]. One may check that there is a unique map π : Ρ χ χ F -> Μ
such that π flu, у]) = ρ (u), and so we have a commutative diagram:
Pri
PxF-^+P
ρ
PxxF^^M
Next we show that (Ρ χ χ F, π, Μ, F) is a fiber bundle (a (G, A)-bundle).
It is said to be associated to the principal bundle P. Bundles constructed
in this way are called associated bundles. More precisely, if the action
λ is not effective, we should say that Ρ χ χ F is weakly associated to
P. This is what lies behind our previously introduced notion of "ineffective
(G,A) bundle".
Theorem 6.87. Referring to the above diagram and notations, Ρ χ χ F is
a smooth manifold and the following hold:
(i) (P хд F, π, Μ, F) is a fiber bundlef and for every principal bundle
atlas {(υα,φα)}, there is a corresponding bundle atlas {(υα,φα)}
for Ρ XxF such that
Φαθφβλ{ρ^) = (ρ, X(gafi(p),y)) ifptUar\Up andyGF,
where the g^ are defined by equation (6.3).
(ii) (Ρ χ F,p,P χχ F,G) is a principal bundle with the right action
given by
fav)-9·= (ug,g~ly).
(iii) Ρ χ F —V Ρ is a principal bundle morphism along π.
Proof. Let {{υα,Φα)} be a principal bundle atlas for ρ : Ρ -> Μ. Note
that ρ {ρ l{Ua) χ F) = K~l(Ua). For each α, define Φα : n~l(Ua) -> F
by requiring that Φα о р(и, у) = Фа(и) · У for all (и, у) Е p~l{Ua) x F and
then let φα :— (π, Φα) on π_1(ί/α). We want to show that фа is bijective by
defining an inverse for фа. For every ρ Ε Ζ7α, let σα(ρ) := Φ^λ{ρ·> e), where e
is the identity element in G. Then we have
Define rja:[/axF-^ π"1(17α) by 7?а(р,у) := p{aa(p),y). We have
?7α ° 0a(p(u> y)) = »fc(p, Ф*(и) - у) = ρ(σα(ρ), Φα(ίΐ) · у)
= Ρ(^α(ρ) " Фа(и) , у) = р{и, у).
Thus τ/α is a right inverse for φα and so φα is injective. It is easily checked
that ηα is also a left inverse for φα. To see this first note that (ρ, Φα(σα(ρ))) —
6.8. Principal and Associated Bundles
301
Φα{°α(ρ)) = (p>e) so Фа{&а(р)) = e. Thus we have
Φα°η*(ρ,υ) = Φα {р(°а(р),у)) = (Ρ,Φα(ρ(*α(ρ),ϊ/)))
= (Ρ>Φα(σα(ρ))·ν) = (ρ,ί/).
Thus </>α is a bijection. Next we check the overlaps. We use Lemma 6.74;
Φα °Φβ1(ρ,ν) = Φα° ηβ(ρ, у) = Фа (рМр), у))
= (ρ, Φα(σβ(ρ)) · У) = (Ρ, Φα{Φβ1{ρ, е)) · у)
= (Р, ΦαΙρΟΦ^Ιρ1^))'!/)
= (Ρ,9αβ(ρ) · е ' У) = {Ρ,9αβ{ρ)ν).
This shows that the transitions mappings have the stated form and that the
overlap maps фа о ф~г are smooth. The family {(υα}φα)} provides the
induced smooth structure and is also a bundle atlas. Since фа о р(щр) =
(π,Φα) ο р(и,р) = (р(и),Фа(и)у) in the domain of every bundle chart
{Uai Φα)) it follows that ρ is smooth.
We leave it to the reader to verify that (Ρ χ F, ρ, Ρ χ д F, G) is a principal
G-bundle. Notice that while the map ртг : Ρ x F —> Ρ is clearly a bundle
map along π, we also have
pri ((w,y) · 5) = pri ((u · g,g~ly)) =u-g = ?тх(и,у) · ρ,
and so pr2 is in fact a principal bundle morphism. D
Clearly what we have is another way of looking at bundle construction.
The principal bundle takes the place of the cocycle of transition maps.
Exercise 6.88. Construct a principal Z2-bundle Ρ and left actions λι and
λ2 of Z2 on Sl and R respectively, such that Ρ Х\г S1 is the twisted torus
and Ρ X\2 R is the Mobius band line bundle.
We have seen that given a principal G-bundle, one may construct various
fiber bundles with G-bundle structures. Let us look at the converse
situation. Suppose that (£7, π, Μ, F) is a fiber bundle. Suppose that this bundle
has a (G, A)-atlas {(Ua, φα)} with associated G-valued cocycle of transition
functions {get/?}. Using Theorem 6.15, one may construct a bundle with
typical fiber G by using left translation as the action. The resulting bundle
is then a principal bundle (P7p,M,G), and it turns out that Ρ χ χ F is
equivalent to the original bundle E.
If (Ε, π, Μ, V) is a vector bundle and we use the standard GL(V)-cocycle
{Φα/?} associated to a VB-atlas, then the principal bundle obtained by the
above construction is (equivalent to) the linear frame bundle F(E). Letting
GL(V) act on V according to the standard action we have F(E) Xgl(v) V,
which is equivalent to the original bundle (Β,π,Μ,ν). More generally, if
302
6. Fiber Bundles
A : G —> GL(V) is a Lie group representation, then by treating λ as a linear
action we can form Ρ χ χ V.
Proposition 6.89. Let Ρ be a principal G-bundle and let λ : G —l· GL(V)
be a representation. Then Px\V has a natural vector bundle structure with
typical fiber V.
Proof. This follows from Theorem 6.87, but we can argue more directly.
Let us denote the total space of Ρ X\ V by В and let Bp be the fiber over
some point ρ € M. Then for each и G Pp there is a map фи := [it, ■] : V —> Bp
given by ν H- [u,v]. We compare фи with фид for g G G and и G Pp. Since
[ug, v] = [u, λ (g) v] for all г; Ε V, the following diagram commutes:
V
A(p)
'Фид X /iu
By,
Prom this it follows that ipu transfers the linear structure of V to Bp
independently of the choice of и G Pp. We leave it to the reader to show that
the local trivializations of Ρ χ л V constructed as in the proof of Theorem
6.87 are linear on each fiber. D
Example 6.90. Let Μ be an η-manifold and let F(M) be the frame bundle
of M. Then, if λο is the standard action of GL(n, R) on Rn, we have the
following vector bundle isomorphisms:
F(M) xXoRn = TM,
F(M) χλ*Κη = Τ*Μ,
F(M) χλο®Λ3 Rn Ξ TM ® T*M.
IfE'^PxAVisan associated vector bundle for λ a representation,
then we can map Ρ into the frame bundle of E. Indeed, the map is just
ip : и и-)- ф(и) = Vuj where фи := [η, ■] as above. Furthermore, ф(ид) =
гр(и)о\ (д) and so we have a principal bundle morphism with respect to the
homomorphism λ:
Ρ >F(B)
Μ
The map φ : Ρ —► F(E) is only injective if the action λ is effective.
Based on what we have seen above we can say that the theory of principal
bundles and associated bundles is an alternative and "invariant" approach
to G-bundles. By "invariant" we mean that the foundations can be laid out
Problems
303
without recourse to strict equivalence classes of G-atlases or the use of co-
cycles (of course, these notions can be brought in as convenient). According
to this approach, the central notion is the principal bundle, and one recovers
the other G-bundles of interest as associated bundles. Developing the
theory in this way has the advantage that much can be accomplished without
the direct need of bundle atlases. It is a more "intrinsic" approach. This
approach seems to have originated with Ehresmann and is the approach
followed by [Hus].
Problems
(1) Show that Sn xR and Sn χ S1 are parallelizable.
(2) Let X := [0,1] χ Rn. Fix a linear isomorphism L : Rn -> Rn and
consider the quotient space Ε — X/~, where the equivalence relation is
given by (0, v) ~ (l,Li;). Show that Ε is the total space of a smooth
vector bundle over the circle Sl.
(3) Exhibit the vector bundle charts for the pull-back bundle construction
of Definition 6.18.
(4) Let ξ = (£",π, M,F) be a G-bundle. Let g^ be cocycles associated to
a G-altas {{Ua9<f>a)} for ξ. Show that ξ is G-equivalent to a product
bundle if and only if there exist functions Xa : Ua ~^ G such that
9fia(x) = ^β(χ)Κλ{χ) for a11 x e Ua Π ϋβ
and all α, β.
(5) Show that the twisted torus of Example 6.17 is trivial as a fiber bundle
but not trivial as a Z2-bundle. (Use Problem 4.)
(6) Show that the space of sections of a vector bundle over a compact base
is a finitely generated module. Show that if the bundle is trivial, then
the space of sections is a finitely generated free module.
(7) Let E\ -> Mi and £2 -> M2 be smooth vector bundles. Show that if F :
Ει ->■ E2 is a vector bundle homomorphism along a map / : M\ —> M%
such that F is an isomorphism of fibers, then E\ -> Mi is isomorphic
to the pull-back bundle /*i?2 -ϊ Μι.
(8) Suppose that ξ = (Ε, π, Μ) is a vector bundle with a positive definite
metric. Show that the metric induces a vector bundle isomorphism
E~E*
(9) Show that the tangent bundle of the real projective plane is a vector
bundle isomorphic to Hom(L(RPn),L(MPTl)-L), where L(RPn) -> RPn
304
6. Fiber Bundles
is the tautological line bundle and L(RPn)± —>· RPn is the rank η vector
bundle whose fiber at I G RPn is {(Ϊ, v) € RPn x Rn+1 : t; ± I}.
(10) Recall Example 6.35. Show that the Whitney sum bundle Ει Θ Ε<ι is
naturally isomorphic to the pull-back Δ*π£ιΧ£2 : Δ* (Εχ χ Ε2) ->■ Μ.
(11) (a) Let π : jE7 —>■ Μ be an F-vector bundle. We wish to show that
Τπ : ТЕ ->■ ГМ is naturally a vector bundle. Consider the maps
α: Εφ Ε -> Ε and μ8 : Ε -> Ε for each s € F given by
α(υρ, ΐϋρ) := г;р + wp for vp, wp € Ep
μβ(βρ) := sep for ep € £?p
Show that we may identify Τ (Ε Θ Ε) with the submanifold of ТЕ х
ТЕ given by
{(v,w) £TExTE:Tn>v = Tn-w}
Now suppose that for v,w G Τϋ7 with Τπ ■ г? = Τπ · w we define
г; ЕВ w := Τα - (ν, ги) and for s G F and ν G ТЕ we define s · ν :
Τμ3 · г>. Show that with these definitions of addition and scalar
multiplication, Τπ : ТЕ -> ГМ is indeed an F-vector bundle,
(b) Let Ε be as above but assume for simplicity that F = R. Let
a:1, ...,a?n be coordinates on U С Μ. Suppose that βι,,.,,β*
is a frame field over U. Let ^1,...,ξη be defined on Ε\υ by
у = Σξι(ί/)β47Γ(2/)) f°r апУ У € Ε. Then, identifying #г with
яг ο π, the functions ж1,..., a;n, ξ1,..., ξη are a coordinate system
for Ε defined on Ε\υ and such that the -Л; are in the kernel of
Τπ. Now if Vj w G ТЕ are such that Τπ ■ ν = Τπ · w, then we may
express ν and w as
a
-Σ·1)* +Σ>*
_0_
and
гт
ι 'У а 'У
Here у and у are the base points of ν and w, and the fact that
the a's are the same for both ν and ги is a result of the condition
Ttt-v = Ttt-w. Show that
vcdw
д
= Еа<7м
dxi
, +Σ ("+*)£ ·
where in ν ЕВ го, the ЕЕ refers to the addition described in part (a).
(12) Exhibit a VB-atlas for the tautological bundle of Example 6.47.
(13) Show that the tautological bundle over RP1 is a Mobius band.
Problems
305
(14) Let Ρ —ϊ Μ be a principal bundle with group G. If Я is a Lie subgroup of
G, then the quotient P/H is an Я-principal bundle. Show that P/H ->
Μ admits a global section if and only if the structure group of Ρ —> Μ
is reducible to H.
(15) Show that the notions of smooth fiber bundle and vector bundle make
sense when the base space is allowed to be a manifold with boundary.
What issues arise if one considers allowing both the base space and
typical fiber to have boundary?
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Chapter 7
Tensors
In this chapter we shall employ the Einstein summation convention. For
example, r*koPvk is taken to be shorthand for
771 П
where the range of summation is understood from the context. Normally,
the repeated indices that are summed over occur once as a subscript and
once as a superscript. For example, if A = (aj) is an η χ m matrix and
В (&*) is an m χ к matrix, where in this case we use upper indices to
indicate rows and lower indices for columns, then С АВ corresponds to
m
1=1
This is reduced by the summation convention to c*· = a]by We will
occasionally include the summation symbol Σ f°r emphasis, or to meet the
demands of clarity.
Tensor fields (often referred to simply as tensors) can be introduced in
a rough and ready way by describing their local expressions in charts and
then going on to explain how such expressions are transformed under a
change of coordinates. With this approach one can gain proficiency with
tensor calculations in short order, and this is usually the way physicists
and engineers are introduced to tensors. However, since this approach hides
much of the underlying algebraic and geometric structure, we will not pursue
it here. Instead, we present tensors in terms of multilinear maps.
307
308
7. Tensors
7.1. Some Multilinear Algebra
It will be convenient to define the notion of an algebraic tensor on a vector
space or module. The reader who has looked over the material in Appendix
D will find this chapter easier to understand. In particular, we assume the
definition of "multilinear" (Definition D.13). In this chapter, if we say that a
module is finite-dimensional,1 we mean that it is free and finitely generated
and thus has a basis. All modules in this chapter are assumed to be over a
commutative ring with unity.
Definition 7.1. Let V and W be modules over a commutative ring R with
unity. Then, an algebraic W-valued tensor on V is a multilinear mapping
of the form
r : Vi x V2 χ · ■ · χ Vm -> W,
where each factor V» is either V or V*. If the number of V* factors occurring
is r and the number of V factors is s, then we say that the tensor is r-
contravariant and s-covariant. We also say that the tensor is of total
*УРе 0-
The most common situation is where W is the ring R itself, in which case
we often drop the adjective "R-valued". Notice that if r : V* x V x V* x V -4
R is a tensor, then we can define a tensor τ : V* x V χ V χ V* -> R by
τ(αι,υι,υ2,α2) :=r(ai,vi,a2,V2).
Although these two tensors clearly contain the same information, they are
nevertheless different. We indicate this with a more specific notation. We
say that τ is a tensor of type (x χ χ), while τ is of type (* 2 *). More
generally, a tensor might, for example, be specified to be of type
Tl Г2 лшага \ / rl ra \
Si 52 SbJ \S\ 52 Sb J '
The general pattern should be clear. If r = r\ Η h ra and s = s\-\ \-$ь
then the tensor would be of total type Q, which we also write as (r,s).
The set of all tensors of fixed type (as above) is easily seen to be an R-module
with the scalar multiplication and addition defined as is usual for spaces of
functions. As another example, a multilinear map
Τ : V χ V* x V χ V* χ V* -> W
is a W-tensor which is of type (χ 1 χ 2) and total type (2) - The set of all W-
valued tensors on V of type (χ ι χ 2) is denoted Τχ * χ 2 (V; W), and we have
analogous notations for other types. In many, if not most, circumstances
we agree to associate to each tensor of total type Q), a unique element of
T^(V; W) by simply keeping the relative order among the V variables and
For modules, what we mean by dimension is what is usually called the rank.
7Λ. Some Multilinear Algebra
309
among the V* variables separately, but shifting all V variables to the right
of the V* variables. Following this procedure, we have, for example, the
map
TiVtVjW^T^VjW).
Maps like this will be called consolidation maps or consolidation
isomorphisms.
Definition 7.2. A tensor
τ : V* χ V* χ - · χ V* χ V χ V χ · · · χ V -> W,
4 ν ' * ν '
r times з times
where all the V factors occur last, is said to be in consolidated form.
The set of all such (consolidated) W-valued tensors on V will be denoted
^(V; W). As a special case we have T^V; R) = V*. We will often
abbreviate ^(V; R) to Trs(V).
For example, elements of T32(V; W) are in consolidated form, while
tensors are unconsolidated.
Remark 7.3. Some authors consolidate by putting all V arguments first.
Also, sometimes it is appropriate to forgo the consolidation especially in
connection with the "type changing" operations introduced later. Our policy
will be to work with tensors in consolidated form whenever convenient.
Example 7.4. One always has the special tensor δ € T\(V; R) defined by
ff(a,v) = a(v)
for a E V* and ν Ε V. This tensor is sometimes referred to as the Kro-
necker delta tensor.
There is a natural map from V to V** given by ν н->- гГ, where ν : α ь->
α(ν). If this map is an isomorphism, we say that V is a reflexive module
and we identify V with V**. Finite-dimensional vector spaces are reflexive.
Exercise 7.5. Show that the C°°(M) module of sections of a vector bundle
Ε -ϊ Μ is a reflexive module. (It is important here that we are only
considering vector bundles with finite-dimensional fibers.)
We now consider the relationship between tensors as defined above and
the abstract tensor product spaces described in Appendix D. We restrict
our discussion to tensors in consolidated form since the implications for the
general situation will be obvious. We specialize to the case of R-valued
tensors where R is the ring. Recall that the fc-th tensor power of an R-
module V is denoted by 0 V := V ® · ■ ■ <8 V. We always have a module
homomorphism
(7.1) ((g)4) (®V)->r.(V;R),
310
7. Tensors
whereby an element u\ ® · · · ® ur ® /31 · · · ® /?5 e (<8>rV) ® (0*v*)
corresponds to the multilinear map given by
(a1,..., a^vi,..., t/e) ь> a1 (Ul)·.· ar (ur) β1 {vi) -'β8 Ы .
We will identify i*i ® ■ · · ® ur ® /31 ® · · · ® /3s with this multilinear map. In
particular, this entails identifying ν G V with the element ν G V** where
ν : а ь-> а(г>). If V is a finite-dimensional vector space, then the map (7.1)
is an isomorphism. In fact, it is also true that if V is the space of sections
of some vector bundle over Μ (with finite-dimensional fibers), then V is a
C°°(M)-module and the map (7.1) is still an isomorphism. A tensor which
can be written in the form г/ι® · - -®β8 is called a simple or decomposable
tensor. Note well that not all tensors are simple.
Remark 7-6. The reader should take careful notice of how we treat the
orders of the factors: An element of V® V* ®V* corresponds to a multilinear
map V* χ V χ V -» R and not to a map VxV*xVMR.
Since the map (7.1) is not always an isomorphism for general modules,
and since no analogous isomorphism exists in the case of tangent spaces to
infinite-dimensional manifolds such as those discussed in [LI], it becomes
important to ask to what extent the map (7.1) is needed in differential
geometry. Serge Lang has written a very fine differential geometry book
for manifolds modeled on Banach spaces [LI] without the help of such an
isomorphism. In any case, we still can and will consider u\ ® · ■ · ® ur ® β1
• · · ® β8 to be an element of T^(V) as described above. Another thing to
notice is that if (7.1) is an isomorphism for all r and 5, then in particular
V = V**, that is, V must be reflexive. Corollary D.33 of Appendix D
states that for a finitely generated free module, being reflexive is enough to
insure that (7.1) is an isomorphism for all r and 5. In the latter case, the
consolidation maps introduced earlier can be described in terms of simple
tensors. For example, the consolidation map
Ti22(V)->r23(V)
is given on simple tensors by
a®v®w®/3®7—»v®w®a®/3®7.
Now let us consider the spaces V ® V* and V ® V* ® V*. By a
straightforward argument using the universal property of tensor product spaces, one
can construct a bilinear map
(V®V*) χ (V®V*®V*)->V®V*®V®V*®V*
such that (v ® a, w ® β\ ® ^2) is mapped to ν ® a ® w ® βι ® β^ This
corresponds to a product map
® : T\ (V) χ Tl2 (V) -> Γ1! l2 (V)
7.1. Some Multilinear Algebra
311
such that (5, T) -> S®T, where
(S ® Τ) (αϊ, vi, a2, v2, v3) - 5 (αϊ, νχ) Γ (α2, v2, ν3).
The general pattern should be clear, but writing down the general case
is notationally onerous. This product is the (unconsolidated) tensor
product of tensors. Note carefully the order of the factors. To simplify the
notation, the tensor product is often defined in a slightly different way when
dealing with tensors which are in consolidated form:
Definition 7.7. For tensors S G Tr^(V) and Τ G TrJ2(V), we define the
(consolidated) tensor product S®T e Tri+r2Sl+S2 (V) by
5®r(^,...,^+r2,Vl,...,vsl+52)
:-S(0\...,^Vl,...,vsjT(^
Whether or not a tensor product is the consolidated version will normally
be clear from the context, and so we will drop the word "consolidated".
We can also extend to products of several tensors at a time. While it is
easy to see that the tensor product defined above is associative, it is not
commutative since the order of the slots is an issue.
Let T*(V) denote the direct sum of all spaces of the form Tr0(V), where
we take T°0(V) := R. The tensor product gives T*(V) the structure of an
algebra over R as long as we make the definition that r ® A :— rA for r G R.
Proposition 7.8. Let V be α free R-module with basis (ei,..., en) and
corresponding dual basis (e1,..., en) for V*. Then the indexed set
{ег1 ® — ®егг ®ejl ® ••■®eJ* : ή,... ,гг,л,... Js - Ι,.,.,η}
is a basis for Tr5(V). If те Tr3(V), then
r _ ru -v^ eii g)... о elr ® ejl ® · · · ® ejs (summation!)
where rll"\ ja = r(eix,..., eS ел,...,eja).
Proof. If r G Trfl(V; R) and we define τίχ~\ js = r{e4,..., e\ en,..., ejs),
then it is easy to check that
r = τ11'·\ ja eh ® · · · (g> eir ® eJ1 ® - · · ® eJs,
and so, in particular, our indexed set spans Trs(V;R). Indeed, if we
denote the right hand side of the above equation by r7, we obtain (using the
312
7. Tensors
summation convention throughout)
r'(ekl,...,ekr,eh,...,eis)
_ Ц...1г
— T 31—Js "ii
= r{ek\ ~k'
C-ir«-i
h...la
,,ε^,β/,,...,^).
Thus rf and τ agree on basis elements, and by multilinearity rf = r.
For independence, suppose Tn"'lr31...j8eii ® · ■ * ® ev ® e·71 ® · · · ® eJa — 0
for some nr+s elements τ'1 ",ir л...^3 of R. This is an equality of multilinear
maps, and if we apply both sides to (e
fcl.../Cf·
, e Γ, βιΎ,..., ei3), then we obtain
«l—l-
U...tr
Л-Js
— 0, Since our choices were arbitrary, we see that all nr+s elements
are equal to 0. Π
As a special case we see that if A £ T\(V; R), then A = Аг3 ег ® eK
where A% j = А(ег,е3). This theorem is a special case of Theorem D.29 of
Appendix D, which we will also invoke below for spaces like W ® V*.
If we are dealing with tensors that are not in consolidated form, it should
still be clear how to obtain a basis. For example, {еъ <g> e? ® e^} is a basis
for Τ } (V; R), and a typical element A would have an expansion
A = A
i fc„
»е7' ® efc.
г к
Notice the purposeful staggered positioning of the indices in A1
Definition 7.9. The elements Ач"Лг jx„.ju from the previous proposition
are called the components of τ with respect to the basis ei,..., en.
Example 7.10. If V = TpM for some smooth manifold M, then we can use
any basis of TpM we please. That said, we realize that if ρ is in a coordinate
chart (?7,x), then the vectors g|r| ,..., ·£^\ form a basis for TjpM, and
we may form a basis for Trs(TpM) consisting of all tensors of the form
д
дх*
д
dxir
<g> dxjl1 ® · · · ® dxjs I
\p \i
For example, an element Ap of Τ11 {TpM) can be expressed in coordinate
form as
A -Ax —
Λρ-Λ3 dxi
® άχ°\
An element Ap of T^I^M) can be written as
A.p — A.
д
Л»*- Qxh
д
дх*
® dxjl I ® ■ · · ® dx?s
and this is called the coordinate expression for Ap.
7.1. Some Multilinear Algebra
313
The components of a tensor depend on the basis chosen, and a different
choice will give new components related to the first by a transformation law.
This is the content of the following exercise:
Exercise 7.11. Let ei,..., en be a basis for V and let e1,..., en be the
corresponding dual basis for V*. If ei,..., en is another basis for V with
ё{ = С $е/г,
then the dual basis ё1,..., ё71 is related to e1,..., en by e1 — (C-1)^ ek,
where С = (Cj). Show that if rl jfc are the components of r with respect to
the first basis (and its dual) and if тг jk are the components with respect to
the second basis, then
τ*** = rV &£% (C-χ (sum over a,b,c).
This is a transformation law. What is the analogous statement for τ G
rs(V;R)?
Example 7.12. It is easy to show that for any basis (with corresponding
dual basis) as above, the Kronecker delta tensor δ has components £*-, where
0^ = 0 if i^ j and 5^ = 1-
It is easy to show that if S € T\{V) and Γ € T22(V), then S ® Γ has
components given by
More generally, if 5 6 Trl1 (V) and Γ € T^fV), then
(7.2) (5 ® τ)*-*!"»™** bl...bnA..A2 = S·1"*! bl...bn ρ*---** Α.^ .
Notice the consolidation. If we choose not to employ consolidation, then the
(unconsolidated) tensor product would be expressed differently in
component form. For example,
This way of treating the position of the indices is a convention that is called
(naturally enough) "positional notation". For more on positional notation,
see [Pe], [Dod-Pos] and [Stern].
If V is a finitely generated free module, then we have a natural
isomorphism V <g> V* = L(V, V). We can be a bit more general. Let W be another
finitely generated free module. Consider the map Φ : W x V* —> L(V, W)
given by (ΐϋ, α) \-> Φ (ιυ, α), where
Φ (w,a) (ν) :— ot{v)w.
314
7. Tensors
This map is multilinear, and so using the universal property of tensor
products we get a map
$:W®V*->L(V,W)
such that Φ : w α ь-» Φ (w,a) Ε L(V, W). Let us show that this map is an
isomorphism. A given element A G W®V* may be written as A = J^ wl®a%,
where the гиг are linearly independent. Indeed, we just write A in terms of
a basis and collect terms. Then if Ф(А) = 0, we have ^ai{v)w% 0 for
all f. Thus аг(у) = 0 for all v. It follows that A = 0. We see that Φ is
injective. Both V and W are finite-dimensional, and W ® V* and L(V, W)
have the same dimension. Indeed, L(V, W) is isomorphic to the space of
πι χ η matrices with entries from R, where m and η are the dimensions
(or ranks) of W and V respectively. If V and W were vector spaces, this
would imply that Φ is onto. However, it remains true that Φ is onto in the
case where W and V are finite-dimensional free modules, but we must argue
differently. In Problem 2 we ask the reader to show that if ei,..., en is a
basis for V and f\,..., fm is a basis for W, then {Ф (/г ® eJ)} is a basis for
L(V, W). Thus Φ : W ® V* -> L(V, W) is an isomorphism. If r = г]/г ® е ,
then
Φ (τ) : t; H> r)€?{v)fi = r^· ^'/г.
The components rj are exactly the entries of the matrix that represents
Φ (τ). When the above conditions hold, so that Φ is an isomorphism, we
often identify W (8) V* with L(V, W) and write r even when we mean Φ (r).
We say that τ has two "interpretations". Under this identification
(w ® a) (v) — a(v)w.
In component form, the two interpretations of r show up as
ν \-t w, where гиг = rlj г^, and
(а, г;) H> r (a, г;), where τ (a, г;) = τ%3ν3α%.
In particular, we identify T\(V) with L(V, V).
Remark 7.13 (Tensions of conventions). Both W ® V* and V* ® W can be
identified with L(V, V). For example, we may also interpret a®w G V*®W
as the map (a®w)(v) = a(v)w. If the reader looks at how we have
consolidated the spaces in the definition of jTrs(V), it will be apparent that
we have preferred W ® V* over V* ® W. However, if one considers the case
where the underlying ring is not commutative, it becomes clear that the
isomorphism V* ® W = L(V, W) is correct for left modules while the other
is correct for right modules (V* is a right module if V is a left module, and
vice versa). On the other hand, the identification W ® V* = X(V, W) is
more natural for the conventions of matrix multiplication. For example, if
we think of elements of Rn as column vectors and elements of (Mn)* as row
7.1. Some Multilinear Algebra
315
vectors, then for w G W1 and α G (Rn)*, the linear map α ® w is indeed
given by the matrix aw. The tension between standard matrix conventions
and left modules is well known to algebraists. We think of modules over
commutative rings as simultaneously both left and right modules.
If V is finite-dimensional, then one can make various other
^interpretations of tensors:
Example 7.14. Suppose that V is finite-dimensional. Then, elements of
Tr$(V) can be interpreted as members of
T°s(V;n(V))
according to the prescription that τ(ν^ ..., vs) acts by
t(vu ..., v8)(al, ...,ar)~ τ(α\ ..., ar, vx,..., vs).
Similarly, elements of Tr3(V) can be interpreted as members of Tq(V; T°3(V)).
Example 7.15. Let V be as in the previous example. Elements of T°SI+S2 (V)
can be interpreted as members of
r0sl(V;r°S2(V))
by the prescription
r(vi,...,vei)(tti,...,ue2) :=r(vi,...,vei,ui,...,uea).
One can easily see from the above examples that many reinterpretations
are possible. One of the most common is where one interprets elements of
T°2(V, R) as elements of L(V, V*) according to
r{v) (u) = τ (υ, и) for и, ν Ε V.
Exercise 7.16. Show that under the identification of T\(V, R) with L(V, V)
we can interpret the Kronecker delta tensor as the identity map.
Definition 7.17. A covariant tensor r G T°S(V, W) is said to be symmetric
if
T(vu...,vs)=T{va{1),...,va{s))
for all vi,..., vs and all permutations σ of the letters {1,2,..., s}. We define
a symmetric contravariant tensor similarly.
Definition 7.18. A covariant tensor r G T°S(V, W) is said to be
alternating if
r(vi, ...,ve)= sgn(a)r(^(1),..., νσ{3))
for all vi,..., vs and all permutations σ of the letters {1,2,..., 5}, where
sgn(a) = 1 if σ is an even permutation and —1 if it is an odd permutation,
sgn σ) is called the sign of the permutation σ; We define an alternating
contravariant tensor similarly.
316
7. Tensors
If £ : U -> V is a linear map, then the map Г : r°s(V; W) -> r°a(U; W)
is defined by
(Γτ)Κ,... ,ue) := r(€ (t*i), · ·. ,1 («,))·
£*r is called the pull-back of r by £ It is easy to show that £* is linear. If
we have linear maps £ : Ui —>■ U2 and A : U2 —» V, then
(Ao*)*:T°(V;W)->I*(Ui;W)
and
(λοί)* = Γολ*.
Thus the pull-back £ —► £* defines a contravariant functor in the category of
W-valued covariant tensors. (Because of this, one might wish that covariant
tensors were called contravariant and vice versa, and indeed some authors
have reversed the traditional terminology.) Suppose that (ei,..., en) is a
basis for V and that (Д,..., fm) is a basis for W. If £ (ei) — $^?/*> ^en
we have
(^)il..^ = (rr)(eilJ...JetJ = r(€(eiJ,...li(e<J))
which gives the component form of the pull-back operation in terms of the
matrix (£%).
Proposition 7.19. IfXe L(V, V), a € T°ei(V) and β G T°S2(V), then
\*(α®β) = \*α®\*β.
Proof. We have
λ*(α ® /3)(wi,..., u81, «β1+ι,..., uSl+32)
= α (8) /3(Aui,..., λιιβ1, λιιβ1+ι,..., AuSl+S2)
= q(Aui, ..., λιιβ1)β (Auei+i,..., AuSl+S2)
= λ*α(«χ,..., η31)\*β (tt51+i,..., u3l+S2)
- λ*α ® λ*/3(«ι,...,uSl,uei+i,. ■ ·,Wsi+s2)· □
Before going on to study tensor fields, we introduce one more notion
from multilinear algebra referred to as contraction.
Definition 7.20. Let (ei,..., en) be a basis for V and (e1,..., en) the dual
basis. If τ Ε Trs(V), then for к < r and / < s, we define Cfr Ε T^iiOO by
η
:=y!r(fl1'···', u e°· · ,..-»er-1»wi,..., ett ,...,ws_i).
^T fe-th position j.th position
7,1. Some Multilinear Algebra
317
This processes is called contraction. Write the components of r with
respect to our basis as тп"Лг jlm„ja- If we pick out an upper index, say ik, and
also a lower index, say jj, then we obtain the components of the contracted
tensor Cjfr by:
(Cfr)tl'"%k"'tr A :- тг'"а-Лг Λ...α..Λ (sum over a).
Here the caret means omission. In practice, one often just writes
rii...tfc...ir
jl—Jl—je
instead of (C^r)4'" k'"%r . - as long as it has been made clear how the
ν * / 31-З1—3s °
contraction was carried out. For example, one often sees expressions like
Rid : R\rj· Notice that we always contract an upper index with a lower
index. The map C* contracts the fc-th upper index with the /-th lower index.
Consider a tensor of the form ν ® w ® 7/ ® 0 Ε T22(V). One can show
that
C{{\ ® w ® η ® Θ) = 7?(v)w ® 0.
Similarly,
C\ (v ® w ® η ® Θ) = e(v)w ® 7?.
In general, C* acts on simple tensors νχ ® v2 ® · · · ® vr ® η1 ® 7?2 ® · · · ® η8
by an obvious extension of the above. Universal mapping properties can be
invoked to give a basis free definition of contraction. Contraction generalizes
the notion of the trace of a linear transformation.
A common use of contraction involves first taking the tensor product of
two tensors, and then performing a contraction of a contravariant slot of one
with a covariant slot of the other. One often performs several contractions.
For example, we may form a tensor that is given in components as
racefg = Sake Tkcfg (sum over k).
Evaluation of a tensor on its arguments is the result of a repeated
contraction. For example, let V have a basis ei,..., en and let e1,..., en be the
dual basis for V* as above. If υ = ντβ^ w = ьУвх, and a = агег, then for
r G T^CV) we easily deduce that
(7.3) r(a, v, ги) = тг j& a{V3wk,
which is the result of a repeated contraction on the tensor r ® a ® ν ® w.
More generally, if we express elements νχ,..., vs € V and αχ,..., ar Ε V* in
terms of our basis and its dual, then for r Ε T^V), we have an analogous
general expression for τ(αχ,..., ar, νχ,..., vs) in terms of the components
of the tensor and its arguments.
318
7. Tensors
7.2. Bottom-Up Approach to Tensor Fields
There are two approaches to tensor fields on smooth manifolds that turn
out to be equivalent (at least for finite-dimensional manifolds). We start
with the "bottom-up" approach where we apply multilinear algebra first to
individual tangent spaces. The second approach directly defines tensors on
Μ as tensors on the module X(M).
Roughly speaking, a smooth (r, s)-tensor field on a manifold Μ assigns
to each ρ Ε Μ an element of Tr$(TpM) in a smooth way. We are interested
in making sense of smoothness for tensor fields, so we wish to view a tensor
field as a section of an appropriate vector bundle (a tensor bundle). Let
us start out being a bit more general by considering a real rank к vector
bundle ξ (Ε1, π, Μ). For convenience, we take the typical fiber to be R*.
Let Trs{E) - \JpeMTr3(Ep). We wish to construct a bundle Trs(£) which
has Trs(E) as total space, Μ as base space, and T^^Ep) as fiber over p. If
(ΙΙ,φ) is a VB-chart for £, then we construct a VB-chart for Ττ3(ξ) in the
following way: Recall that φ has the form φ = (π, Φ), where Φ : π~ιΙΙ -t Rk
and where Φρ := Φ\Ε : Ep -» Rk is a linear isomorphism for each p. We
obtain a map Фг/ : Trs(Ep) -» T^R*) by
(Φρ'βΤρ)(αι,...,θ!Γ,7;ι,...57;θ)
:= τρ((Φρ)* αϊ,..., (Φρ)* αΓ> Φρ 4,..., Φ~ 4).
These maps combine to give a map ΦΓ,δ : π-1 С/ —> T^R^) which is smooth
(exercise). Our chart for Ττ3(ξ) is
0r'5 := (π, Фг'*) : тг λϋ -+ U x Trs(Rk).
If desired, one can choose, once and for all, an isomorphism T^R*) = Rk \
A VB-atlas {{υα,φα)} for ξ = (Ε,π, Μ) gives a VB-atlas {(υα,φ%8)} for
Exercise 7.21. Show that there is a natural vector bundle isomorphism
Τ$(ξ)^(®τΕ)®{®*Ε*).
We leave it to the interested reader to prove the following useful theorem.
Proposition 7.22. Let ξ = (Ε, π, Μ) be a vector bundle as above and let
Τ : Μ -> ^(E) be a map which assigns to each ρ Ε Μ an element of
Trs (Ep). Then Τ is smooth if and only if
ρ н> Τ(ρ)(α1(ρ),..., ar(p),Xi(p),... ,Xs(p))
is smooth for all smooth sections ри аг(р) and ρ н> Хг(р) of Ε* -ϊ Μ and
Ε -> Μ respectively. The same statement is true if we use local sections.
7.2. Bottom-Up Approach to Tensor Fields
319
The set of smooth sections of Ττ3(ξ)) is denoted T(Trs{£)). If Τ G
r(Trs(£)), then for ΛΊ,..., X3 any smooth sections of Ε —> Μ and αϊ,..., аг
smooth sections of Ε* -> Μ, define Τ (αϊ,..., αΓ, Χχ,..., Xs) G C°°(M) by
Τ(αι,..., or,Χι,..., -Χ,)(ρ) :- Τρ(αι(ρ),..., αΓ(ρ), Χι(ρ),..., -Χ, (ρ)).
Now we have a map Τ : (ГЕ*)к χ (ТЕ)1 -► С°°(АГ). This map is clearly
multilinear over C°°(M), and we see that we can interpret elements of Г(ТГ3(£))
as such maps when convenient. This extends the idea of thinking of a 1-form
α as a C™(M) linear map X(M) -> C°°{M).
Like most linear algebraic structures existing at the level of a single fiber
Ep> the notion of tensor product is easily extended to the level of sections:
For τ e Τ(ΤΓ^(ξ)) and η G Г(ТГ22(£)), we define the (consolidated) tensor
product τ <g) r? € Г(:Г5+;252(0) by (r ® η) {ρ) := rp (8) τ^. Thus
(r ® η) (ρ)(α1,..., аГ1+Г2, νχ,..., vSl+S2)
= τ(α\..., аг\ νι,..., ν3ι)η(ανι+ι,..., аГ1+Г2, νβ1+ι,.. ·, VS1+S2)
for all аг G £"* and v% G £?p.
Let (si,..., Sk) be a local frame field for ξ over an open set U and let
σ^.,.,σ* be the dual frame field of the dual bundle E* —У М so that
σ*(βί) ^}· Consider the set
{σ11 ® · · ■ ® σ* О 5Л ® · · ■ ® sJs : гъ ..., гг, jb ..., js = 1,..., *}.
If τ G Г(Тг5(0), then we have functions τ11-* л...л G C°°(17) defined
^У rti...»r ^ ^ — τ(σ»ι,..., σ*Γ, 5jx,.. ·, 5je). It follows from Proposition 7.8
that r (restricted to U) has the expansion
T _. rii...ir ^ ^ σ*ι ® ... ® σ*ν ® «^ ® ... ® 5js.
Also, applying equation (7.2) in each fiber Ep, we see that the component
functions for r ® 77 are given by
(r®T?)11"'iri+r2 . ■
V ^ '^ Jl-J«i+*2
_ «-*1».»Γχ . . „iri+1—*r2 .
- T 13i...js1 V j*1+i...jea "
Here, and wherever convenient, we use the consolidated tensor product.
Notation 7.23. Whenever there is no chance of confusion, we will refer to
Ττ8(ξ) by Trs(E) -> Μ or even just Tr3(E) (the latter is the notation for the
total space of the bundle).
In the case of the tangent bundle TM, we have special terminology and
notation:
Definition 7.24. The bundle T^TM) -^ Mis called the (r,s)-tensor
bundle on M.
320
7. Tensors
By Exercise 7.21, Tr8(TM) 2* ((gfTM) ® ((g)sT*M) and this natural
isomorphism is taken as an identification so the latter bundle is also referred
to as a tensor bundle. We now restrict ourselves to the case of the tangent
bundle of a manifold but note that much of what follows makes sense for
general vector bundles.
Definition 7.25. The space of sections T(Trs(TM)) is denoted by TS{M)
and its elements are referred to as r-contravariant s-covariant tensor fields
or just type (r, s)-tensor fields. The space 7~o(M) is denoted by ^(M)
and T°S{M) by TS(M).
In summary, a smooth tensor field Л is a smooth assignment of a
multilinear map on each tangent space of the manifold. Thus for each p, A(p)
is a multilinear map
A{p): (т;М)гх(ТрМ)8^Ш,
or in other words, an element of Trs(TpM). Elements of Trs(TpM) are called
tensors at p. We also write Ap for A(p).
Example 7.26. In Definition 6.42, we introduced the notion of a Riemann-
ian metric on a real vector bundle. We saw that such metrics always exist.
The most important case is where the bundle is the tangent bundle TM of
a manifold M. In this case, we say that we have a Riemannian metric
on M. Thus a Riemannian metric on Μ is an element of 7*2 (M) which is
symmetric and positive definite at each point.
Of course the manifold in question could be an open submanifold U
of Μ so we have C°°(f7)-module (r, s)-tensor fields over that set denoted
7J{U). The open subsets are partially ordered by inclusion V С U and the
tensor fields on these are related by restriction. Let Гу : Τζ(ΙΙ) —> ТЦУ)
denote the restriction map. The assignment U —> Τζ{11) is an example of a
presheaf and in fact a sheaf.
We will also sometimes deal with tensors with values in TM (or in
T*M). First note that the space Tr8(TpM]TpM) of all multilinear maps
(T*M)r χ (TpM)s -> TPM is a vector space. The set Trs(TM;TM) :=
\JpTrs(TpM] TPM) can be given a smooth vector bundle structure in a way
that is closely analogous to Trs(TM) -^ M.
Definition 7.27. The space of sections T{TTS(TM\TM)) is denoted by
Τζ(Μ]ΤΜ) and its elements are referred to as r-contravariant s-covariant
TM-valued tensor fields. Similarly, we may define T*M-valued tensor
fields.
Note that ΓΜ-valued tensor fields can be associated in a natural manner
with ordinary tensor fields. For example, if A e T\(TM\ Τ Μ), then using
7,2. Bottom-Up Approach to Tensor Fields
321
the same letter A by abuse of notation, we may define an element A G
T\(TM) by
Αρ(θρ, vp, Wp) = θρ (Ap(vp, wp)) for θρ G TpM and vp, wp G TpM,
Many such reinterpretations are possible. For this reason, we shall stick to
studying ordinary tensors and tensor fields in what follows.
We shall define several operations on spaces of tensor fields. We would
like each of these to be natural with respect to restriction. We already have
one such operation: the tensor product. If A G ТЦ{и) and В G ТЦ{11)
and V С С/, then r% {А® В) = r^A ® r$B. A (ft,i)~tensor field A may
generally be expressed in a chart (?7, x) as
(7.4) A = Ah"ir. , —- <g> · · · ® —- ® dxjl - - - ® dx3*,
v } n-JsQxn дхг'
where A%1"%%T . are functions, ^ G X(U) and d^ G 3£*(i7). Actually, it
is the restriction of r to U that can be written in this way, but because of
the naturality of all the operations we introduce, it is generally safe to use
the same letter to denote a tensor field and its restriction to an open set
such as a chart domain. It is easy to show that
^ы-*{*г--*--£е ще)·
and so the components of a smooth tensor field are C°° for every choice of
coordinates ([/, x) by Proposition 7.22. Conversely, one can obviously define
tensors that are not necessarily smooth sections of the appropriate tensor
bundle, and then a tensor will be smooth exactly when its components with
respect to every chart in an atlas are smooth. Evaluating the expression
(7.4) above at a point ρ G U results in an expression such as that given at
the end of Example 7.10.
Exercise 7.28 (Transformation laws). Suppose that we have two charts
(17, x) and (V,x). If A G 7~2 (M) has components Агк in the first chart and
Ajk in the second chart, then on the overlap U f\V we have
-и. _ а дхг дхь дхс
where
,-г дх* , а , д дхь д
ах = л ах and ^r-^ =
дха дх1 дх{ дхь'
This last exercise reveals the transformation law for tensor fields in
T\{M), and there is obviously an analogous law for tensor fields from TTS(M)
for any values of r and s. In some presentations, tensor fields are defined in
terms of such transformations laws (see [L-R] for this approach). It should
be emphasized again that there are two slightly different ways of reading
322
7. Tensors
local expressions like the above. We may think of all of these functions as
living on the manifold in the domain U C\V. In this interpretation, we read
the above as
3>) - ^(Ρ) |£(Ρ)|ϊ(Ρ) |ί Μ for each PtUnV.
This is the default modern viewpoint. Alternatively, we could take J^
to be functions on x(U Π V) and write ^ (z1,..., xn). Then, |^ would
(я1,... ,xn) so that both sides of the equation are
refer to |4
ox1
ο χ ο χ
«-ι
functions of variables which we abusively write as (a;1,. · · ,жп). The first
version seems theoretically pleasing, but for specific calculations that use
familiar coordinates such as polar coordinates, the second version is often
convenient. For example, suppose that a tensor r has components with
respect to rectangular coordinates on R2 given by A3k and we wish to find the
components in polar coordinates. For indexing purposes, we take (#, у) -
(и1, и2) and (г, θ) = (ν1, ν2). Then we have
which can be read so that both sides are functions of (v1^2) by writing
ul and u2 as a function of (v1^2), etc. Of course, the charts are there to
"identify" open sets in Euclidean space with open sets on the manifold so
these viewpoints are really somehow the same after all. Using (x,y) and
(r, 0), the transformation (7.5) is given in matrix form as
An Ли
Au A22
cos θ —r sin θ
sin θ r cos θ
An A12
A12 A22
COS0
—r sin θ
sin0
rcos0
We now introduce the pull-back of a covariant tensor field, which will
play a big role in the next chapter.
Definition 7.29. If / : Μ —> N is a smooth map and r G Ts{N)y then we
define the pull-back /*r e T3(M) by
/*r(vi,...,t;e)(p)-r(T/.i;b.
for all vi,..., vs G TpM and any ρ G M.
>Tf.va)
Notice the connection of this with the pull-back defined earlier in a
purely algebraic context. It is not hard to see that /* : TS(N) —> TS{M) is
linear over R, and for any h Ε C°°(N) and r € TS(N) we have /* (hr =
(h о /) /*т. If / : Μ —► N and g : Ν -¥ Ρ are smooth maps, then of course
(9°f)*-r°9*.
Let us discover the local expression for pull-back. Choose a chart (ϋ,χ)
on Μ and a chart (V,y) on JV and assume that f(U) С V, Let us denote
7.3. Тор-Down Approach to Tensor Fields
323
*£β- by g for simplicity. We have Tpf ■ &\p = Ε & (p)&|/w and
(TT)tl..
ω-<™(**
= r(r/
д
0z*i
' дх
дх1'
= T(dJLkl
, (r>) ——
\дх^ к ' dykl
-<
д
дук
m
д
№
'02/*
= rkl...k3(fip))-^(p)
al)
дук"
\dyki
дук' м
*гСр>
9
gyfcs
(Ρ)
/О»)
<9yfcs
(Р)
Thus we have
(/*г)п...г. = (7*i»-fc.°/) ^Ϊ7
ay
Kg
дхг*
This looks similar to a transformation law for a tensor, but here / is not
a change of coordinates and need not even be a diffeomorphism. Pull-back
respects tensor products:
Exercise 7.30. Let / : Μ -¥ N be as above. Show that for τ\ G TSl (N)
and r2 G TS2{N) we have /* (n <g> T2) = f*n /*7^.
In the case that / : Μ —у N is a diffeomorphism, the notion of pull-back
can be extended to contravariant tensors and tensors of mixed covariance.
For such a diffeomorphism, let (Tf *)* : T*M -> T£N denote the dual of
the map Tf
-1
TPN
TPM.
Definition 7.31. If / : Μ —> N is a diffeomorphism and τ is an (r, s)-tensor
field on N, then define the pull-back f*r G Tr $ (M) by
/*τ(αι,...,αΓ,νι,...,υβ)(ρ)
:=г((ГГ!)*01 (Г/ 1)*аг,Г/^ь...,Г/.г;5)
for all fi,..., vs G TPM and αϊ,..., ar G T^Μ and any ρ Ε Μ. The push-
forward is then defined for τ e Tr s (M) as /*r := (/ 1)*r.
7.3. Top-Down Approach to Tensor Fields
Specializing what we learned from the discussion following Proposition 7.22
to the case of the tangent bundle, we see that a tensor field gives us a
C°°(M)-multilinear map based on the module X(M). This observation leads
324
7. Tensors
to an alternative definition of a tensor field over M. In this "top-down" view,
we simply define an (r, s)-tensor field to be a C°°(M)-multilinear map
X*(M)r χ X(M)8 -> C°°(M).
In this view, a tensor field is an element of Τ* s (X(M)). For example, a
global covariant 2-tensor field on a manifold Μ is a map τ : X(M) χ X(M) ->
C°°(M) such that
TtflXl + /2-У2.У) = flT(XuY) + f2T(X2,Y),
t(Y, fiXi + /2X2) = fir(Yt Χι) + f2T(Y, X2)
for all /i, /2 € C°°(M) and all Xi,X2,Y € X(M). As we shall see, it turns
out that such C°°(M)-multilinear maps determine tensor fields in the sense
of the previous section.
If we take a top-down approach to tensor fields, then we must work to
recover the presheaf/sheaf aspects. Indeed, it is not obvious what is the
relation between Trs(X(M)) and ^(^(i/)) for some proper open subset
U С Μ. Indeed, thinking purely in terms of modules makes the issue clear,
The module X(M) is not the same module as X(U) unless U = M. A
priori, there is no immediate reason to think that a multilinear map with
arguments from the module X(M) should be able to take elements of X(U)
as arguments! For instance, from the top-down viewpoint, how can we
insert coordinate fields -^ and dx% into an element of Trs(X(M)) to get
coordinate expressions if the chart domain is not all of M? We address this
in the next section indirectly by showing how the top-down approach gives
back tensors as sections (the bottom-up approach). Another comment is
that both X(U) and T7^(£([/)) are finite-dimensional free modules over the
ring C°°(U) whenever U is a chart domain or, more generally, the domain of
a frame field. The reason is that a local frame field and its dual frame field
provide a module basis for X(U) and 3£*(U) and the latter really is the dual
of the first in the module sense. On the other hand, the C°°(M)-modules
X(M) and Trs(X(M)) are not generally free unless Μ is parallelizable.
7.4. Matching the Two Approaches to Tensor Fields
If we define a tensor field as we first did, that is, as a field of tensors in
tangent spaces, then we immediately obtain a tensor as defined in the top-
down approach. On the other hand, if r is initially defined as a C°°(M)-
multilinear map, then how should we recover a field of tensors on the tangent
spaces?2 Answering this is our next goal.
2This is exactly where things might not go so well if the manifold is not finite-dimensional.
What we need is the existence of smooth cut-off functions. Some Banach manifolds support cut-off
functions but not all do.
7.4. Matching the Two Approaches to Tensor Fields
325
Proposition 7,32. Let ρ G Μ and τ e Trs(X(M)). Let 0b...,0r and
01,..., 0r be smooth 1-forms such that θ{{ρ) = θχ{ρ) for 1 < г < г; also let
X\}.«., Xs and X\,..., Xs be smooth vector fields such that Хг(р) = Хг(р)
for 1 < г < s. Then we have that
г(0ь...,0г,Х1?...,Х5)(р)=г(0ь...,0г,Хь...,Хв)(р).
Proof, The result will follow easily if we can show that
т(ви...,вг,Хи...,Ха)(р)=0
whenever one of 0i(p),..., 0r(p), Χχ(ρ), · - -, Xs(p) is zero. We shall assume
for simplicity of notation that r = 1 and s = 2. Now suppose that Xi(p) = 0.
If (i7,x), with χ = (ж1,... ,яп), is a chart with ρ G J7, then -Xil^· = S^S?
for some smooth functions £l Ε C°°(U). Let /3 be a cut-off function with
support in U and /3(p) = 1. Then for any smooth vector field X defined
on U we can consider both βΧ and β2Χ to be globally defined and zero
outside of U. Similarly, if / is a smooth function defined on U, then /3/
can be taken to be globally defined on Μ and zero outside of U. Now
β2Χι = Σ {PC) {β^)' (Notice that in this last expression we have used β
to extend both the functions ξ1 and the coordinate fields ^, which is why
we used β2 rather than just β.) Thus
β2τ(θ1,Χ1,Χ2)=τ(θ1,β2Χ1,Χ2)
(Notice that the point of the above expression is that, at this moment, τ
is defined only for global sections and is linear over global functions. For
example, A is not a global section while β-^χ is a global section.) Since
Xi(p) ~~ 0j we must have ξι(ρ) = 0 for all г. Also recall that β(ρ) = 1.
Plugging ρ into the formula above we obtain
т(виХиХа)(р)=0.
A similar argument holds when X2(p) = 0 or θι(ρ) = 0.
Assume that θχ{ρ) - θχ(ρ), Χχ{ρ) = Χι(ρ) and X2(p) = X2(p). Then
we have
τ(Θ1,Χ1,Χ2)-τ(Θ1,Χ1,Χ2)
= τ(θ1-θ1,Χι,Χ2)+τ(θ1,Χ1-Χ1,Χ2)+τ{θ1,Χ1,Χ2-Χ2).
326
7. Tensors
Since 0i — 0i, ΛΊ — -ΧΊ, and X2 — X2 axe all zero at p, we obtain the result
that т(въХиХ2)(р) = τ{θι,Χι,Χ2)(ρ). □
Thus we have a natural correspondence between Trs(X(M)) and ТЦМ)
(the latter being smooth sections of the bundle Trs(TM) -¥ M). For
example, if A E 7~з(М), then we obtain an element of T\(X(M)), also
denoted by A, by defining a smooth function Α(Θ,Χ, У, Ζ) for given fields
(9,X,Y,Z)by
Α(θ, Χ, Υ, Z)(jp) := Α(ρ)(θ(ρ),Χ(ρ), Υ(ρ), Ζ (ρ)).
Conversely, if A 6 Т\(ЩМ)), then we can use the above proposition to
define an element of T3 (M), which we denote by the same letter. Given A e
T\{?t(M)), define A(p) e T\{TpM) for each ρ as follows: For XP,YP,ZP £
TPM and θρ G T*M we let
Α(ρ)(θρ,Χρ,Υρ,Ζρ) :=Α(θ,Χ,Υ,Ζ)(ρ),
where 0, X, У, Ζ are any fields chosen so that 0(p) = 0Р, X(p) = ^p, V(p)
1^,, and Z(p) = Zp. By Lemma 6.37 we can always find such extensions, and
by Proposition 7.32 above, A is well-defined. That A so defined is smooth
follows from Proposition 7.22. The general case should be clear and, all said,
we end up with a natural isomorphism of C°°(M)-modules:
T75(X(M))^7T(M).
Similar reasoning shows that there is a correspondence between fields of
TM-valued tensors and X(M)-valued tensors on X(M). For example, we
have
(7.6) Г°2(£(М);£(М)) ~ 7^(М;ГМ).
Elements of T°2(X(M); X(M)) are C°°(M)-bilineax maps X{M) χ Χ Μ
-ϊ X(M), while elements of T2(M;TM) are sections of the bundle whose
fiber at ρ is Γ°2(ΓρΜ;ΓρΜ). So, if A e 7^(M;TM), then for each p, A{p)
is an Ж-multilinear map TPM χ TPM -» TpM. Similarly, we have
(7.7) T%(X{M);X(M)) * Гз(М;ГМ).
In fact, later, when we define the curvature tensor on a semi-Riemannian
manifold, it will initially be given as a multilinear map on modules of fields
with values in X(M). The correspondence is then invoked to get a tenser
field (as defined in the bottom-up approach) with values in TM. It is just as
easy to give a similar correspondence between ΤΓ3(Γ(ξ)) and Τ(Ττ8(ξ)) for
some vector bundle ξ = (Ε, π, Μ), where we view Γ (ξ) as a C°°(M) module.
Exercise 7.33. Exhibit the isomorphism (7.7) in detail.
7.5. Tensor Derivations
327
Exercise 7.34. Suppose that S and Τ are tensors of the same type and
we wish to show that they are equal. Then it is enough to check equality
under the assumption that the vector fields inserted into the slots of S and
Τ are locally defined and have vanishing Lie brackets. Hint: Think about
coordinate vector fields.
We end this section with some warnings. It may seem that there is a
simple way to obtain a pull-back by a smooth map / : Μ —► N entirely from
the top-down or module-theoretic view. In fact, one often sees expressions
like
/*r (-ΧΊ,..., X8) = r(/*Xi,..., f*X8) (problematic expression!).
This looks cute, but invites misunderstanding. The left hand side takes fields
ΛΊ,... ,X3 as arguments, while on the right hand side, if we consider r as
a multilinear map X(N) χ ... χ X(N) -* C°°(N), then f*Xx must be fields.
But the push-forward map /* is generally not defined on fields, and even
if it were, the above expression would seem to be an equality of a function
on Μ with a function on N. Note that X(M) is a C°°(M)-module, while
X(N) is a C°°(N)~mod\ile. The above expression may be taken to mean
something like f*r(Xu ..., Xs){p) = r(Tf ·Хг(р),...,Tf · X5(p)), but now
the right hand side has tangent vectors as arguments, and we are back to
the bottom-up approach! A correct statement is the following:
Proposition 7.35. Let f : Μ —» N be α smooth map. Let r be a (0,5)-
tensor field. If τ and /*r are interpreted as elements of T°S(X(N)) and
T°S(X(M)) respectively7 then
f*T(X1,...,Xa) = T(Yu...,Ys)of
whenever Y% is f-related to Хг for г = 1,..., s.
Of course, we can use Definition 7.31 to make sense of both push-forward
and pull-back in the case that / is a diffeomorphism.
7.5. Tensor Derivations
We would like to be able to differentiate tensor fields. In particular, we
would like to extend the Lie derivative to tensor fields. For this purpose we
introduce the following definition, which will be useful not only for extending
the Lie derivative, but also in several other contexts. Recall the presheaf of
tensor fields U »->· Τζ(ΙΙ) on a manifold M.
Definition 7.36. A tensor derivation is a collection of maps Vrs jj :
Tl(U) —> 7J(I7), all denoted by V for convenience, such that
328
7. Tensors
(1) V is a presheaf map for 7J considered as a presheaf of vector spaces
over R. In particular, for all open U and V with V С U we have
X>A|V=X>(4V)
for all A 6 7^(C/), i.e., the restriction of VA to V is just Ρ (A\v).
(2) Ρ commutes with contractions.
(3) V satisfies a derivation law. Specifically, for A G Τ£(Ϊ7) and В 6
Tjk(U) we have
X>(A ®B) = VA®B + A® VB.
For smooth η-manifolds, the conditions (2) and (3) imply that for A 6
TS{U), αϊ,.... or € £*(E7) and Xb ..., Xa € £(17), we have
DtA^!,...,^,*!,...,^))
= (DA)(ai,...,ar,Xi,...,Xs)
r
(7.8) + 53i4(ai' · · ·'Pai' ■ ■ ·, <*r, Xb ..., Xs)
i=l
+ 53^(аь..., αΓ> ΛΊι ■ · ·, X>Ai,.. ·, ^s).
2=1
This follows by noticing that
A(ax, ...,аг,Хи...,Х3) = С {А (g> (αϊ ® · · · (8) ar ® X\ ® ■ · · <g> Xe))
(where С is the repeated contraction) and then applying (2) and (3).
Note that V stands for a family of maps whose domains T^{U) depend
not only on r and 5, but also on [/. The next proposition considers the
situation where we only have derivations defined for U = Μ (the global
case).
Proposition 7.37. Let Μ be a smooth manifold and suppose we have a map
on globally defined tensor fields V : 7Ζ{Μ) —> T£{M) for all nonnegative
integers r, s such that (2) and (3) above hold for the case U = M. Then there
is a unique induced tensor derivation that agrees with V on global sections,
that is, on the various Τζ(Μ).
Proof. We need to define V : TJ(I7) -* 7^(17) for arbitrary open U as a
derivation. Let δ be a function in C°°(U) that vanishes on a neighborhood
VofpeU. We claim that (VS)(p) = 0. To see this, let β be a cut-off
function equal to 1 on a neighborhood of ρ and zero outside of V. Then
δ — (1 — β) δ and so
νδ(ρ)=ν((1-β)δ)(ρ)
= δ(ρ)Ό(1 - β)(ρ) + (1 - β(ρ))νδ(ρ) = 0.
7.5. Tensor Derivations
329
Now given τ € 7^(C7), let β be a cut-off function with support in U and
equal to 1 on a neighborhood of ρ € U. Then βτ £ TJ(Af) after extending
by zero. Define (Vr){p) := Ί)(βτ)(ρ). To show that this is well-defined
let /?2 be any other cut-off function with support in U and equal to 1 on a
neighborhood of ρ Ε U. Then we have
ν(βτ)(ρ)-ν(β2τ)(ρ)
= (Ό(βτ) - V(JhT))(p) = Τ>((β - Α)τ)(ρ) = О,
where the last equality follows from our claim above with δ = β — 02. Thus
V is well-defined on T%(U). We now show that Vr so defined is an element
of Tr8{U). Let (J7',x) be a chart with ρ G Uf С U. Then we can write
τ\υ, e TS{V) as
τυ< = ri^-'fdx11 ® ■ ■ · ® dxia Θ —-τ- ® ■ · ■ ® —;-.
We can use this to show that Vr as defined agrees with a global section in a
neighborhood О of ρ and so must be a smooth section itself since the choice
of ρ € U was arbitrary. To save on notation, let us take the case r = 1,
s = 1. Then тца = Tjdxi ® ^-. Let β be a cut-off function equal to 1 in the
neighborhood О of ρ and zero outside of Ul. Extend each of the sections /3rj,
βάχ?> and β-^ to global sections and apply V to β3τ = (β-ή) {βάχ^)®{β^)
to get
ν{βζτ)=ν{βτ}βά3Ρ®β-^)
= ν(βτί)βάχ3 ®β-^ + fhpifitot) ® β§-
+ βήβώ*ΐ9ν(β±).
By assumption, V takes smooth global sections to smooth global sections,
so both sides of the above equation are smooth. On the other hand, we have
Ί){β3τ){ς) = V(r){q) by definition and valid for all q € O. Thus V{r) is
smooth and is the restriction of a smooth global section. This gives a unique
derivation V : ТЦи) -> ТЦР) for all U satisfying the naturality conditions
(1), (2) and (3). We leave it to the reader to check this last statement. D
Exercise 7.38. Let V\ and T>2 be two tensor derivations (so satisfying
conditions (1), (2) and 3 of Definition 7.36) that agree on functions and
vector fields. Then Vx = V2. [Hint: If α Ε £*(!/) = 7?(17), we must have
(Да) (X) = Vi {<*{X)) - a{ViX) for г = 1,2. Then both Vx and V2 must
obey formula (7.8) above.]
330
7. Tensors
Theorem 7.39. IfV\j can be defined on C°°(U) and X(U) for each open
U С М so that
(1) Vu(fg) = (Vuf) g + fVu9 for all f,ge C°°(U),
(2) {VMf)\u = VV f\v for each f Ε С00(Μ),
(3) Vu(fX) = {Vvf) X + fVxjX for all f G C°°{U) and X e £([/),
(4) {VMX)\U = VUX\U for each XeX{M),
then there is a unique tensor derivation V that is equal to Vjj on C°°(U)
and X(U) for all U.
Sketch of proof. We wish to define V on X*(i7) so that
(7.9) Vu{a X)-Vua®X + a VVX.
By contraction we see that we must have {T>uot){X) = Vjj{a{X)) -a(V{jX)>
which we take as the definition. Then check that (7.9) holds. Now define
V\j by formula (7.8) and verify that we really have a map TJ(U) —> Tl{U .
Check that Т>ц commutes with contraction С : Tl(U) -¥ C°°(U) for simple
tensors a0lG T\ (U). Use the fact that, locally, every element of T\ can
be written as a sum of simple tensors. Next extend to 7J along the lines
exemplified by the case of T\ (U) and the contraction C\ as follows: For
τ Ξ T\ (C/), we have
{VuClr) (X) = Vu {{C\t) (X)) - φίτ) VVX
= 2^(σ(τ(-,χ,-)))-σ(τ(.,2?^,.))
= CVu(T(;Xr)-T(^VuX,·))
= C{{Vut){^X,.))={C\Vut){X).
The general case would involve an inconvenient profusion of parentheses.
Uniqueness follows from Exercise 7.38. Finally check by direct calculation
that (3) of Definition 7.36 holds.
Corollary 7.40. The Lie derivative Cx can be extended to a tensor denva-
tion for any X G X(M).
The last corollary extends the Lie derivative to tensor fields. It follows
from formula (7.8) that we have
(CxS)(Yu...,Ys) X(S(ru...,Ye))
(7'10) -^5(Уь...,Уг-1,£хУг5Уг+1,...5П).
ι=1
We now present a different way of extending the Lie derivative to tensor
fields that is equivalent to what we have just done. First let A € Τζ{Μ
7.6. Metric Tensors
331
and recall that if / : Μ —> Μ is a diffeomorphism, then we can define
Г А 6 TS(M) by
(f*A)(p)(a1,...,ar,v1,...,vs)
= A(f(p)) ({ΤρΓΎ (α1), · ■ · > (V"1)* (αΓ), ΤΡ/(«ι),... ,Гр/ы)
for all α1,..., ar e (TPM)* and vi,..., t>s e TPM. If X is a vector field on
Μ (possibly locally defined), we can define
(7.11) {CxA){p)=j^ ((^)*л)(р),
just as we did for vector fields. We leave it as a project for the reader to
show that this definition agrees with our first definition of the Lie derivative
of a tensor field.
The Lie derivative on tensor fields is natural with respect to diffeomor-
phisrns in the sense that for any diffeomorphism f : Μ —ϊ Ν and any vector
field X we have
(7.12) Cfmxf*A = UCXA.
This property is not shared by some other important derivations such as the
covariant derivative, which we define later in this book.
Exercise 7.41. Show that the Lie derivative on tensor fields is natural with
respect to diffeomorphisms in the above sense of equation (7.12) by using
the fact that it is natural on functions and vector fields.
7.6. Metric Tensors
We start out again considering some linear algebra that we wish to globalize.
Thus the vector space V that we discuss next should be thought of as a
tangent space of a manifold or a fiber of some vector bundle.
We recall the following definitions: A symmetric bilinear form g on a
finite-dimensional vector space V is nondegenerate if and only if g(v, w) =
0 for all w Ε V implies that ν = 0. A (real) scalar product on a (real)
finite-dimensional vector space V is a nondegenerate symmetric bilinear form
g : V χ V —> R. A scalar product space is a pair (V,#) where V is a
vector space and ρ is a scalar product. We say that g is positive (resp.
negative) definite if g(v,v) > 0 (resp. g(v,v) < 0) for all ν Ε V and
g(v, v) = 0 => ν = 0. In case the scalar product is positive definite, we also
refer to it as an inner product and the pair (V,^) as an inner product
space. Otherwise we say that the scalar product is indefinite. A scalar
product on V is sometimes called a metric tensor on V. We now need to
introduce quite a few more definitions.
332
7. Tensors
Definition 7.42. The index of a symmetric bilinear form g on V is the
dimension of the largest subspace W С V such that the restriction g\w is
negative definite. The index is denoted ind(<?).
Definition 7.43. Let (V,p) be a scalar product space. We say that ν and
w are mutually orthogonal if and only if g(v, w) = 0. Furthermore, given
two subspaces Wi and W2 of V we say that Wi is orthogonal to W2 and
write Wi J- W2 if and only if every element of Wi is orthogonal to every
element of W2.
Since, in general, g is not necessarily positive definite or negative definite,
there may be nonzero elements that are orthogonal to themselves.
Definition 7.44. Given a subspace W of a scalar product space V, we
define the orthogonal complement as W1- = {v G V : g(v, w) = 0 for all
weW}.
Exercise 7.45. We always have dim(W) + dim(W_L) = dim(V), but unless
g is definite, we may not have WnW1-^}.
Definition 7.46. A subspace W of a scalar product space (V,g) is called
nondegenerate if #|w is nondegenerate.
Lemma 7.47. A subspace W С (V,p) is nondegenerate if and only ifV =
W Θ W1- (inner direct sum).
Proof. This an easy exercise. One uses the standard fact that
dim W + dim W^ = dim(W + W^) + dim(W П Wx). D
It is a standard fact from linear algebra, already mentioned in Chapter 5,
that if g is a scalar product, then there exists a basis βχ,..., en for V such
that the matrix representative of g with respect to this basis is a diagonal
matrix with ones or minus ones along the diagonal. Such a basis is called
an orthonormal basis for (V, g). The number of minus ones appearing is the
index ind(g) and so is independent of the orthonormal basis chosen. It is
easy to see that the index ind(g) is zero if and only if g is positive definite.
Definition 7.48- For each г; Ε V with {ν, ν) φ 0, let e(v) := sgn{v,v).
Then if β 1,..., e,ji are orthonormal, we have ez = e(i) := €(e;).
Thus if ei,..., en is an orthonormal basis for (V, 3), then g{e^ e3) — t%StJ)
where ег = д(еиег) = ±1 are the entries of the diagonal matrix ind(g) of
which are equal to —1 and the remaining are equal to 1. Let us refer to the
list of ±l's given by (ei,..., en) as the signature. We may arrange for the
—l's to come first by permuting the elements of the basis. For example, if
(-1, —1,1,1) is the signature, then the index is 2.
7.6. Metric Tensors
333
Remark 7.49. From now on, whenever context allows, we shall always
assume that by "orthonormal basis" we mean an orthonormal basis that is
arranged so that the —l's come first as described above. The convention of
putting the minus signs first is not universal, and in fact we used the opposite
convention in Chapter 5. The negative signs first convention is popular in
relativity theory and semi-Riemannian geometry, but the reverse convention
is perhaps more common in Lie group theory and quantum field theory. It
makes no difference in the final analysis as long as one is consistent, but it
can be confusing when comparing references in the literature.
Another difference between the theory of positive definite scalar products
and indefinite scalar products is the appearance of the e;'s from the signature
in formulas that would be familiar in the positive definite case. For example,
we have the following:
Proposition 7.50. Let ei,...,en be an orthonormal basis for (V,p). For
any ν Ε V7 we have a unique expansion given by ν = J2i ei(v, е*)е*.
Proof. The usual proof works. One just has to notice the appearance of
the ej's. D
Definition 7.51. If ν Ε V, then let ||v|| denote the nonnegative number
|(/(г>,г>)| ' and call this the (absolute or positive) length or norm of v.
Some authors call g(v,v) or giv^v)1^2 the norm, which would make it
possible for the norm to be negative or even complex-valued. We will avoid
this.
Just as for positive definite inner product spaces, we call a linear
isomorphism Φ : (Vi,5i) —> (V2,g2) from one scalar product space to another
an isometry if g\(v, w) = 52(Φ^, Фги). It is not hard to show that if such
an isometry exists, then g\ and #2 have the same index and signature.
Let (Vf, ft) be scalar product spaces for i = 1,..., k. By Corollary D.35
of Appendix D, there is a unique bilinear form
φ : <g)LiVi x ®tiVi -> К
such that for Vi Ε V* and W{ Ε Wi,
φ Οι (g) · · · ® vk, wi (8) ■ · · (8) wk) = gi{vi, w\) · · ■ 9k(vhj u>k)·
The form φ is clearly symmetric and by Problem 3, it is a scalar product
on ®i=1 V* that is positive definite if each gi is positive definite. If (V, g) is
a scalar product space, then we use the above to endow each of the various
tensor spaces Trs(V) with a scalar product. First consider V*. Since g is
nondegenerate, there is a linear isomorphism g\> : V —> V* defined by
9b{y){w) =g(v,w).
334
7. Tensors
Denote the inverse by g$ : V* —> V. We force this to be an isometry by
defining the scalar product on V* to be
9*(α,β) = 9{^(α),^(β))·
Under this prescription, the dual basis (e1,..., en) corresponding to an or-
thonormal basis (ei,..., en) for V will also be orthonormal. The signatures
(and hence the indexes) of g* and g are the same.
Notation 7.52. When convenient, we shall also denote g\>(v) by either к
or t>b and similarly for 5"(a).
The above procedure now applies to give a scalar product on any tensor
space Trs(V). For example, consider T\(V) = V®V*. Then there is a unique
scalar product g\ on V V* such that for v\ ol\ and V2 #2 £ V (g) V* we
have
9i(vi ® «ι, Щ ® a2) = g{v\, ν2)ρ*(αι, α2).
One can then see that for orthonormal ei,..., en we have that
{ei e>h
is an orthonormal basis for (T\(V),g{). In general, if we endow Trs V)
with a scalar product as above, then the natural basis for Trs(V) formed
from the orthonormal basis (ei,..., en) (and its dual (e1,..., en)) will also
be orthonormal.
Notation 7·53. In order to reduce notational clutter, let us reserve the
option to denote all these scalar products coming from g by the same letter
g or, even more conveniently, by (·, ·). So, for example, {v\ «1,^2 0*2)
(^1,^2)(αϊ,#2) by definition.
Exercise 7.54. Show that under the natural identification of V ® V* with
L(V, V), the scalar product of linear transformations A and В is given by
(A,B) -trace(i4*B).
The maps g\, and g$ are called musical isomorphisms. Let us see how
things look in terms of components. Let /1,..., fn be an arbitrary basis of
V and let /*,..., fn be the dual basis for V*. The components of g are
given by gij :— g(f%, f3). So if ν ν ft and w = wlfly then
g{y,w) =g(v%fuw3f3) v%vPg(fuf3) =gJvtwj,
where we continue to use the Einstein summation convention. There must
be a matrix (A%j) such that \>f% — Afafk. On the other hand,
<ь,=0(Л,Л)-(ЬЛ)(/,) = 4ь/*(/,)
= Afadj = Aj{.
7.6. Metric Tensors
335
So we have
b/» = &*/*■
Thus if г; — г^/г, then Ьг; = v3bf3 = vig^f1, so the components of bv
are (bv)t — v*9ji — 9ijV·*· It is a common convention that if vl are the
components of ν with respect to a basis, then the components of bv are
denoted simply by lowering the index:
Vi g%dvK
The map g\> is called the flatting operator and the effect of this operator
is sometimes described as "index lowering". If we write Ц/г — дгзfj for
some matrix {g%3), then (gtj)~ (дгз) so that
Λ* - ή.
This follows from f* = Цр = gtk\>fk gikgkjfj. If ω € V* is written as
ω = utfl, then an easy calculation shows that the components of (Jo; are
given by
ω* :- (Мг А*·
The isomorphism 3" is called the sharping operator and its effect is
referred to as "index raising". The scalar product 3* on V* introduced above
has components дг° with respect to the dual basis. Indeed,
9*(f\fJ)=9ttf\m-9(9lkfk,93lfl)
= 9ikS>l9kl = 9jl5\ д>* 9ij.
Next we see how to extend the notions of index raising and lowering to
tensors. Suppose we have a tensor A € T22(V) and we wish to obtain a new
related tensor Af whose final slot takes elements of V* rather than elements
of V. Thus we want A! be of type (2 γ l). The trick is to define A! using )J
as follows:
Α!(ω,η,ν,α) := Α(ω,η,ν,$α).
Let us compute the components of A! with respect to our basis. We have
{A'f k' A'(f\ Ρ, Λ, f) = A(f\ f>, Л,fl/«)
= A{f\P,h,gbfkr)=9trAl\r.
It is common to write AlJ k in place of (А')гз к when the context makes the
meaning clear. In other words, we use the same root symbol but reposition
the indices. This is an instance of index raising. Similarly we might use the
flatting operation to obtain a new tensor from A. For ω G V* and u, v, w € V
we could define A! by
Af(u,uj,v,w) :— A(bu,uj,v,w)
336
7. Tensors
and the components of Af would be given by
\A)% ki:=girJ^ k£>
and again it is common to see simply A% 3 ы. This process of raising and
lowering indices is called type changing. Notice that we often obtain tensors
that are not in consolidated form. However, one may simply apply
consolidation as desired. But, notice that the staggering of position makes the
relation of Ai J ki to the original tensor Αυ ki clear. This is where positional
index notation excels.
The above can be approached in a slightly different way. We can take
tensor products of various combinations of #[,, φ and the identity maps idv
and idy*. For example,
& ® idv ®9\> 5* ® idv*: V ® V ® V ® V* V* -» V* V ® V* ® V ® V*.
Depending on convenience, this map might then be followed by the
consolidation isomorphism
V* ® V ® V* ® V ® V* -> V* ® V* ® V* ® V ® V.
Exercise 7.55. Show that the map g\, ® idv 9b $ ® idv* effects three
iterated type changes and is given in component form by
A im *~* ^i к т : = 9ia9kb9 A cm.
In the presence of a scalar product, a type-changed tensor is considered
to be just a different manifestation of the original tensor. We say that it is
metrically equivalent. The reader should expect to see some slight variability
with regard to how index positioning and order of slots is handled when type
changing is done. For example, there is nothing stopping us from raising
the α-th lower index into the 6-th upper position while keeping everything
in consolidated form:
V71 ) Jl-Je lJa+l».J· '~ Л 31-За im3a+1^3s9 ·
Invariantly, this is described as
:= ^(а1,..., α6,..., αρ+1, vu ..., να ι, ^(α6), να+ι,..., ν8).
Notice that if we raise all the lower indices and lower all the upper indices
on a tensor, then we can "completely contract" against another tensor of
the original type. We leave it to the reader to show that the result is the
scalar product of tensors defined earlier. For example, let χ = Σ Xijf1 f
and τ = ΣτυΡ ® Я- We may apply two type changes to τ that are given
7.6. Metric Tensors
337
in component form as тц ι-> τ*. ι-> ru. In other words, rZJ = g%kg^r\a* Then
we have
See Problem 18.
7.6.1. Metrics on manifolds. If g Ε 75 (Μ) is nondegenerate,
symmetric and positive definite at every tangent space, we call g a Riemannian
metric (tensor). If g is a Riemannian metric, then we call the pair (M, g)
a Riemannian manifold. For example, in Chapter 4, we saw how a hy~
persurface in Rn inherits a Riemannian metric. This works just as well for
a regular submanifold Μ С W1 of arbitrary codimension.
In Riemannian geometry, it is the metric that is the basis for
generalizations of length, volume and so on. Motivated by a desire to generalize and
to include the mathematics needed for general relativity, we also allow the
metric to be indefinite. In this case, some nonzero tangent vectors ν might
have zero or negative self-scalar product (υ, υ). If ρ € 7г(Л/) is a symmetric
tensor field, then we say that it is nondegenerate if gp is nondegenerate on
TPM for every p. If furthermore gp has the same index for all p, then we say
it has constant index.
Definition 7.56. If g Ε ТЦМ) is symmetric nondegenerate and has
constant index on M, then we call g a semi-Riemannian metric and (M,g)
a semi-Riemannian manifold or pseudo-Riemannian manifold. The
index is called the index of (M, g) and denoted ind(<j) or ind(M). The
signature is also constant and so the manifold has a signature also. If the signature
of a semi-Riemannian manifold (with dim(M) > 2) is (—1, +1, +1, +1,...)
(or according to some conventions (1,-1,-1,-1,...)), then the manifold is
called a Lorentz manifold.
The simplest semi-Riemannian manifolds are the spaces RJJ, which are
the spaces Rn endowed with the scalar products given by
ν η
г 1 г ι/+1
Since ordinary Euclidean geometry does not use indefinite scalar products,
we shall call the spaces RJJ semi-Euclidean spaces when the index ν =
ind(g) is not zero. If we write just Rn, then either we are not concerned
with a scalar product at all, or the scalar product is assumed to be the usual
inner product [y = 0). Thus a Riemannian metric is just the special case of
index 0. The space Rf is called the Minkowski space.
We will usually write (XP,YP) or g(Xp,Yp) in place of g(p)(Xp,Xp). Also,
for a pair of vector fields X and Υ, we define the function (X, Y) which is
given by (X, Y)[p) — {Xp, Yp). In local coordinates (я1,..., xn) on U С М,
338
7. Tensors
we have that g\v = gl3dxx <8> dx^, where gtJ = {-^j, ^-). Thus if X X g|*
and Υ = Y1^ on U, then
(7.13) (X,Y)=9ijXlY\
which is a smooth function defined on U. The expression (X, Y) — g%jXlY
means that for all ρ G U we have (X(p),Y(p)) g%3(p)Xl(jp)Yl{p). As we
know, the functions Xх and Yl are given by X1 dxl{X) and Y% — dxl(Y).
On a semi Riemannian manifold, the musical isomorphisms are
globalized in the obvious way to act on tensor fields. We simply apply the type
change at each point in the domain of a given tensor field. For example,
if A is a tensor field of type (2,2), then we may obtain a new metrically
equivalent tensor field Af of type (1,3) by the rule that for any ρ e M, we
have A'(p)(a,u, v,w) := A(p)(a,\>u,v,w) for α e Τ*Μ and u,v,w Ε ΤρΜ.
If we choose a chart (Ϊ7,χ), then in terms of the coordinate frames and the
corresponding g%3, we have
(7.14) А' = Агзк1-Ц-г dx* dxk dxl,
where
Of course, each of the possible conventions for consolidation and index
position globalize accordingly. The reader is invited to compare our treatment
with those found in [Pe], [Dod-Pos], [ONI] and [Stern].
We have been using coordinate frame fields, but there is nothing
preventing us from giving local components of tensor fields with respect to arbitrary
smooth frame fields. For example, if we choose a frame field {E\,..., En
and the corresponding dual frame field, then we may define дц := {EUE3
with the corresponding дгз, and then local expressions analogous to those
above hold with the ^»'s and dxrs replaced by the EiS and E3\ where
the components of the tensor are obtained by evaluating on these frame
fields. In particular, if the frame field is an orthonormal frame field, then
g%3 — ±1 for г = j and g%3 0 for г ф j. This can result in a good deal of
simplification.
We now say a few words about the appropriate notion of equivalence of
semi-Riemannian manifolds.
Definition 7.57. Let (M, g) and (iV, h) be two semi-Riemannian manifolds.
A diffeomorphism Φ : Μ —> N is called an isometry if Φ*/ι — д. Thus for an
isometry Φ : Μ —► N we have g(v, w) = /ι(ΤΦ · υ, ΤΦ * w) for all v, w £ TM.
If Φ : Μ —> iV is a local diffeomorphism such that Φ*Λ, #, then Φ is called
a local isometry. If there is an isometry Φ : Μ —у TV, then we say that
(M, g) and (TV, h) are isometric.
7.6. Metric Tensors
339
Definition 7.58. The set of all isometries of a semi-Riemannian manifold
Μ to itself is a group called the isometry group. It is denoted by Isom(M).
The isometry group of a generic manifold is most likely trivial, but
examples of manifolds with relatively large isometry groups are easy to find
using Lie group theory. Also, Myers and Steenrod showed that the
isometry group of a compact Riemannian manifold is a Lie group (see [My-St]).
Recall from Chapter 5 that associated to RJJ we have the matrix groups
0(y,n — v) and SO(^,η — и). The isometry group of RJJ is given by
Iso(z/, η — и) = {L : L(x) = Qx + x$
for some Q G 0(ι/, η — и) and xq € R™}.
This is the group of semi-Euclidean motions.
Example 7.59. We have seen that a regular submanifold of a EucUdean
space Rn is a Riemannian manifold with the metric inherited from Rn. In
particular, the sphere Sn l С Rn is a Riemannian manifold. Every isometry
of £η_1 is the restriction to Sn г of an isometry of Rn that fixes the origin.
Definition 7.60. Let Μ and Μ be semi-Riemannian manifolds. lip: Μ -^
Μ is a covering map such that ρ is a local isometry, we call p:M-)Ma
semi-Riemannian covering.
If we have a local isometry φ : N —^ M, then any lift φ : N —> Μ is also
a local isometry (Problem 16). Deck transformations are lifts of the identity
map Μ —► Μ, and so are diffeomorphisms which are local isometries. Thus
deck transformations are, in fact, isometries. We conclude that the group of
deck transformations of a semi-Riemannian cover is a subgroup of the group
of isometries Isom(M).
Let us consider here the case of a discrete group G and a discrete group
action λ : G χ Μ —У М that is smooth, proper, and free. We have already
seen that the quotient space M/G has a unique structure as a smooth
manifold such that the projection « : Μ —> M/G is a covering. Let us now assume
that G acts by isometries so that λ*(·, ·) = (·, ·) for all g 6 G. The tangent
map Τ к : TpM -+ T^p){M/G) is onto. For χ Ε M/G, let vuv2e TX(M/G).
Define hx(vuV2) = (vi?^), where v\ and v% are chosen at the same point
and such that Τκ-Vi = щ. We wish to show that this is well-defined. Indeed,
if Vi € TpM and W{ G TqM are such that Трк · Vi = TqK · Wi = щ for г = 1,2,
then there is an isometry Xg with Xgp = q. Furthermore, since Xg is a deck
transformation and curves representing Vi and Wi must be related by this
deck transformation, we also have Tp\g · Vi — wi* Thus
(V1,U2> = (Tp\gVi,TpXgV2) = <Wi,U72),
340
7. Tensors
which means hx is well-defined. It is easy to show that χ ι-» hx is smooth
and defines a metric on M/G with the same signature as that of {·, ·) and
that further, K*h ={·>·)· In ^ас*> we wiU use the same notation for either
the metric on M/G or on M.
Definition 7.61. A lattice of rank к in Rn is a set of the form
Г := {χ e Rn : x = ΣΧΛ where n* e Ζ} ,
where /i,..., Д are linearly independent elements of Rn. The Д,..., Д are
called the generators of the lattice.
The lattice Zn С Rn is the standard rank η lattice, and it is generated
by the standard basis. A lattice is a subgroup of Rn and so acts on W1 by
a \-¥ a + ν for ν Ε Γ. This is a discrete, free and proper action, and so the
quotient Шп/Г provides a simple example of the above construction and so
has a metric induced from Rn. If the lattice has full rank n, then Κη/Γ
is called a flat torus (or flat η-torus) and is diffeomorphic to the product
of η copies of the circle 51. Each of these n-dimensional flat tori is locally
isometric, but may not be globally isometric. To be more precise, suppose
that Д, Д,..., fn is a basis for Rn which is not necessarily orthonormal. Let
Γ/ be the lattice consisting of integer linear combinations of Д, Д,..., Д.
Now suppose we have two such lattices Γ/ and Γ^. When is Rn/Tf isometric
to Rn/ry? It may seem that, since these are clearly diffeomorphic and since
they are locally isometric, they must be (globally) isometric. But this is
not the case (see Problem 17). The study of the global geometry of flat
tori is quite interesting and even has deep connections with fields outside of
geometry such as arithmetic, which we shall not have the space to pursue.
We know from Chapter 4 that there are surfaces in R3 that are
diffeomorphic to a torus Sl xS1. Such surfaces inherit a metric from the ambient
space, but it turns out that the Riemannian surface obtained in this way
cannot be isometric to one of the flat 2-tori introduced here. In Chapter 13
we will see how each metric on a manifold gives rise to an associated
curvature tensor. The reason the tori just introduced are referred to as flat is
because (being locally isometric to some Rn) they have vanishing curvature
tensor.
If we have semi-Riemannian manifolds (M,g) and (N:h), then we can
consider the product manifold MxN and the projections prx : MxN -»Μ
and pr2 : Μ χ Ν —> N. The tensor g χ h = prjg + prjn, provides a semi-
Riemannian metric on the manifold MxN, which is then called the semi-
Riemannian product of (М,д) and (Nyh). Let (Ε/Ί,χ) and([/2,y) denote
charts on Μ and N respectively. Then we may form a product chart for
MxN defined on U\ x C/2. The coordinate functions of this chart are given
by x1 — x% о pr! and у* = у1 о pr2. We have the associated frame fields 4·*
7.6. Metric Tensors
341
and ^7. The components of g χ h = pr|p + prijh in these coordinates are
discovered by choosing a point (px,P2) € U\ χ U2 and then calculating. We
have
j д I a I >
Pfi5
I(pi«)
i-
l(p>«)
(ρ,9)'^
a
a
(p.e)
)+ρΓ^(^ >Ш )
' Kc,x I(p,9) °у \ш'
д
дуз
д
д
= 9(Трчж\ 'T?Vldtf\ )+ЧГрт2Ы 'ГрГ2Ы )
а
=аШ^)+н^м\)
дх
= 0 + 0 = 0,
and (abbreviating a bit)
д I д
ХОХ Ι(Λί) UXJ\{p,qY ΚΟΧ Ιρ ΟΧ V
Similarly g χ h(-jf=-, ^j)(p, g) = hlj(q). In practice, the coordinate functions
constructed above are often abusively denoted by (я1,..., жП1, у1,..., у™2)
and the frame field,s by g|r,..., ^^, ^r > · · -} g^j ■ So with respect to these
coordinates, the matrix of g x h is of the form
(?я)
where G = (gij о prx) and Η = (/i^· о pr2).
Notation 7.62. The product metric is often denoted by g + h or in
coordinates by ds2 = gijdx%dx3 + hkidykdyK
Every smooth manifold that admits partitions of unity also admits at
least one (in fact infinitely many) Riemannian metric. This includes all
(finite-dimensional) paracompact manifolds. The reason for this is that the
set of all Riemannian metric tensors is, in an appropriate sense, convex. We
record this as a proposition.
Proposition 7.63. Every smooth (paracompact) manifold admits a
Riemannian metric.
Proof. This is a special case of Proposition 6.45. D
If Μ is a regular submanifold of a Riemannian manifold (Ν, /ι), then Μ
inherits a Riemannian metric g := г*Л-, where г : Μ <-»- N is the inclusion
map. We have already used this idea for submanifolds of the Euclidean
342
7. Tensors
space Rd. More generally, if / : Μ —> N is an immersion, then (M, f*h) is
a Riemannian manifold. In particular, if / : Μ —> Rd is an immersion, then
we obtain a Riemannian metric on M. It turns out that every Riemannian
metric on Μ can be obtained in this way. Actually, more is true! For
any Riemannian manifold (M,g) there is an embedding f : Μ —> Rd, for
some d, such that g := г*go, where go denotes the standard metric on Rd.
What this means is that f(M) is a regular submanifold, and if we give
f(M) the metric induced from the ambient space Rd, then / becomes an
isometry when viewed as a map into /(M). We say that such an / is an
isometric embedding of Μ into Rd. In short, the result is that every
Riemannian manifold can be isometrically embedded into some Euclidean
space of sufficiently high dimension. This difficult theorem is called the
Nash embedding theorem and is due to John Forbes Nash (see [Nashl]
and [Nash2]). Note that d must be quite large in general (d = (dimM) +
5(dimM) 4- 3 is sufficient). For an indefinite semi-Riemannian manifold
(iV, h), the pull-back f*h by a smooth map / : Μ —> N may not be a metric
because there may be points ρ Ε Μ such that Tpf(TpM) is a degenerate
subspace of Tf^N* In particular, not every embedding of a manifold Μ
into a semi-Euclidean space R^ (with 1 < ν < d) induces a metric on M.
Nevertheless, every metric on Μ of any index can be obtained using an
appropriate embedding into some Rj5 (see [Clark]).
Problems
(1) Show that if r G V ® V* has the same components rj with respect to
every basis, then rj = αδι for some a G R.
(2) If ei,...,en is a basis for V and /i,...,/m is a basis for W, then
{-E5-}i=i1„.1TE j=l,...,m is a basis for L(V,W), where E)(v) :- ei{y)fj.
Show this directly without assuming the isomorphism of W (8) V* with
L(V,W).
(3) Let (Vi,5i) be scalar product spaces for г — 1,..., fc. By Corollary D.35
of Appendix D, there is a unique bilinear form
ψ ■ <8>*=iVi χ 0f=1Vi -+ ж
such that for Vi G Vi and W{ Ε Wj,
φ (νι ® ■ · · ® Vfc, w\ ® · ■ · ® wk) = φι (vi, w\) · · · <РкЫ, Wk).
Show that φ is nondegenerate and that it is positive definite if each g
is positive definite.
Problems
343
(4) Define τ : Χ(Μ) χ Χ(Μ) -> C°°(M) by τ(X, У) = ХУ/. Show that r
does not define a tensor field.
(5) Let bzj and b^ be the components of a bilinear form b with respect to
bases ei,..., en and e'1?..., efn respectively. Show that in general det(by)
does not equal det(b^). Show that if det(b^) is nonzero, then the same
is true of det(6^·).
(6) Show that while a single algebraic tensor rp at a point on a manifold can
always be extended to a smooth tensor field, it is not the case that one
may always extend a (smooth) tensor field defined on an open subset to
a smooth tensor field on the whole manifold.
(7) Let φ : R2 -> R2 be defined by (x,у) н> (χ + 2y,у). Let τ := x^®dy +
щ <g> dy. Compute φ*τ and φ*τ.
(8) Prove Proposition 7.35.
(9) Let V be a tensor derivation on Μ and suppose that in a local chart we
have Χ>(^τ) = Ε ЩШ for smooth functions D\. Show that V(dxj) =
— Σ^άχ1· Let X be a fixed vector field with components Xх in our
chart. Find the D\ in the case that V = £χ.
(10) Let A € 7^(M). Show that the component form of the Lie derivative
with respect to a chart is given as
//» -луб- дА<хЬ yh dxa Ahb_2*L_A*h
{LxA) ~ дх*Х ~д^А дх*А
(where we use the Einstein summation convention). Show that if Л €
^(M), then the formula becomes
dAab h dXh dXh
{CxA) = Ί^χ + ~d^Ahb + ~ΜΑαΗ·
Find a formula for A e 7|(M).
(11) Show that our two definitions of the Lie derivative of a tensor field agree
with each other.
(12) In some chart (£7, (x, y)) on a 2-manifold, let A = χβ- ® dx ® dy + ^ ®
dy ® dy and let X = ^ + χ-β-. Compute the coordinate expression for
CXA.
13) Suppose that for every chart (C/, x) in an atlas for a smooth n-manifold
Μ we have assigned n3 smooth functions Γ^·, which we call Christoffel
symbols. Suppose that rather than obeying the transformation law
expected for a tensor, we have the following horrible formula relating the
Christoffel symbols IV* on a chart (Uf,y) to the symbols Г^. :
,k d2xl dyk t дх^дх^ду^
ij д^дуз дх1 rs dy* dyi dxt {SUm,t
344
7. Tensors
Assume that such a transformation law holds between the Christoffel
symbol functions for all pairs of intersecting charts. For any pair of
vector fields X,7g X(M), consider the functions (DxY)k given in
every chart by the formula {DxY)k := ^Xh + Τ^ΧΎΚ Show that
the local vector fields of the form (DxY)k-^: defined on each chart,
are the restrictions of a single global vector field ΌχΥ. Show that
Dx : Υ м- DXY is a derivation of X(M) and that with Dxf := Xf for
smooth functions, we may extend to a tensor derivation; ϋχ is called a
covariant derivative with respect to X. There are many possible covari-
ant derivatives.
(14) Continuing on the last problem, show that Dfx+gyT = JDxT + gDyT
for all /,5 e C°°(M) and Χ,Υ Ε X(M) and Τ € 7^(M).
(15) Show that if g|r,..., -^ are coordinate vector fields from some chart,
then [д!^, g|j] ξ 0. Consider the vector fields ^ and щ arising from
standard coordinates on R2 and also the ^ and щ from polar
coordinates. Show that [Jj, ·§ρ\ is not identically zero by explicit computation.
(16) Let Μ —> Μ be a semi-Riemannian cover. Prove that if we have a local
isometry φ : N -> Μ, then any lift φ : N —> Μ is also a local isometry.
(17) Suppose we have two lattices Γ/ and Γ^ in Rn. Let Rn have the standard
metric. Describe the induced metrics on Жп/Г/ and Rn/Tf and provide
a necessary and sufficient condition for the existence of an isometry
Rn/Tf -> Rn/Tf.
(18) Show that if α, β £ 7fc(V) where V is a scalar product space, then the
scalar product on 2fc(V) is given in terms of index raising and contraction
by
(a,/3> = £ail...ifc/?il-"<fc.
Chapter 8
Differential jbrms
In one guise, a differential form is nothing but an alternating
(antisymmetric) tensor field. What is new is the introduction of an antisymmetrized
version of the tensor product and also a natural differential operator called
the exterior derivative. We start off with some more multilinear algebra.
8.1. More Multilinear Algebra
Definition 8.1. Let V and W be real finite-dimensional vector spaces. A k-
multilinear map a : V x - · ■ x V —> W is called alternating if a(v\,..., Vk) =
0 whenever V{ = Vj for some i ^ j. The space of all alternating k-
multilinear maps into W will be denoted by L*lt(V;W) or by L^lt(V) ^
W = R. By convention, L^lt(V;W) is taken to be W and in particular,
^t(V) = R.
Since we are dealing with the field R (which has characteristic zero), it
is easy to see that alternating ^-multilinear maps are the same as
(completely) antisymmetric fc-multilinear maps which are defined by the
property that for any permutation σ of the letters 1,2,..., к we have
ω(υι, V2,.. ·, vk) = 8^(σ)ω(ϋσ(ΐ), νσ(2), ·.., υσ^).
Let us denote the group of permutations of the к letters 1,2,..., к by Sfc.
In what follows, we will occasionally write σ* in place of σ{ι).
Definition 8.2. The antisymmetrization map Alt* : T°fc(V) -¥ L^t(V)
is defined by
Altk(u))(vuv2,...,Vk) := ^ί Σ sgn^u;^,^,...,^).
345
346
8. Differential Forms
Lemma 8.3. For α € T°fci(V) and β G T°k2(V), we have
Altfci+fc2(Altfcia ® д) = Aitfci+fc2 (α ® β),
Altfel+fc2(a® Alt*2/3) = Altfcl+*2 {α®β),
and
Altfel+fc2(Altfel (a) ® Alt*2 (/?)) = Altfcl+*2 (a ® /3)
Proof. For a permutation σ € 5& and any Τ G T°fc(V), let σΤ denote
the element of T°fc(V) given by (σΤ) (vu ..., vk) := Τ(νσ(ΐ),..., υσ(Λ)). We
then have Altfc(aT) = sgn(a)Altfc(T) as may easily be checked. Also, by
definition Altfc(T) = 5>βη(σ)σΤ. We have
= Altfcl+fc2 \\-r-; Σ sgna (σα) ) ®
= Altfcl+fc2 I -^ Σ sgn σ (σα ® /9) ]
\ 1'<T6Sfcl /
= Λ Vs sgaa Alt*1**2 (σα ® β).
fci! ^^
Let us examine the expression sgnσ we extend each
σ Ε Sk± to a corresponding element σ' G Sfcj+j^ by letting σ\ι) = σ(ί) for
г < кх and а'(г) = г for г > fci, then we have σα® β = σ'(α ® /3) and also
sgn(a) = sgn(a'). Thus sgnaAltfcl+A;2 (σα ® /3) = sgn σ' Altfcl+fcV(a ® jfl
and so
Altfcl+fc2(Altfcla®)8)
= FT Σ sgn^Altfcl+A:2 (</ (α ® /3))
= tt 7 sgn σ'sgn
fci! ^—/
<reSfcl
= Α1^1+*2(α®/3)Λ V 1 = Altfcl+fe2 (a ® β),
65fcl
We arrive at Alt*1+*2 (Alt*1 (a) ® β) = Altfcl+fc2 (a ® β). In a similar way,
Altfel+fe2(a® Alt*2/?) = Altfcl+fe2 (α® β), and so the last part of the theorem
follows. D
8.1. More Multilinear Algebra
347
Given ω G l£ft(V) and η £ L^t(V), we define their exterior product
or wedge product ω Λ η € ΐ!^~ 2 (V) by the formula
Written out, this is
ω Α η(ντ ,...,vkl, vfcl+i,..., vfcl+fc2)
(8Л) =к№ Σ 88η(σΜν*1.···.νσ*1Μνσ^1+1,...,νβ^+42).
Warning: The factor in front of Alt in the definition of the exterior
product is a convention but not the only convention in use. This choice
has an effect on many of the formulas to follow which differ by a factor from
the corresponding formulas written by authors following other conventions.
It is an exercise in combinatorics that we also have
(ω Λ т/Ж,..., vfel, vfel+b ..., v^+jtj
= > sgn(a)o;(v )r/(v
(fci,/c2)-shuffles σ
In the latter formula, we sum over all permutations such that σ (1) < σ(2) <
... < а{к\) and а(к\ + 1) < σ(&ι + 2) < ■ · ■ < a{fa + fa). This kind of
permutation is called a (fci,A;2)-shuffle as indicated in the summation.
The most important case of (8.1) is for ω,η € L^t(V), in which case
(ω Λη)(υ,ιν) = ω(ν)η(νυ) — ω(ιν)η(ν).
This clearly defines an antisymmetric bilinear map.
Proposition 8.4. For α Ε L^V), β e LkJt{V), and 7 Ε L&(V), we have
(i) Λ : L*ft(V) x L5t(V) -> L5+fa(V) г5 R-btKnear;
(ii) αΛβ = (-1)Ηι**βΛα;
(iii) α Λ (/? Λ 7) = (α Λ /3) Λ 7.
Proof, We leave the proof of (i) as an easy exercise. For (ii), we consider
the special permutation / given by (/(1), /(2),..., /(fci + fe)) = (&i + l,...,
*ι + *&, 1, ·.., fa). We have that α®β = / (β ® α). Also sgn(/) = (-l)*1*2.
So we have
Alt*1+*2 (α ® β) = Alt*1+fe2 (/ (0 ® a)) = (-I)***2 Altfel+*2 (0 ® a),
which gives (ii).
348
8. Differential Forms
For (iii), we compute
fci
(fci+fc2 + fc3)!(fc2 + fc3)!
*i! (*2 + Аз)! к2\к3\
(h + k2 + k3)\
»Л0Л7)-Й|±£±Мл1.<«,.0Л7))
Alt(a®Alt(0®7))
By Lemma 8.3, we know that Alt (a ® Alt (β ® 7)) = Alt (a ® (/? ® 7)), and
so we arrive at
« Λ 09 Л7) = fcL|+M!Alt(a β 05 β7)).
By a symmetric computation, we also have
(α Λ β) Λ 7 = (^!W3)!Alt((a ® ^ ® 7)'
and so by the associativity of the tensor product we obtain the result. D
Example 8.5. Let V have a basis ех,е2,ез with dual basis e^e^e3. Let
а = 2ε1 Λ e2 + e1 Λ e2 and β = e1 — e3. Then as a sample calculation we
have
α Λ β = (2el Л е2 + e1 Л е3) Л (е1 - e3)
= 2el Л е2 Л e1 + e1 Л е3 Л e1
= -2ε1 Λ e2 Л е3 - e1 Л е3 Л е3
= -2ex Л е2 Л e3,
where we have used that e1 Л е2 Л e1 = —e1 Л е1 Л e2 = 0, etc.
Lemma 8.6. Let a1,..., ak be elements of V* — L^t(V) and /ei wi,...,v*
G V. TAen we /шг>е
α1 Λ · · · Л ak (vi,..., v*) = det A,
where A = (a*·) is the к х к matrix whose ij-th entry is oj = a*(vi)-
Proof. Prom the proof of the last theorem we have
« Λ (β Λ 7) = ί^±|^3)!Alt(a ® 0? ® 7».
By inductive application of this we have
α1 Λ · · · Л ак = k\ Alt^1 <g> · · ■ <8> afc).
Thus
ο^Λ-Λα'ίοι ι») = ^sgn(a)a1(wffl)---aA:(i;(Tfc) = detA D
8Λ. More Multilinear Algebra
349
Let us define
( 1 if ji,..., jk is an even permutation of ii,...,г&,
—1 if ji,...,^ is an odd permutation of ii,..., i/e,
0 otherwise.
(8·2) «H
Then we have
Corollary 8.7. Let ei,..., en be α basis for V and e1, e2,..., en the dual
basis for V*. Then we have
е<1Л...Ле**(ел>...|ел)=^.
Since any a G L^t(V) is also a member of T°k(V), we may write
a = 5^atl...lfceil®---®e<fc,
where an...ifc = «(e^,..., eifc). By Alt (a) = a and the linearity of Alt we
have
a = Σ<*ίι...<*Α1ΐ (eix О · · ■ ® elfc) = ^ J^...^1 Л · · · Л eifc.
We conclude that the set of elements of the form e*1 Л ■ ■ ■ Aelk spans i*Jt(V).
Furthermore, if we use the fact that both <Χίσι„Λσ = sgn aatl,„ik and ег<т1 Л
... Л ег<7* = sgn σ en Λ · · - Λ егк for any permutation σ Ε Sfc, we see that we
can permute the indices into increasing order and collect terms to get
a = fc[ Σ α4-**βίι Λ * · · Λ eifc = ^ θιι»2,...ί*βίι Λ е'2 Л ■ - · Λ eifc,
U<i2<-<*fc
where in the last expression we sum only over strictly increasing indices.
We can check that the set of (£) elements of the form епЛег2Л--Л e4fc
with 1 < ίχ < %2 < - · · < ik < η is linearly independent as follows: Suppose
* = Ег1<г2<-<г, βίΑ,-Α^1 Л e« Л ■ - ■ Л e* = 0. Let Л < j2 < · ■ · < J*.
Then we have
0 = а(ел,...,ел)
Σ au»2...tfccil Λ ei2 Λ - · · Λ е<к(ел, · - ·»^)
*1<12< —<»fe
— / v й»11'2...»ьСч\ ή. — Q>i-
П-1к _
'Ui2-*fecjx..Jfc — wJU2».Jfc·
U<i2<—<ifc
We have used that ег\'''1к is zero unless j\.,. j* is a permutation of i\ ... i*,
and then in this case, since both are increasing, we must have ir = jr for
r = l,...,fc. Thus we get 0 — аяЛ...Ль> anc* s^nce the choice of j's was
arbitrary, we have shown independence. Thus, we have proved the following
theorem.
350
8. Differential Forms
Theorem 8.8. // (e1, e2,..., en) is α basis for V*, then the set of elements
{eh A ei2 Л ■ ■ · Л eik : 1 < h < i2 < · · ■ < tfc < n}
is a basis for L^t(V). Thus dim(L^t(V)) = (£) = щп-к)\* ^п Раг^си^аг>
L^(y) = 0fork>n.
Corollary 8.9· /fa1,..., ak G V*? then a1,..., ak are linearly independent
if and only if
a1 Л · · ■ Л ak φ 0.
Proof. If a1,..., ak are independent, then there are elements afe+1,..., otn
such that a1,..., an is a basis for V*. Let v\,..., vn be the basis for V dual
to the above basis. Then since α1 Λ · · · Л ак is a basis element for L^lt(V),
it cannot be zero.
For the other direction, suppose that, after rearranging if needed, we
have
a1 = c2a2 + hCfccA
Then we would have
a1 A · · ■ Л ak = (c2a2 + h ckak\ Λα2Λ···Λ^ = 0.
Thus we see that а1 A ■ ■ ■ Λ ak φ 0 implies that a1,..., ak are linearly
independent. D
Notation 8.10. In order to facilitate notation, we will sometimes
abbreviate a sequence of к integers, say ii, 12,..., ik > from the set {1,2,..., dim(V)}
as /, and en Л e12 A · - · Л е%к will be written as e1 and e£ — c£"*£* · Also, if
we require that i\ < i2 < · · ■ < г&, then we will write /. We will freely use
similar self-explanatory notation as we go along without further comment,
For example, we may write
<*=Σα/β/
to mean α = Егкг2<-<гк ain2^keh A e'2 Л ■ ■ ■ Л еЧ
Whenever we have α = Σ afeIι where af = щ^лк and i\ < · · · < i*, we
can define aj for any fc-tuple of indices J — (ji,..., j*) by requiring that
aj = Σ б 5α/· Then we have aj = 0 when the entries of J are indices that
are not distinct. Otherwise, aj = e^aj where J is J rearranged in increasing
order. But, it is also easy to see that а(е^,..., e^) = Σ^αρ We then
have
a = Σ αΓβΙ = й Σ а'е' = й Σ ^.«i*^1 A · · · Л elfc
8.1. More Multilinear Algebra
351
Exercise 8.11. Show that for α = ^ Σ aleI and β = h Σ bjeJ with I and
J as above, we have
tl...tfc+ί
where
(8.3) (а Л /3)il...ifc+€ = j^j- 2^ ah-Jkbjk+i->-jk+t%„.ik+i ■
Definition 8.12. Let t; € V and ω € I&tOO· Define ί,,ω Ε /^(V) by
ivo;(vi,..., Vk-ι) := w(v, vi, ..., ^-ι);
ίυω is called the interior product of ν with ω or the contraction of ω by
v. By convention г^а = 0 for a Ε L^t(V) := R. Thus we obtain a linear
map^iZ^OO-Hi^CV).
It is clear that г^ depends linearly on ν and that м^а; = —iwivu for all
v, w Ε V and hence ivoiv = 0.
With £У V) = V* and L2it(V) = R, the sum
dim(V)
^alt(V) = 0 1&(V)
k=0
is made into an R-algebra via the exterior product just defined. Since αί\β Ε
Lk^k2(V) whenever a Ε i#t(V) and β € L^2t(V), the algebra Lait(V) is a
graded algebra (see Definition D.43).
The contraction map iv for υ Ε V extends to a map Lalt(V) —► Lait(V).
Since iv(L^t(V)) С Z^^V), we say that iv is a map of degree —1.
Proposition 8.13. For ν Ε V, the contraction map iv satisfies the product
rule
iv(a Λβ) = (г„а) Λ β + (-1)ка Λ {ζνβ) for a E L*it(V).
The map iv : I/ait(V) —> Lait(V) is the unique degree —1 map satisfying the
above product rule and satisfying iva = 0 for a E R and
г„0 = β(ν) /or 0 Ε V* = lit(V) and г; Ε V.
Proof. Let a E U^(V) (M) and /3 Ε Lalt(V) (M). In the following
computation we use the permutation σ which is given by
(2,3,..., к + 1,1, к + 2,..., к + ί) *-> (1,2,..., к + i).
352
8. Differential Forms
The sign of σ is (—l)fc. We compute as follows:
(ιυα Λ β + (-1)ка Λ ιυβ)(ν2,..., vk+e)
(k-l)W.
-A\t(iva®0)(v2,...,vk+e)
+ (-1)кУщ-$Ща ® ιυβ)(ν2, ■ · ·, vk+1)
= (fc-1)111 Σ 8бп(<г)<*0>,ν^» ■ ■ ■ > *ч)£(^+1, ■.., <w)
+ (~1)fefe;(^-i)i Σ8ёп(™М^ ν**, - · -,*w)/3(^fc+1, · ··,νσ,+ί)
= (fe-1)111 Σ^(^Μ^ *>σ» ■ ■ ■ > νσΛ)^(νσΛ+1, · ■ ·, νσΜ)
+ fcl(l-l)l Σ *Μ<τ)α{ν, νσ2,..., ν^+1 )0(νσΛ+1,..., V(7fc+i)
= (fc-l)ll! Σα(ν' ^2, · · · >^(fc+i))β(υ<τ*+ι> · · · >^fc+£)
= α Λ 0 (ν, va,..., Vfc+i) = г„(а Λ β)(ν2,. ■., ufc+i).
We leave the proof of the remaining statements of the theorem to the reader
(see Problem 14). Π
It follows from the above that if 0i,..., Ok € V* and ν € V, then
fc
ΐ^6ιΛ-.·Λ^) = 5^(-1)*+1^(ι;)βιΛ..-Λ^Λ·-·Λ^ι
1=1
where the caret over 0£ denotes omission.
Proposition 8.14. If A E L(V, V), a e LkJt(V) and β Ε L%t(V), then
λ*(αΛ/?) = λ*αΛλ*/3.
Proof. This follows from Proposition 1.83 and the definition of the exterior
product. D
We look more closely at L^t(V) where η = dimV. The dimension
of LJjt(V) is one, and any nonzero element of LJt(V) provides a basis.
If λ e £(V,V), then λ* : I£t(V) -► £^(ν) is a linear transformation
8.1. More Multilinear Algebra
353
between one-dimensional vector spaces, and so it must be multiplication by
an element of R. Thus there is a unique number det (A) £ R such that
Χ*ω = det(A)o;
for any ω £ LJt(V). This number is called the determinant of λ. This
provides a definition of determinant that does not involve a choice of basis.
We will show that if A = (aj) represents A £ L(V, V) with respect to a basis
for V, then det(A) = det (A), where the determinant of a matrix is given by
the standard definition. Let A £ L(V,V) and suppose that А(е») = J2aiej
for some basis (ei,..., en) with dual (e1,..., en). Then A = (aj) represents
A, and we have
A*(ex Л · ■ · Л е*)(еь ..., еп) = (е1 Л ■ · - Л еп) (Аеь · · ■, Аеп)
= det(e<(AeJ-)) = detA
On the other hand, A*(ex Л ■ · · Л en) = (detA) (e1 Л · ■ · Л en), and since
e1 Л · ■ · Л en(ei,..., en) = 1, it must be that det(A) = det (A).
Exercise 8.15. Show (without using a basis) that if ί, λ £ L(V, V), then
(i) det (* ο λ) = det* det λ;
(ii) det(id) = 1;
(iii) A £ GL(V) if and only if det λ φ 0;
(iv) if A £ GL(V), then det (A"1) = (detA)-1.
We now briefly discuss orientations of real vector spaces. In what follows,
let V be a real vector space with η = dimV. The nonzero elements of
Z£lt(V) are sometimes referred to as volume elements (although this term
will also apply to a global object later on). Two nonzero elements ω\ and
u)2 of L™lt(V) are said to be equivalent if there is a scalar с > 0 such that
o>i = OJ2- The equivalence class of ω will be denoted [ω]. There are clearly
exactly two equivalence classes. An equivalence class is referred to as an
orientation for V. An oriented vector space is a vector space with a
choice of orientation and is sometimes written as a pair (V, [ω]).
An ordered basis (ei,..., en) for an oriented real vector space (V, [ω])
is said to be positive with respect to the orientation if ω(βχ,... ,en) > 0
for some and hence any ω Ε [ω]. Equivalently, (ei,,..,en) is positive if
e1 Λ * ■ · Λ en € [ω]. Actually a choice of ordered basis for a real vector space
determines an orientation with respect to which it is positive. Indeed, if
e1,..., en is dual to such a basis, then choose [ω] where ω = e1 A · · · Λ en.
Definition 8.16. Let (Vi, [ωχ]) and (V2, [0/2]) be oriented real vector spaces.
A linear isomorphism A : Vi —> V2 is said to be orientation preserving if
A*o;2 = cu)\ for some с > 0 and some, and hence any, choices ω\ £ [ωι] and
^2 £ [ω^].
354
8. Differential Forms
When one talks about an element λ G GL(V) being orientation
preserving, one means that detA > 0 and this is tantamount to λ : (V, [ω]) -l·
(V, [ω]) being orientation preserving for any choice of orientation [ω].
8.1.1. The abstract Grassmann algebra. We now take a very abstract
approach to constructing an algebra that will be seen to be isomorphic to
^ait(V). We wish to construct a space that is universal with respect to
alternating multilinear maps. We work in the category of real vector spaces,
although much of what we do here makes sense for modules. Consider the
tensor space Tfc(V) := 0 V (take any realization of the abstract tensor
product as in Definition D.17). Let A be the submodule of Tk(V) generated
by elements of the form
V\ ® · ■ · Vi ® ' · · ® Vi · ■ ' ® Vk-
In other words, A is generated by simple tensors with two (or more) equal
factors. Recall that associated to Tk(V) we have the canonical map ® :
Vx-'-xV^ Tfc(V) defined so that ®(vi,..., Vfe) = vi® · · ·®ν*. We define
the space of Ar-vectors to be
V Λ · · · Λ V := Д* V := Tk(V)/A.
Let Л& : V χ · - χ V —>· Д Vbe the composition of the canonical map ® with
quotient map of Tk(V) onto Д V. This map turns out to be an alternating
multilinear map. We will denote Λ*(νι,... ,ufc) by v\ Λ · · · Л Vk- Using
the universal property of Tk(V) as described in Appendix D, one can show
that the pair (Д V, Л&) is universal with respect to alternating fc-multilinear
maps: That is, given any alternating fc-multilinear map а : V x · - ■ x V -► W,
there is a unique linear map αΛ : Д V —► W such that а = aAoAk', that is,
the following diagram commutes:
Vx -·■ χ V >W
| yS OLA
Afcv
Notice that we also have that ν\/\· · ·Λ^ is the image of vi®· · ·®ν& under the
quotient map Tfc(V) -> /\hV. Next we define /\V := ©^LqA^V, which is
a direct sum, and we take Д0 V := R. We impose on Д V the multiplication
generated by the rule
(υι Λ · · · Λ v^ χ (v[ A · ■ ■ Λ Vj) H- vi A · · · Λ V{ A v[ A · · ■ Λ vfj € Д V.
The resulting graded algebra is called the Grassmann algebra or exterior
algebra. (Of course, the definition of Λ here is different from what we
defined previously.) If we need to have a Z grading rather than an N grading,
8.1. More Multilinear Algebra
355
we may define Д V := 0 for к < 0 and extend the multiplication in the
obvious way. Elements of Д V are called multivectors and specifically,
elements of /\k V are called fc-multivectors.
Notice that since (v + w)A(w + v) = 0, it follows that vAw = —wAv. In
fact, any transposition of the factors in a simple element such as v\ A · · · Л Vk,
introduces a change of sign:
v\ A · ■ · Л Vi A ■ · · Л Vj A · · ■ Л Vk
= —V\ A · · ■ Л Vj A · · ■ Л V{ A · · ■ Л Vk-
Lemma 8.17. //V has dimension n, then /\k V = 0 for к > n. Ife\}..., en
is a basis for V, then the set
{e^ A ■ · · Л eik : 1 < %i < ■ · ■ < ik < n}
is a basis for Д V where we agree that e^ Л · ·· Л ejfc = 1 if к — 0.
Proof. The first statement is easy and we leave it to the reader. We will
show that the set above is indeed a basis. First note that Д71 V is spanned
by ei Л · ■ ■ Л en. To see that e\ A · · ■ Л еп is not zero we let
det : V χ · ■ · χ V -> R
be the multilinear map given by representing the arguments as column
vectors of components with respect to the given basis and then taking the
determinant of the η χ η matrix built from these column vectors. Then
det(ei,..., en) = 1. But by the universal property above there is a linear
map detA such that det = detA о Ак and so
detA(ei Л ■ · · Л еп) = detA о Ак (еь ..., еп)
= det(ei,...,en) = 1;
thus, we conclude that ei Л ·· ■ Л en is not zero (and is a basis for Дп V).
L.
Now it is easy to see that the elements of the form e^ Л · ■ · Л eih span Д V.
To see that we have linear independence, suppose that
У^ ач~лкеп A"'Aeifc=0.
l<ti<—<ife<n
Now multiply both sides by e^ Л · ·· Л ejn_A, where {ji,... ,jn-k} is the
complement of {ii,..., ik}· Then we obtain
=bu*i...ifcei Л · · · Л en = 0,
from which we conclude that а%1ш.лк = 0? and since ii,. -., ifc was arbitrary,
we are done. D
Exercise 8.18. Show that υχ,... ,ν& Ε V are linearly independent if and
only if v\ A · · · Λ Vk φ 0. Compare this with Corollary 8.9.
356
8. Differential Forms
Definition 8.19. An element ξ G Д V is called decomposable if £ -
^i Λ ■ ■ ■ Λ Vk for some νχ,..., ν* Ε V.
Let us gain a little practice dealing with multivectors by proving the
following proposition:
Proposition 8.20. Let ξ Ε f\2 V ωίίΛ ξφΟ. Then there is α basis vi,..., vn
of Υ such that
Furthermore, in this case the r-fold product ξ Λ · · - Λ ξ is nonzero and
decomposable while the r + l-fold product is zero.
Proof. First we prove that there exists such a decomposition for ζ. Let
ei,..., en be a basis for V. We have
ξ = 2_j aijei Λ ej = αχ2βι Λ β2 + ai3ei Λ β2 Η l· αιηβι Λ en
+ а,2зе2 Λ ез + 024^2 Λ е± Η l· θ2ηβ2 Λ en + ξ',
where ξ' does not involve e\ or β2· By renumbering if necessary we may
assume that a\2 is not zero. But then if
vi '·= ei + (023/012) ез Η h (θ2η/οΐ2) en
^2 := οΐ2β2 + αΐ3β3 Η \- а\пеп,
we have that v\,v%, ез,..., en are linearly independent and
ζ = vi Λ V2 + υ,
where υ does not involve ei or ег- If ν = 0, then we are done. Otherwise
we may repeat the process with
ν = ^2 btjeiAej.
%<3
*J>3
Clearly a simple induction gives
ξ = νχ Λ V2 + V3 Л Uj Η h V2r-1 Λ V2r
for some nonzero vi,..., V2r such that vi,..., V2r, e2r+i?..., en is the desired
basis.
Next we consider the r-fold product. If we set ф^ := V2h-i A V2k for
Ar = 1,... ,r, then
r
ξ = ^ 0i and φχ/\φί— 0.
t=l
8.1. More Multilinear Algebra
357
On the other hand, if г φ j, then φι A <l>j = <f>j Α φι is plus or minus one times
one of the elements in the linearly independent set
{2v&1 A V£2 A vt3 А ы4 :£i<£2<h<h and 4 € {1,..., 2r}}.
Thus
l<i<j<r
In general, it is easy to see that since all of the φι commute with each other,
we have
ίΗ = Σ ^i Л^2 А"'АФч =к}· Σ ^ι Λ<^2 А"'АФч·
Then, it is easy to see that
С = г\ф\ Αφ2Α···Αφτ = r\v\ A V2 A ■ · ■ Λ v2r φ Ο,
while £r+1 = 0. D
The number r from the previous proposition is called the rank of the
element ξ 6 Д V and the definition works just as well for elements of Д V*.
It can be shown that if
s — / v aijei A 6j — у ^ aij6i A ej,
where (aij) is an antisymmetric matrix, then the rank of ξ is the rank of the
matrix (a^·).
The following proposition follows easily from the universal properties of
the exterior product.
Remark 8.21. There is a natural isomorphism
LyV;W)-L(y\4w).
Lemma 8.22. In particular,
^t(v)=(y\fcv
We would now like to embed /\kV* into §£)fcV*, and this involves a
choice. For each fc, let Ak : V* χ · · · x V* -4 <g)fe V* be defined by
Ak(ai,..., ak) := ^sgn(a)a£Tl ® · · · <g> ασ&.
σ
By the universal property of Д V* we obtain an induced map
358
8. Differential Forms
If we identify <g)fc V* with Tk(V), then we get a map Ak : /\k V* -)· Tfc(V).
Proposition 8.23. The image of the map Ak : Afe V* -► Tfc(V) is a linear
isomorphism with image equal to L^t(V) such that
Ak{ax Λ · ■ · Λ α*)(νι,..., vfc) = det(a» (u,)).
Proof· We leave the proof as Problem 5. D
We combine these maps to obtain a linear isomorphism
A:/\Y*^L<(V).
Now both Lajt(V) and f\Y* have already independently been given the
exterior algebra structures via their respective wedge products. One may
check that A has been defined in such a way as to be an isomorphism of
these algebras:
Д V* = iait(V) (as exterior algebras).
Also notice that
AV-Lyv)-(Afcv)\
which allows us to think of Д V* as dual to Д V in such a way that
(αχ Λ ■ ■ ■ Λ Qife)(vi Λ · ■ · Λ vk) = det (α» (v3)).
Remark 8.24 (An identification). In what follows, we will identify f\k V*
with L^t(V) and hence /\Y* with Lait(V) whenever convenient. In other
words, we freely treat elements of Д V* as alternating multilinear forms.
8.2. Differential Forms
Let Μ be an η-manifold. We now bundle together the various spaces
L%Lt(TpM). That is, we form the natural bundle Lklt(TM) that has as
its fiber at p, the space L^{TPM). Thus L^(TM) = \JpEM Lk<(TpM).
Exercise 8.25. Exhibit the smooth structure and vector bundle structure
on Lllt{TM) = \JpeMLllt(TpM). [Hint: Let (Ϊ7,χ) be a chart of Μ and
x1,..., xn the coordinate functions. Let U := \JpEu L^t(TpM) and let d =
(£). Then we have a map U —> U x Rd given by ap ·->► (p, a) where a is
the d-tuple of components (in some fixed order) of ap given by its local
coordinate representation.]
Let the smooth sections of this bundle be denoted by
(8.4) nk(M)^T(M;Lkalt(TM)),
and sections over U С Μ by Ω^τ(ί7).
8.2. Differential Forms
359
Definition 8.26. Elements of Qk(M) are called differential A:-forms or
just /c-forms.
The space Qk(M) is a module over the algebra of smooth functions
C°°(M) = J"(M). If η — dim Μ then we have the direct sum
Ω(Μ) = 0Ω*(Μ),
fc=0
with a similar decomposition for any open U С М.
Exercise 8.27. Show that there is a module isomorphism
©П*(М)вг(ф^(™))·
fc=0 \fc=0 /
Definition 8.28. Let Μ be an η-manifold. The elements of Ω(Μ) are
called differential forms on M. We identify Ω*(Μ) with the obvious
subspace of Ω(Μ) = 0fc ilk(M). A differential form in ftk(M) is said to be
homogeneous of degree fc.
If ω Ε Ω(Μ), then we can uniquely write ω = £fc 1ujc where ω& Ε
Ω^Μ) and the ω^ are called homogeneous components of ω.
Definition 8.29. For ω Ε ΩΛι(Μ), and η € Ω*2(Μ), we define the exterior
product α; Λ η Ε Ω*1+*2(Μ) by
(ω Λ τ/)(ρ) :=ω(ρ)Λ»7(ρ).
It is easy to see that Ω(Μ) is a ring under the wedge product and, in
fact, a C°°(M)-algebra. Whenever convenient, we may extend this to a sum
over all n e Ζ by defining (as before) ttk(M) := 0 for к < 0 and Пк(М) := 0
if к > dim(M). We have made the trivial extension of Л to a Z-graded
algebra by declaring that ω Α η = 0 if either 77 or ω is homogeneous of
negative degree. Thus Ω(Μ) is a graded algebra and is said to be graded
commutative because αΛβ = {-ΐ)Ηβ/\α for а е ilk(M) and β € Ω*(Μ).
Sometimes one see this notion referred to as skew-commutativity.
Just as a tangent vector is the infinitesimal version of a (parametrized)
curve through a point ρ Ε Μ, so a covector at ρ Ε Μ is the infinitesimal
version of a function defined near p. At this point one must be careful. It
is true that for any single covector ap G TPM there always exists a function
/ such dfp = ap. But as we saw in Chapter 2, if α Ε Ω1 (Μ), then it is not
necessarily true that there is a function / Ε C°°(M) such that df = a. If
/b /25. ■ ·, fk же smooth functions, then one way to picture df\A- - -Adfk is by
thinking of the intersecting family of level sets of the functions /1, /2, · · -, Λ,
which in some cases can be pictured as a sort of "egg crate", structure. For
360
8. Differential Forms
Figure 8.1. 2-form as "flux tubes"
a 2-form in a 3-manifold, one obtains "flux tubes" as shown in Figure 8.1.
The infinitesimal version of this is a sort of straightened out "linear egg
crate structure," which may be thought of as existing in the tangent space
at a point. This is the rough intuition for df\\pA ■ ■ · Λ dfk\p, and the Morm
df\ A · · · Λ dfk is a field of such structures which somehow fit the level sets of
the family Д, /2,..., Д. Of course, dfi A · · · Λ dfk is a very special kind of
fc-form. In general, a fc-form over U may not arise from a family of functions.
In local coordinates, calculation is often quite easy and formal. Fo
example, in R3 with standard coordinates x,2/,z, a simple wedge product
calculation is as follows:
(xydx + zdy + dz) A (xdy + zdz)
= xydx A xdy + xydx A zdz + zdy A xdy
+ zdy A zdz + dz A xdy + dz A zdz
= x2ydx A dy + xyzdx Adz + z2dy Adz + xdz A dy
= x2ydx Ady + xyzdx Adz + (z2 — x)dy A dz.
An equally trivial calculation shows that
(xyz2dx Ady + dy A dz) A (dx + xdy + zdz) = {xyzz + l)dx Ady A dz.
8.2.1. Pull-back of a differential form. Since we treat differential forms
as alternating covariant tensor fields, we have a notion of pull-back already
defined. It is easy to see that the pull-back of an alternating tensor field is
also an alternating tensor field, and so given any smooth map f : Μ -¥ N,
8.2. Differential Forms
361
we get a map /* : Q,k(N) -» Qk(M). We recall here the definition:
(f*v) (P)(vi, · · ·, Vfc) = Vf(P)(TPf -vu ... ,Tp/ ■ vfc),
for tangent vectors νχ,..., Vk Ε TPM. The pull-back extends in the obvious
way to a map /* : Ω(Ν) -> Ω(Μ).
Proposition 8.30. Let f : Μ —► JV be α smooth map and let 771,772 € Ω(Ν).
Γ/ien we /mt>e
/*(τ/ιΛ772) = /*77ιΛ/ν
Proof. This follows directly from Proposition 8.14. D
Proposition 8.31. Let f : Μ -¥ N and g : Ν -¥ Ρ be smooth maps. Then
for any smooth differential form η Ε ЩР) we have (/оg)*η = g*(f*^.
Thus(fog)* = g*of*.
Proof. We prove only the case 77 Ε Ω,1 (Ν). The general case is entirely
similar and is left to the reader. For ν Ε ΤρΜ, we have
(f о g)* η(ν) = V(T(f о g)v) = V(Tf oTg(v))
= f*v(Tg.v)=g*(f*v)(v),
which completes the proof for the considered case. D
From the above propositions we see that we have a contravariant functor
from the category of smooth manifolds and smooth maps to the category
of rings which assigns to each smooth manifold Μ the space Ω(Μ) and to
each smooth map / the ring homomorphism /*.
In case S is a regular submanifold of M, we have the inclusion map
l: S <—> M, which maps ρ Ε S to the very same point ρ Ε Μ. As mentioned
before, it is natural to identify TPS with Tpl(TpS) for any ρ Ε S. In other
words, we often do not distinguish between a vector vp and Tpl(vp). Thus
we view the tangent bundle of S as a subset of TM. With this in mind we
must realize that for any a E Vlk(M) the form l*a is just the restriction of
a to vectors tangent to S. In particular, if U С Μ is open and ι: U «->■ M,
then 6*a = а\ц.
The local expression for the pull-back is described as follows. Let us
abbreviate: б£ = б£"'^fc (recall the definition given by equation (8.2)). Let
(U: x) be a chart on Μ and (V, y) a chart on N with / (U) С V. Then,
362
8. Differential Forms
writing η = Y^bjdyJ and abbreviating ^ζχ1 ' to simply jfe, etc. we have
rV = Y^bjofd{^of)A...Ad (у** о /)
-Σ('/·Λ (ς£**)
^ Я^ь
ifc
=Σ(^/)Σ E4Sr
-cte*1 Λ ■ ■ ■ Λ dzlfc
ώΛ Λ · · ■ Λ dx&k
Qyji dyh
θ&ι dxb
where
dyJ = д(у>\...,и**)
= det
аул
5?T
ду3к
dxei
дхек
ду3к
дх€к
Since the above formula is a bit intimidating at first sight, we work out the
case where dim Μ = 2, dimiV = 3 and к = 2. As a warm up, notice that
since dx% Л dxl = 0, we have
dx* дя·?' ^ dz* dxi
ox1 ox2 ox2· ox1
'ду2ду* ду2дУ*\, , 2
Λ αχ
cte1 Λ dx7
■(
дх1 дх2 дх2 дх1
= dJMdxlAdx*
d{x\x2)dX ЛйХ ·
jdx1
Using similar expressions, we have
ί*η = f* {bndy2 A dy3 + bndyl A dy3 + bX2dyl A dy2)
dy2
dy
dy1
dy3
= b230fY^dxiA^dxJ + b13ofY?LdxiA%LdxJ
dxi
dxi
dxi
^о/^^о/^^о/*
d(x\x2)
d(x\x2)
д{х\х2)ГХ
Adx2.
8.3. Exterior Derivative
363
Remark 8.32. Notice that the space Ω°(Μ) is just the space of smooth
functions C°° (Μ), and so unfortunately we now have several notations for
the same space: C°°{M) = Ω°(Μ) = Τ§{Μ).
8.3. Exterior Derivative
Here we will define and study the exterior derivative d. For 0-forms, exterior
differentiation is just the operation of taking the differential: / »->■ df. Let us
start right out with giving an idea of what the exterior derivative looks like
for fc-forms defined on an open set in U С Rn. Using standard rectangular
coordinates жг, all fc-forms can be written as sums of terms of the form
fdx4 Л · · ■ Л dx%k for some / G C°°(U). We know that the differential of a
0-form is a l-form: d: f H> §£zdxl. We inductively extend the definition of
d to an operator that takes fc-forms to k + 1 forms. We declare d to be linear
over real numbers and then define d(fdx4A· · -Adxlk) = dfAdxnA- · -Adxlk.
It is an easy exercise to show that if the latter formula holds for increasing
indices i\ < · · - < ik, then it holds for all choices of indices. For example, if
in R2 we have a l-form а = x2dx + xydy, then
da = d(x2dx + xydy)
= d (x2) Adx + d (xy) A dy
= 2x dx A dx + (y dx + χ dy) A dy
= ydx A dy.
We now develop the general theory on manifolds. For the next theorem we
think of Ωμ as an assignment Ω^ : U η* Ωμ(^) = Ω(ί7) for open U С М.
Thus we are simply thinking in terms of (pre)sheaves. We will drop the
subscript Μ on Ω,μ when there is no chance of confusion.
Definition 8.33. A (natural) graded derivation of degree r on Ω^ is a
family of maps, one for each open set U С Μ, denoted Vjj : Ω,μ(U) —>
Ωμ(^)ι such that for each U С М,
and such that
(1) T>u is R linear;
(2) Vv{a Αβ)= Vva Αβ+ (-l)kra Λ Όνβ for a G flk{U) and β G
Ωμ(*7);
(3) T>u is natural with respect to restriction:
Ω*(17) Έ»> Qk+r(U)
4 4
n*(V) ^ Qk+r{V)
364
8. Differential Forms
As usual we will denote all of the maps by a single symbol Z>. In
summary, we have a map of (pre)sheaves V : Ωμ —> Ωλ/. Along the lines similar
to our study of tensor derivations, one can show that a graded derivation
of Ωμ is completely determined by, and can be defined by its action on 0-
forms (functions) and 1-forms. In fact, since every form can be locally built
out of functions and exact 1-forms, i.e. differentials, we only need to know
the action on O-forms and exact 1-form to determine the graded derivation.
Recall that an element of fll(U) is said to be exact if it is the differential of
a smooth function.
Remark 8.34. If one has a map V : Ω(Μ) -¥ Ω(Μ) that satisfies (1) and
(2) of the previous definition, then it is just called a graded derivation of
degree k. But, when we meet derivations below, they will also be defined
on open submanifolds and will all give natural derivations.
Proposition 8.35. IfV\ andV2 are (natural) graded derivations of degrees
7*1 and Г2 respectively then the operator
[DUV2] :- DioD2- (-1)Г1Г2Р2 ° £>!
is a (natural) graded derivation of degree r\ + r2.
Proof. See Problem 15. D
Lemma 8.36. Suppose Vx : tokM{U) -> Ω^"Γ([7) and V2 : ^(U) -»
Ω^(ί7) are defined for each open setU С Μ and both satisfy (1), (2) and
(3) of Definition 8.33 above. IfV\ andV2 agree when applied to functions
and exact forms, then V\ =Т>2*
Proof. By (3), if T>i and T>2 agree on chart domains, then they agree
globally. Let x1,..., xn be local coordinates on U. Then every element of Ω^(£/)
is a sum of elements of the form fdxn Λ ■ · ■ Λ dx%k. But by (2) we have
Vi (fdxil Λ · - ■ Λ dxlk) = Vxf A dxh A--Adxik± fVi (dxh Λ · · · Λ dxlk)
= V2f A dxh A--Adxtk± fVi (dxh A · · · Λ dx%k).
The last term can be expanded using (2), and then the elements V\dxx can
be replaced by V2dxl*. The result is equal to V2 (fdx11 A · ■ · Λ dx%k). D
The differential d defined by
(8.5) df{X) = Xf for X e %m{U) and / € C°°{U)
gives a map Ω^ —> illM. Next we show that this map can be extended to a
degree one graded derivation.
8.3. Exterior Derivative
365
Theorem 8.37. Let Μ be α smooth manifold. There is a unique degree one
graded derivation d : Ω,μ —> Ωμ such that
dod = 0
and such that for each open U С Μ and f Ε C°°(U) = Ω^(ί7)? the 1-form
df coincides with the usual differential. Furthermore, for any chart (U,x)
for M, we have the following local formula:
d 2_\ otfdx = YJ {daf) Λ dx ·
ι
Proof. We define an operator dx for each chart (f/, x). For a 0-form on
U (i.e. a smooth function), we just define dxf to be the usual differential
given by df = Σ β^άχ1. For a € Ω^(ί7), we have a = Σ otfdx1 and we
define dxa = Y^dotfAdx1. To show the product rule ((2) of Definition 8.33),
consider a = Σ afdxr G Q^(U) and β = £ pfdxf e ΩέΜ{ϋ). Then
dx (αΛβ) = dx (У2 <*jdxfA J^ β^άχΛ
= 4 (У2 <xjPjdxT л dx,/)
= Σ ((rfar) 0 /+ar idh))άχί Λ άχ3
= i^2darAdxr\ A^jSjda/
+ Y^afdxTf\ Ι {-ΙΫ^άβ^Αάχ^
ι \ J
since άβ^Α dx1 = (—l^dx1 A dfij due to the к interchanges of the basic
differentials dx1. This means that the product rule holds for each dx. For any
function /, we have dxdxf = dxdf = JV \SL·) da;i л dx* = ° since 5^7
is symmetric in i, j and dx* Л do* is antisymmetric in i, j. More generally,
for any functions /, 5 € C°°(U) we have dx(d/ Л d#) = 0 because of the
graded commutativity. Inductively we get dx(dfi A dfe A ■ · · Л d/*) = 0 for
any functions /, Ε C°°(i7). From this it follows that for any a = Σαϊ^χΙ
€ fij^(I7) we have dxdxX^a^dx7 = dx^^a/л dx1 = 0 since dxdaf — 0
and dxdx7 = 0. We have now defined, for each coordinate chart (i/, x), an
operator dx that clearly has the desired properties on that chart. Consider
two different charts (E7, x) and (V, y) such that С/ П У ^ 0. We need to show
that dx restricted to U Π V coincides with dy restricted to U Π V, but it is
366
8. Differential Forms
clear that these restrictions of dx and dy satisfy the hypothesis of Lemma
8.36 and so they must agree on U Π V.
It is now clear that the individual operators on coordinate charts fit
together to give a well-defined operator with the desired properties. Π
Definition 8.38. The degree one graded derivation just introduced is called
the exterior derivative.
Another approach to the existence of the exterior derivative is to exhibit
a global coordinate free formula. Let ω € Q,k(M) and view ω as an
alternating multilinear map on 3t(M). Then for Xq? ^i> · · · ι Xk € X(M), define
du)(X$,X\,...,Xk)
= 5^(-1)%их0,...,^,.-.,-У*))
0<i<fc
+ Σ (-1Г+м№,^],^о,...Д,...,^,...,^)-
0<Kj<k
One can check that άω is an alternating C°°(M)-multilinear map on X(M)
and so defines a differential form of degree к + 1. By applying this formula
to coordinate fields one obtains the same local operator defined previously.
Lemma 8.39. Given any smooth map f : Μ —> N', we have that d is natural
with respect to the pull-back:
Г (*?) = d(f*V).
Proof. By Lemma 2.119 we know the result is true if 77 is a 1-form. Because
d is natural with respect to restriction, we need only prove the formula for
a differential form defined in the domain of a chart (C/,x). By linearity we
may assume that η = g dx%1 Λ ■ · · Λ dxlk since an arbitrary form on U is a
sum of forms of this type:
/*(<fe;) - f*(d(gdxh Λ - · ■ Λ dxik)) = f*(dg Λ dxh Λ · · · Λ dxik)
= d (f*g) A d (/V1) Λ ... Λ d (/V*)
= d(gof)Ad (xh of) A---Ad (xik о f)
= d{{g о f) d (xh о /) Л ■ · · Л d (xik о /)) - ά(Γη). Ώ
Definition 8.40. A smooth differential form a is called closed if da = 0
and exact if a = άβ for some differential form β.
Notice that since d о d = 0, every exact form is closed. In general, the
converse is not true. The extent to which the converse fails is a topological
property of the manifold. This is the point of the de Rham1 cohomology
Georges de Rham 1903-1990.
8.4. Vector-Valued and Algebra-Valued Forms
367
to be studied in detail in Chapter 10. Here we just give the following basic
definitions. The set of closed forms of degree fcona smooth manifold Μ
is the kernel of d : Пк(М) -> Ω*+1(Μ) and is denoted Zk{M). The set of
exact fc-forms is the image of the map d : fifc-1(M) ->■ Qk{M) and is denoted
Bk{M). Since d о d = 0, we have Bk(M) С Zk{M).
Definition 8.41. The fc-th de Rham cohomology group (actually a
vector space) is given by
In other words, we look at closed forms and identify any two whose difference
is an exact form.
If α Ε Zk(M) С Ω^(Μ), then the equivalence class that contains α is
denoted [a] and called the cohomology class of a. If / : Μ —> N is smooth
and [β] Ε Hk(N)y then it is easy to see that /*β is closed since β is closed.
Thus we obtain a cohomology class [/*/?]. Also, [/*/?] depends only on the
equivalence class of β. Indeed, if β — β( = άη, then /*β — /*β* = ά/*η. Thus
we may define a linear map /* : Hk(N) -> Hk(M) by /* [β] := [/*/?]. We
return to this topic later.
8.4. Vector-Valued and Algebra-Valued Forms
Now we consider a straightforward generalization. Let V and W be real
vector spaces so that we have Lkh(V; W) (Definition 8.1). For ω Ε L*lt(V;W)
and η Ε L^t(V; W), we define the exterior product using the same formula
as before except that we use the tensor product so that ω Λ η is an element
ofi^(V;W®W):
(ω Λ η)(υι, V2, · · ·, Vfc, Щ+ъ ^fc+2, · · ·, уш)
щ-у Σ 8βη(σ)α;(7;σι, νσ%,..., ν^) ® η{νσ{^λ), νσ{,+2),..., νσ{^ζ)).
We want to globalize this algebra. Let Μ be a smooth η-manifold and
consider the set
L*lt(TM;W) = U^t(rpM;W).
рем
This set can easily be given a rather obvious vector bundle structure. In this
setting, it is convenient to identify L^t(TpM\ W) with W®(f\k T*M), so that
our bundle be identified with the vector bundle W® (Д* T*M) whose fiber at
ρ is W® {/\k T*M). The C°° (M)-module of sections of this bundle is denoted
Qfc(M, W). Elements of Ω*(Μ, W) are called (smooth) W-valued fc-forms.
368
8. Differential Forms
We obtain an exterior product Ω*(Μ, W) x Ω'(Μ, W) -> Ω*+*(Μ, W<g>W)
as usual by (α Λ β)(ρ) := α(ρ) Λ /?(ρ).
We give an alternative definition of Λ and leave it to the assiduous reader
to show that the result is the same. Let wi,..., wm be a basis for W and
ω e Ω*(Μ, W) and η € Ω*(Μ,W). We have
m тп
ω = VJ w^o/ and 7/ = Y^ Wjff7
t-l j=l
for some ω% Ε Ω* (Μ) and rf € Ω*(Μ), where we write wjo/ rather than
wl ® ω\ etc. Then,
m m
ω /\η — Y^У^Wi <g> Wjω1 Λ r^.
t-lj 1
One can show that this definition is independent of the choices and is
consistent with the basis free definition.
We still have a pull-back operation defined as before so that if / : N -*
Μ is smooth and ω — Y^L\ WiCJ* € Ω*(Μ, W) for smooth A:-forms ω1, then
m m
i 1 г 1
is an element of ttk(N, W). We also define a natural exterior derivative
d : Ω*(Μ, W) -+ Ω^+1(Μ, W)
as follows: If ω Ε Ω* (Μ, W), then choose a basis as above and write ω =
EI^WzuA Then
m
άω := y^Wiciu/.
i ι
Κ ω = ΣΓ x WjU? where Wi = 53 wj^> *^en we must bave
m tn / тп ч m / m \
ς^-ς(ς^ρ=ς^ ς^ί ,
г 1 rPj 1 / j 1 \г 1 /
from which it follows that
m
=Σ^
α;·
г 1
and so
i ι j-i г ι i=i
8Λ. Vector-Valued and Algebra-Valued Forms
369
Thus the definition does not depend on the choices. It turns out that the
invariant definition is
du)(Xo,Xi,...,Xk)
= Y^(-iyxMx0,...X,...,xk))
0<%<k
+ Σ (-1Г+М[^,^],Х0,...^г,...Д,...,^),
0<Kj<k
where now ω(Χο,..., X^..., Xk) is a W-valued function. (A vector field X
acts on a W-valued function as Xf := df(X).)
If W happens to be an algebra, then the algebra product WxW^W
is bilinear, and so it gives rise to a linear map m: W ® W —> W. We
compose the exterior product with this map to get an exterior product
Λ : fik{M,W) x Ω*(Μ,W) -* nk+i(M,W). If ω e fi*(M,W) and η G
Ω^(Λί, W), then as above we may write
m τη
ω = Y^ Witt/ and η = ^ w^V,
г=1 j=l
for some ω1 Ε Пк(М) and rf € Ω£(Μ) and
ω Α η = Y^ 2J m (w* <8> w?) u/ Λ 7/^.
г=1 J-l
Using a dot for the multiplication, we also have
(ω /\η)(Χχ,Χ2,- . . ,ХГ5-^г+ъ^г+2, •••iXr+s)
= *Γ7αΓ7 Σ sgn(a)o;(X(7l, Χσ2,..., Χστ) · η(ΧσΓ+1, ΧσΓ+2,..., Χσν+3).
aG5fc1+fc2
In this case d is also defined as before. A particularly important case is
when W is a Lie algebra g with bracket [·,·]. Then we write the resulting
771
product Λ as [·, -]A or just [·, ■] when there is no risk of confusion. Thus if
ω, η Ε Ω1 (С/, q) are Lie algebra-valued 1-forms, then
[ω,ηηΧ,Υ) = [ω{Χ),η{Υ)) + Ш)МУ)}-
In particular, ^[ω,ω]Λ(Χ, Υ) = [ω(Χ),ω(Υ)], which might not be zero in
general!
Example 8.42. The Maurer-Cartan forms are β-valued 1-forms.
370
8. Differential Forms
8.5. Bundle-Valued Forms
It is convenient in several contexts to have on hand the notion of a differential
form with values in a vector bundle. Let £ = (£*, π, Μ) be a smooth real
vector bundle of rank r. We can consider the vector bundle L^t(TM,E)
over Μ whose fiber at ρ is Lklt(TpM, Ep). We identify L^t(TpM,Ep) with
Ep <g> /\kT*M and thus the bundle is identified with Ε ® /\kT*M.
Definition 8.43. Let ξ = (Ε, π, Μ) be a smooth vector bundle. A smooth
differential fc-form with values in ξ (or values in E) is a smooth section of
the bundle Ε ® /\kT*M. These are denoted by Ω*(Μ; Ε).
Remark 8.44. The reader should avoid confusion between Qk(M]E) and
the space of sections Г(М, /\кЕ).
Theorem 8.45. There is α natural C°°(M) module isomorphism Uk(M] E)
— ^ait(3*(^)>r(£0)· If this isomorphism is taken as an identification^ then
μ(Χ1,...,ΧΙ()(ρ) = μ(Χ1(ρ),...,Χι<(ρ))
for μ e Пк(М;Е) and Хъ...,Хк€ L^(3i(M),T(E)).
Proof. We use Proposition 6.55. The reader should check each of the
following:
Пк(М; E) Si T(E <g> /\kT*M) * Г (^(ТМ, Е))
S Г (Eom(/\kTM, Ε)} S Нот (Г (д*™) , Т(Е))
<* Нот(Л*£(М), Г(Я)) = ^4(*(М), Г(Е)). О
In order to get a grip on the meaning of fifc(M; £"), let us exhibit
transition functions. For a vector bundle, knowing the transition functions is
tantamount to knowing how local expressions with respect to a frame
transform as we change frames. A local frame for Ε ® /\kT*M can be given by
combining a local frame for Ε with a local frame for Д T*M. Let (ei,..., tr)
be a frame field for Ε defined on an open set U. We may as well take U
to also be a chart domain for the manifold M. Then any local section of
Qh(M;E) defined on U has the form
s = 2_2 aJfej ® dx
for some smooth functions a3-* = a? 4 defined in U. Then for a new local
set up with frames (Д,..., fr) and dy1 = dyn Λ · ■ · Λ dy%k {i\ < · · ■ < i*) we
have
8.5. Bundle-Valued Forms
371
for some a3-> and the transformation law
where C\ is defined by fsCfj = ej>
Exercise 8.46. Derive the above transformation law.
Note that if we write uP = ^ a^dx1, then we may write 5 in U as
s = Σ e, ® ujj for up G Qk(U).
Example 8.47, If Ε is a trivial product bundle MxV -> M, then iife(M; £")
is canonically isomorphic to Ω*(Μ; V).
Example 8,48. For an η-manifold M, the bundle map I : TM -ϊ ΤΜ
which is the identity on each fiber can be interpreted as an element of
аг{М\тм).
Example 8.49. If / : Μ —> N is a smooth map, then the tangent map
Γ/ : Τ Μ -у Τ N can be interpreted as an element of ΩΧ(Μ; f*TN), where
fTJV is the pull-back bundle (Definition 6Л8).
Now we want to define an important graded module structure on the
direct sum
Ω(Μ;£) = 0Ω*(Μ;£).
к
This will be a module over the graded algebra Ω(Μ). The action of Ω(Μ)
on Ω(Μ; Ε) is given by maps Λ : Ω*(Μ) χ Ω* (Μ; Ε) -> Ω*+*(Μ; Ε), which
in turn are defined by extending the following rule linearly:
ω1 Λ (з ® ω2) := s ® ω1 Λ ω2 for ω1 G Ω*(Μ), α;2 G ίϊ£(Μ) and 5 G Г(Я)
(with the same formula for local sections). Actually we can also define the
analogous right multiplication Λ : Ωέ(Μ\Ε) χ Ω* (Μ) -> ПЫ(М;Е),
($ ® ω1) Λ ω2 := 5 ® ω1 Λ ω2,
and then we have
rj Λ μ = {-1)Ημ Λ τ? for μ G Ω£(Μ; Д), τ? G Ω*(Μ).
If the vector bundle is actually an algebra bundle, say Λ —> Μ, then we
may turn Λ ® AT* Μ := Σρ=ο^ ® f\pT*M into an algebra bundle whose
sections can be multiplied: For ω1 G ΩΑ;(Μ), ω2 G Ω£(Μ), and 5χ, S2 G Г(Д),
define
(si ® ω1) ® (β2 Θ ω2) := si · S2 ® cj1 Λ ω2,
372
8. Differential Forms
where ■ is the product in Д. This extends linearly on (possibly locally
defined) sections:
a)s% (g> ujj ® (У2 b£sk ® ωΊ := ^ affisiSk ® ω* Λ u/,
where α* and 6* are smooth functions. We obtain a product that is natural
with respect to restriction to open sets. Prom this the sections Ω(Μ, Α) =
Г(М, А ® AT*M) become an algebra over the ring of smooth functions. We
can think of elements of Г1к(М,А) as C°°(M) multilinear maps on X(M).
Then the invariant formula for φ G Ω*(Μ, A) and φ 6 Ω^(Μ, A) is
φ*ψ(Χι,...,ΧΗ+ι)
= Ш Σ ^Ы^)Ф(^г , · - - , ΧσΗ) ' Φ(Χσ fc+1), ■ · , *σ(Λ+ί))
σ
for Χ\,... ,-Xfc+i Ε 3Ε(Μ). Depending on the context, the symbol may be
chosen to be the same as that for the product in Л, or it may just be Λ
(especially if A is commutative), or it may be some hybrid symbol.
Another important example is where A = End(E). Locally, say on
Z7, sections μι and β2 of Ω(Μ,Εηά(25)) take the form μι = ΣΑ% ® ax
and β2 = ΣΒι ® /Зг, where A; and B* are maps /7 -^ End(E'). Thus for
each χ Ε U, the Aj and Л» evaluate to give Д(ж),В»(х) 6 End(^). The
multiplication is then
(J2 Μ ® οή a(^B3® βή = Y^AiB3 ®al/\ ft,
where the A{Bj : U —> End(l£) are local sections given by composition:
AiBj : χ \-> Ai(x) о Bj(x).
Exercise 8.50. Show that Ω(Μ, End{E)) acts on Ω(Μ, Ε) making Ω(Μ, £)
a module over the algebra Ω(Μ, End(£")).
Perhaps it would help to think as follows: We have a cover of a manifold
Μ by open sets {Ua} that simultaneously locally trivialize both Ε and T*M.
Then these also give local trivializations over these open sets of the bundles
End(E) and ДГ*М. Associated with each local trivialization is a frame
field for Ε -4 Μ, say (ei,..., er), which allows us to associate with each
section σ G Vtk(MyE) an r-tuple of A;-forms συ = (a\j) for each U such
that σ — Υ^σ%υει. Similarly, a section A 6 Ωί(M,End(£,)) is equivalent to
assigning to each open set U G {Ua} a matrix of ^-forms A\j. The algebra
structure on Ω(Μ, End(i?)) is then just matrix multiplication, where the
entries are multiplied using the exterior product A\j A Bjj,
(ΑυΛΒυ)ί = ΣΑ\ΛΒ].
Σ
8.6. Operator Interactions
373
The module structure of the above exercise is given locally by ац ι-> Ац Л
σ£/. Where did the bundle go? The global topology is now encoded in the
transformation laws, which tell us what the same section looks like when we
change to a new frame field on an overlap υαΠΐ7β. In this sense, the bundle
is a combinatorial recipe for pasting together local objects.
Recall that with the product [A, B] := А о В — В о A, the bundle
Rom(E,E) is written as Ql(E) rather than End(iJ), and then our general
construction gives Ω(Μ, gl{E)) an algebra structure, whose product is
denoted [0, φ]Α or something similar.
8.6. Operator Interactions
The Lie derivative acts on differential forms since the latter are, from one
viewpoint, alternating tensor fields. When we apply the Lie derivative to a
differential form, we get a differential form, so we should think about the
Lie derivative in the context of differential forms.
Lemma 8.51, For any Χ Ε X(M) and any f Ε Ω°(Μ), we have Cxdf =
dCxf.
Proof. For a function /, we compute as
d
IMM-Qtf™)™-*
df(Ttf-Y)
о
d_
dt
Υ{{ψ?)Ί)
0
=Y(jtl (^r/)= nCxf)=<£χΜγϊ>
where Υ Ε Χ(Μ) is arbitrary. D
We now have two ways to differentiate sections in Ω(Μ). First, there is
the Lie derivative Cx : Ω2(Μ) —> Ω*(Μ), which turns out to be a graded
derivation of degree zero,
(8.7) Cx {α Λ β) = Cxa Λ β + а Л £χβ.
We may apply £χ to elements of Ω(ί7) for CicM, and it is easy to see that
we obtain a natural derivation in the sense of Definition 8.33.
Exercise 8.52. Prove the above product rule.
Second, there is the exterior derivative d which is a graded derivation of
degree 1. In order to relate the two operations, we need a third map, which,
like the Lie derivative, is taken with respect to a given field Χ Ε £(Μ).
This map is defined using the interior product given in Definition 8.12 by
letting
(8.8) irfi Xw)(p) :=ίχρωρ{Χ1{ν),...,Χι 1{ρ)).
374
8. Differential Forms
Alternatively, if ω G Ω,%(Μ) is viewed as a skew-symmetric multilinear map
from Χ (Μ) χ···χϊ (Μ) to C°° (M), then we simply define
ίχω(Χχ,..., Xi-ι) := ω(Χ, Χι,..., -Xi-i).
By convention %xf = 0 for / G C°°(M). We will call this operator the
interior product or contraction operator. This operator is clearly linear
over R. Notice that for any / G C°°(M) we have
if χω = fixu, and ixdf = df(X) = Cxf.
Proposition 8.53. %χ is α graded derivation ofil(M) of degree —1:
ix(a Λ β) = (ίχα) Λ β + {-1)ка Λ (ίχβ) for α Ε Ω* (М).
It is the unique degree —1 graded derivation ofQ(M) such that ixf = 0 for
/Gfi° (M) and
гх0 = Θ(Χ) for веП1 (Μ) and Χ € £ (Μ).
Proof. This follows from Proposition 8.13. D
Actually, %χ is natural with respect to restriction, so it is a natural
graded derivation in the sense of Definition 8.33. Formulas developed for
the interior product in the vector space category also hold for vector fields
and differential forms. For example, if 0χ,..., θ^ € Ω1 (Μ) and X EX (M),
then
к
ΐχ{θ1Λ·.·Λθίζ) = Σ{-1)Ι<+1θ£{Χ)θ1Λ···ΛθϊΛ.··Λθίζ.
1=1
Notation 8.54. Other notations for ίχω include Χ\ω and (Χ,ω). These
notations make the following theorem look more natural:
Theorem 8.55. The Lie derivative is a derivation with respect to the pairing
(Χ,ω) H> (Χ,ω). That is,
Cx(iYu) = %£χγω + ίγ€χω,
or in alternative notations,
£χ(Υ\ω) = (£χΥ)\ω + Υ\(εχω),
£χ(Υ,ω) = (£χΥ,ω) + (Υ,£χω).
Proof. Exercise. D
Now we can relate the Lie derivative, the exterior derivative and the
contraction operator.
Theorem 8.56. Let X G Хм- Then we have Cartan's formula,
(8.9) Cx = d ο %χ + ix о d.
8.7. Orientation
375
Proof. Both sides of the equation define derivations of degree zero (use
Proposition 8.35). So by Lemma 8.36 we just have to check that they agree
on functions and exact 1-forms. On functions we have %xf = 0 and %xdf =
Xf = £xf so the formula holds. On differentials of functions we have
(άοίχ + ίχο d)df = (rf о ix)df = dCxf = Cxdf,
where we have used Lemma 8.51 in the last step. D
As a corollary, we can extend Lemma 8.51:
Corollary 8.57. d о Cx = €χ ο d.
Proof. We have
ά£χα = d(dix + %χά)(α) = άίχάα
= άίχάα + ίχάάα = (Cx ο d) α. D
Corollary 8.58. We have the following formulas:
(i) i[xy] = £χ °iy -iyoCx;
(ii) C/χω = fCxu + df Λ ίχω for all ω € Ω(Μ).
Proof. We leave (i) as Problem 9. For (ii), we compute:
CfXu = ζ/χάω + d{ifXu) = ifXdu + d{f%xu)
= / ίχάω + df Λ ίχω + fd (ίχω)
= f (ίχάω + d (ίχω)) + df Λ ίχω = f (Ιχω + df Λ ίχω, D
8.7. Orientation
A vector bundle Ε —> Μ is called oriented if every fiber Ep is given a
smooth choice of orientation. There are several equivalent ways to make a
rigorous definition:
Proposition 8.59. Let Ε —> Μ be a rank к real vector bundle with typical
fiber V. The following are equivalent:
(i) There is a smooth global section ω of the bundle f\k E* = ^alt(^) ~^
Μ such that ω is nowhere vanishing.
(ii) There is a smooth global section s of the bundle Д ВчМ such
that s is nowhere vanishing.
(iii) The vector bundle has an atlas of VB-charts (local trivializations)
such that the corresponding transition maps take values in GL+(V),
the group of positive determinant elements o/GL(V). This means
that the standard GL(V) -structure on Ε -ϊ Μ can be reduced to a
GL+iy)-structure (refer to Chapter 6).
376
8. Differential Forms
Proof. We show that (i) is equivalent to (iii) and leave the rest as an easy
exercise. Suppose that (i) holds and that ω is a nonvanishing section of
L^t(E), Now fix a basis (ei,... ,e*) on V and recall that with this basis
fixed, each VB-chart (f7, φ) for Ε —>■ Μ corresponds to a local frame field.
Indeed, we let е$(р) := </>-1(р, е;). The transition maps between two charts
will have values in GL+CV) exactly when the matrix function that relates
the corresponding frame fields has positive determinant (check this). Given
a VB-atlas we construct a new atlas. We retain those VB-charts (υ,φ)
whose corresponding frame field ei,..., е& satisfies ω(βι,..., e^) > 0. For
the charts for which ω(βι,..., ejt) < 0, we replace ei,..., e* by — ei,..., e*,
and the resulting chart will be included in our new atlas. Now if ei,... ,е*
and /i,..., fk are two frame fields coming from this atlas, then /i = Σ С3ге3
and
ω(/ι,..., fk) = (det С) ш{еъ ..., ек).
We conclude that det С > 0.
Conversely, suppose the vector bundle has an atlas {{ΙΙ^φα)} taking
values in GL+(V). We will use the frame fields coming from this atlas to
construct a nowhere vanishing section of L^t(E) —> M. If ei,... ,е* and
/i,..., fk are two frame fields coming from this atlas, then let /*,..., fk be
dual to /i,..., fk and consider /* Λ · · · Л fk. We have /i = Σ ^1ез апс^
(/X Λ · · · Λ /*) (еь ..., ек) = det C> 0.
For each chart (ί/α, φα) in our VB-atlas, let /f,..., /£ be the corresponding
frame field and let (/^,..., /£) be the dual frame field. Then let {pa} be a
partition of unity subordinate to the cover {Ua}. Let
a
Then, ω is nowhere vanishing. To see this let ρ G Μ and suppose that
ρ Ε Uβ for some chart (ΙΙβ,φβ) from the GL+(V)-valued atlas. Then
^(/f,...,/f)(p)-X;Pa(p)detC^(p)>0,
where Οβα is the matrix that relates /f,..., /£ and /f,..., /£. D
Definition 8.60. If any one (and hence all) of the conditions in Proposition
8.59 hold, then Ε —> Μ is said to be orientable. A VB-atlas that satisfies
(iii) will be called an oriented atlas.
If Ε —► Μ is orientable as above, then two nowhere vanishing sections
of Lklt(E), say ωι and ω2, are said to be equivalent if ωι = fu<i, where /
is a smooth positive function. We denote the equivalence class of such a
nowhere vanishing ω by [ω].
8.7. Orientation
377
Definition 8.61. An orientation for an orientable vector bundle of rank
к is an equivalence class [ω] of nowhere vanishing sections of Д E*. If such
an orientation is chosen, then the vector bundle is said to be oriented by
м-
Notice that if we have two oriented VB-atlases on a vector bundle, then
we know what it means for them to determine the same GL+(V)-structure.
This was the notion of strict equivalence from Chapter 6. The next exercise
shows that the notion of a reduction to a GL+(V)-structure is equivalent to
the notion of an orientation as we have defined it.
Exercise 8.62. Recall the construction of a nowhere vanishing section ω in
the proof of Proposition 8.59. Show that the class [ω] does not depend on the
partition of unity used in the construction. Show that if two oriented VB-
atlases determine the same GL+(V)-structure, then the constructed sections
are equivalent and so determine the same orientation. Conversely, show
that an orientation as we have defined it determines a unique reduction to
a GL+(V)-structure on the vector bundle.
Let Ε —> Μ be oriented by [ω]. A frame (νι,...,ν*) of fiber Ep is
positively oriented (or just positive) with respect to [ω] if and only if
ω(ρ){νΐι · ■ · >^A;) > 0. This condition is independent of the choice of
representative ω for the class [ω].
Definition 8.63. Let π : Ε —> Μ be an oriented vector bundle. A frame
field (/i,..., fk) over an open set U is called a positively oriented frame
field if (/i(p),..., fkip)) Is a positively oriented basis of Ep for each ρ EU.
Exercise 8.64. Let π : Ε —^ Μ be an oriented vector bundle. Show that
if Μ is connected, then there are exactly two possible orientations for the
vector bundle.
Exercise 8.65. If πχ : Ει -> Μ and π2 : #2 —> Μ are orientable, then so is
the Whitney sum πι Θ π2 : Ε\®Ε2-ϊ Μ.
Definition 8.66. A smooth manifold Μ is said to be orientable if TM
is orientable. An orientation for the vector bundle TM is also called an
orientation for M. A manifold Μ together with an orientation for Μ is
said to be an oriented manifold.
Definition 8.67. An atlas {(Ua, x<*)} for Μ is said to be an oriented atlas
if the associated frame fields axe positively oriented. If this atlas is positively
oriented with respect to an orientation on Μ (an orientation [ω] of TM),
then we call {(C/a,xa)} a positively oriented atlas.
It follows from the definitions that an oriented atlas induces an orientar
tion for which it is a positively oriented atlas. For this reason, a choice of
378
8. Differential Forms
oriented atlas is equivalent to a choice of orientation, and so one often sees
an orientation of an orientable manifold defined as simply being given by a
choice of oriented atlas.
Now let Μ be an η-manifold. Consider a top form, i.e. an reform
vj G Ωη(Μ), and assume that w is nowhere vanishing. Thus Μ must
be orientable. We call such a nonvanishing vj a volume form for M,
and every such volume form obviously determines an orientation for M. If
φ : Μ —> Μ is a diffeomorphism, then we must have that φ*τσ = 5w for
some δ € C°°(M), which we will call the Jacobian determinant of φ with
respect to the volume element w\
Clearly Jw(^) is a nowhere vanishing smooth function.
Proposition 8.68. Let (M, [w]) be an oriented η-manifold. The sign of
Jw{y>) is independent of the choice of volume form vj in the orientation
class [vj].
Proof. Let vj1 6 Ωη(Μ). We have w — avj1 for some function α that is
never zero on U. Furthermore,
J(ip)w = {φ*νσ) — (ao{p)(ip*mr) = (αοφ)3ΤΌι{φ)χυί = 1*7,
a
and since ^^ > 0 and vj is nonzero, the conclusion follows. D
Let us consider a very important special case of this: Suppose that ψ ;
U -¥ U is a diffeomorphism and i/cRn, Then letting ^o = du1 A · - · Adun
we have for any χ G Uy
φ*τπο(χ) = ip*dul A · · · Λ (p*dun(x)
—^—(x) J шо(х) = J4>{x)wq(x).
So in this case, JmQ{4>) is just the usual Jacobian determinant of φ. More
generally, let a nonvanishing top form w be defined on Μ and let wf be
another such form defined on N. Then we say that a diffeomorphism
φ : Μ —> N is orientation preserving (or positive) with respect to the
orientations determined by vj and vj1 if the unique function Jw^ such that
φ*ζυ' = Jw<U3*vj is strictly positive on M.
Exercise 8.69. An open subset of an oriented manifold Μ inherits the
orientation from Μ since we can just restrict a defining volume form. Show
that a chart (f/,x) on an oriented manifold is positive if and only if
8.7. Orientation
379
χ : U —> χ (U) is orientation preserving. Here, χ (U) inherits its
orientation from the ambient Euclidean space with its standard orientation.
We now construct a two-fold covering manifold Mor for any
manifold M. The orientation cover will itself always be orientable. Recall that
the zero section of a vector bundle over Μ is a submanifold of the total
space diffeomorphic to M. Consider the vector bundle whose total space is
f\nT*M and remove the zero section to obtain
(ДПГ*М)Х := (ДПТ*М) \{zero section}.
Define an equivalence relation on (/\nT*M)x by declaring v\ ~ v^ if and
only if v\ and ν<χ are in the same fiber and if v\ = αν<χ with а > 0. The space
of equivalence classes is denoted Mor and we will show that it is a smooth
manifold. Let q :(ДПТ*М)Х -► Mor be the quotient map and give Mor the
quotient topology. There is a unique smooth map π0Γ making the following
diagram commute:
(/\nT*M)x ^(Mor)
^"\^ TTor
Μ
It is easy to see that for each ρ G M, the set π^.1(ρ) contains exactly two
elements, which are the two orientations of TPM. We give the set MOT a
smooth structure. First let [μο] be the standard orientation of Rn and choose
a fixed orientation reversing linear involution ro : Mn —> Mn. Let {(?7α,χα)}
be an atlas for M. By composing some of the charts with ro and adding
the resulting charts to the atlas, we may suppose that {(Ua^a)} has the
property that for every chart (t/, x) in the atlas there is a chart (f/, y) in the
atlas with the same domain such that у о х-1 is orientation reversing. Let
us say that such an atlas is "balanced". (The maximal atlas is obviously
balanced.) Now for each chart (E7, x), where χ = (ж1,..., xn), define a map
фх:х(и)-+Мот by
фх(и) := [^(x"1^)) Λ ... Λ dxn(x-\u))] G M°\
Because we have assumed that the atlas is balanced, it is easy to see that
each element of M0T is in the image of some фх. The фх are injective and
so are bijections onto their images. These maps are local parameterizations
of MOT and the inverses of these maps are charts on Mor. Let us denote the
chart arising from (i/,x) by (f/,x). Thus U = φχ(ϋ) and χ = φ~λ. In this
way the balanced atlas {(Ua) xa)} gives an atlas on Mor. We must check that
the overlaps are smooth. So let υαΠΐ/β φ 0 and consider x^ox"1 = ЩофХа.
For uEXq (Ua Π Uβ) and w = Χβ ο x^l(u) we have
dxj^x"1^)) Л .·. Л dx^{x~l{u)) = Xdxfa^iw)) A · · ■ Λ άχηβ(χ^ι{νύ))
380
8. Differential Forms
for some λ > 0. Indeed, we must have λ = det(D(xa о х^1)(г/)). Then,
x/3 ° Фх*Ы = χβ ([da£(xa(u)) Λ ■ ■ · Λ dxJJ(xa(«))])
= χβ {[άχβ{χβ(ίν)) Λ · · · Λ (^(х/зН)])
= гу = х^ох^1(гб).
Thus х/з οχ"1 = χ^ οχ"1 on xa(f/aΠ t/β). Note that for each α and β the set
Χα(ίΛ* Π Uβ) consists of exactly those connected components of χα (ϋα П ΙΙβ)
on which det (-D(xp о χ"1)) is positive. Thus χα(ί7α Π E/jg) is open. One may
check that the topology induced by this atlas coincides with the quotient
topology and that the quotient map is smooth. Also, note that if ϋαΓ\ϋβ φ 0,
then det (ΰ(χβ οχ"1)) > 0 and so our atlas is oriented! We thus have a
canonical orientation of Mor.
For each admissible chart (17, x), we obtain a section U -¥ MOT given by
p^ [ώ^Λ-Λώ»].
It is easy to see that such sections are smooth. Prom the existence of these
sections, one surmises that the connected components of the chart domains
are evenly covered and so ποτ : MOT —> Μ is a two-fold covering map (but
MOT may not be connected). This two-fold covering map is called the
orientation covering map. The space Mor itself is called the orientation
(double) cover.
Exercise 8.70. Let R+ be the multiplicative Lie group of positive real
numbers. Show that i?+ acts freely and properly on (ДПТ*М)Х and that
the quotient manifold is Mor. Show that the canonical orientation on Mor
can be described as follows: Each у G Mor is an orientation of Τποτ^Μ.
Since Ty7Tor : TyMor —> ΓποΓ(ν)Μ is an isomorphism, we may transfer the
orientation у on Τποτ^Μ to an orientation for TyMOT. This gives a canonical
orientation on each fiber of TMOT which agrees with the orientation derived
from the oriented atlas described above. How can we get a smooth global
nonvanishing top form that induced this orientation?
Exercise 8.71. A manifold Μ is orientable if and only if MOT is
disconnected. If Μ is oriented, then Mor has exactly two components and an
orientation of Μ corresponds to a choice of connected component of Mor.
8.7.1. Orientation induced on a boundary. Now let Μ be a manifold
with boundary. According to Problem 15, we can consider vector bundles
over M. The definitions of orientable and orientation make sense for M.
Here we wish to consider orientations of dM. If [ω] is an orientation for
an orientable vector bundle π : Ε —> Μ, then [Нам] ^s an orientation on
E\dM -> dM where E\dM = π"1(9Μ). In particular, if Μ is oriented by
8.7. Orientation
381
[ω], then we may obtain an orientation on TM\dM —> dM. But what we
would really like to obtain at this time is an orientation on dM. In other
words, we need an orientation on the bundle T(dM). First notice that by our
conventions, an atlas for Μ consists of charts that take values in half-spaces
of the form
pn
Exercise 8.72. Consider vp G TPM for some boundary point ρ G 9M.
Let (?7, x) be a half-space chart containing ρ and taking values in Мд>с and
let (V, y) be another chart containing p, but taking values in M™>d. Then
dxody-1 maps M™<0 diffeomorphically onto K™<0· Thus \od~x.(vp) < 0 if and
only if μ ο dy(vp) < 0. Similarly, λ ο dx(vp) > 0 if and only if μ ο dy(vp) > 0.
In light of the exercise above, the following definition makes sense:
Definition 8.73. Let ρ G dM. A vector vp G TMp С ТМ\Ш is called
outward pointing if in some R^>c-valued chart (U, x) we have Xodx{vp) <
0. A smooth section χ of TM\dM is called outward pointing if χ(ρ) is
outward pointing for each p. Inward pointing is defined analogously.
To clarify the situation, let us consider the special case of an M^fe<0-
valued chart (i7,x) for fixed к G {1,2, ...,n} (thus λ = —uk). lip e U,
then vp is outward pointing exactly when dxk(vp) > 0. In particular, ^|
is outward pointing. For an R^A>0-valued chart (C/,χ), the reverse is true;
vp is outward pointing exactly when dxk(vp) < 0. We shall see that
is special, so keep the corresponding criterion dxl(vp) > 0 in mind.
η
^<0
R«l<0
L·,
fei-.* 1
Figure 8.2. Outward pointing
Notice that the notion of outward pointing on dM does not depend on
Μ being orientable (see Figure 8.3). In fact, we have the following:
Lemma 8.74. Outward pointing sections of TM\dM always exist
382
S. Differential Forms
outward
Figure 8.3. Outward at boundary of Mobius band
Proof. We may assume that we are dealing with, an R™i<0-valued atlas
{(C/Q,xa)} for M. Let {pa} be a partition of unity subordinate to {Ua}.
Define
a a
Then for ρ Ε дМ we have
д
Χ(ρ) = ΣΡα(ρ)
дх\
To see if χ(ρ) is outward pointing, let (ίΤ/з, х/з) be a given R"1<0-valued chart
from the atlas. Then
^(χ(ρ)) = Σ*(ρ)^(^γ| )>0·
since pa(p) > 0 for all a, we have Ρα(ρ) > 0 for at least one a, and
V(^(p))>Oforalla. D
In the case that Μ is oriented we can use the orientation plus the notion
of outward to define an orientation on the boundary. Let (Μ, [ω]) be an
oriented smooth manifold with boundary and suppose that χ is an outward
pointing section of TM\dM. Let ω G [ω]\ then we define ίχω £ Ωη_1(<9Μ)
by
ιχω(ρ)(υ2,... ,un) = ω(ρ)(χ(ρ), tfc, · ■ ■»vn).
It is clear that ίχω is nowhere vanishing. If also ω\ Ε [ω], then it is easy to
see that ίχω = /&χωι for a positive function /. Thus [ίχω] depends only on
[ω] and provides an orientation on dM,
Definition 8.75. The orientation on dM defined above is called the
induced orientation.
It follows that if (Д, Д,..., /n) is a positively oriented frame field on
an open set U С Μ with nonempty intersection with dM and if Д(р) is
8.7. Orientation
383
outward pointing for all ρ G U Π дМ, then (/2,..., fn) restricted to U Π дМ
is a positively oriented frame with respect to the induced orientation on dM.
The case η = 1 needs some interpretation. Here dM is a discrete set
of points dM = {pi,... ,Pfc}, and an orientation is an assignment of +1 or
-1 to each point. For pi € dM, we assign +1 if ω(ρί)(χ(ρ)) > 0 for some
outward pointing vector χ(ρ).
If every chart in an atlas takes values in a fixed half-space Мд>с, then
we say the atlas is R™>c-valued. It is not true generally that an oriented
η-manifold has an atlas of positively oriented charts with values in a fixed
half-space. However, it is almost true:
Lemma 8.76. If Μ is an oriented n-manifold with nonempty boundary and
n>2, then there is an atlas of positively oriented charts.
Proof. If ([/, x) is not positively oriented, then replace χ = (χ1, χ2,..., χη)
by у — (у1, у2,..., уп) := (ж1, —χ2,..., хп) to obtain a positively oriented
chart. Notice that this does not work if η = 1. D
Exercise 8.77. Show that although, as a manifold with boundary, the
interval Μ — [0,1] is oriented in the standard way, there is no atlas consisting
of positively oriented R*1<0-valued charts. Trace the problem to the fact
that we have defined an atlas for a manifold with boundary as having charts
with values in a fixed half-space (in this case R^i<0 ). Hint: There are
no positively oriented charts containing 0. Changing to K^i>0 pushes the
problem to the other endpoint.
This last exercise exhibits a fact that is an annoyance if one wants to
work with an atlas taking values in a fixed half-space such as the ever popular
upper half-space Ш^п>^ It is an issue often overlooked in the literature.
Definition 8.78. A nice chart on a smooth manifold (possibly with
boundary) is a chart (t/,x) such that x(U) = R£1<0 if U Π дМ ф 0 or where
τ{ϋ) = Rn if U Π дМ = 0.
Lemma 8.79. The following assertions hold:
(i) Every smooth manifold has an atlas consisting of nice charts.
(ii) Every oriented smooth manifold without boundary has an atlas
consisting of positively oriented nice charts.
(iii) Every oriented smooth manifold with boundary of dimension η > 2
has an atlas consisting of positively oriented nice charts.
Proof, (i) If (£7, x) is a chart with range in the interior of the left half-space
M"1<0, then we can find a ball В inside x(?7) in R£i 0 and then we form a
new chart on х_1(Б) with range B. But a ball is diffeomorphic to Rn (by
384
8. Differential Forms
an orientation preserving diffeomorphism). If (f7,x) is a chart with range
meeting the boundary of the left half-space R£i<0, then we can find a half-
ball B- in RlJx^n with center on R*tl n. Reduce the chart domain as before
to have range equal to this half-ball. But every half-ball is diffeomorphic to
the half-space R™i<0 so we can proceed by composition as before.
(ii) For this, we repeat the procedure of (i) and notice that since Rn is
diffeomorphic to В by an orientation preserving diffeomorphism, the new
nice chart will be positively oriented if (i/,x) is positive.
(iii) If (ί7,χ) is a chart with range in the interior of the left half-space
R£i<0, we proceed as in (ii). If (C/,χ) is a chart with range meeting the
boundary of the left half-space R"i<0 that is already positively oriented,
then we proceed as in (i). If ({/, x) is not positively oriented, then replace
χ = (χι,χ2,...,χη) by у = (y\y2,...,yn) := (хг,-х2,...,хп) as in the
proof of Lemma 8.76 and proceed as before. D
Exercise 8.80. If χ = (ж1,..., xn) is an oriented R™1<0-valued chart on a
neighborhood of ρ on the boundary of an oriented manifold with boundary,
then the vectors -^ > ■ · · > gfn form a positive basis for TpdM with respect
to the induced orientation on dM and thus (x2,..., xn) restricts to dM to
give a positively oriented chart.
8.8. Invariant Forms
Throughout this section G is an η-dimensional Lie group.
Definition 8.81. An element ω € Q,k(G) is called left invariant if L*ui - ω
for all g e G.
It is easy to see that a left invariant fc-form is determined by its value
uj{e) at the identity element. Now suppose that cj1,...,^71 are invariant
1-forms such that cj1(e),... ,u;n(e) is a basis of T£G. The ω1,... ,wn are
independent everywhere (why?). If Χχ,..., Xn is the frame field dual to
ω1,... ,α/\ then each Хг is a left invariant vector field as may be easily
checked. By definition, a left invariant form satisfies
ω(χ)(νχ}..., vk) = u>(gx)(TLgvu ..., TLgvk)
for all χ 6 Μ and g EG. This would make sense even if ω were not smooth,
but ω(Χΐ)... ,Xk) is not only smooth but constant, so any left invariant
form must be smooth after all. Every left invariant fc-form ω can be written
as
(8.10) ω= Σ <Hi...ikuh Λ···Λ^
1<ύ<—<ik<n
for unique constants α^...^.
8.8. Invariant Forms
385
The exterior derivative of a left invariant form is left invariant since for
any J Ε G, we have
L*<L· = dL*u = duj.
Furthermore for any left invariant vector fields Χ, Υ we have
(8.11) άω(Χ, Υ) = Χω(Υ) - Υω(Χ) - ω([Χ, Υ])
(8.12) = -w([X,Y]),
and so in particular
(8.13) du\e(v,w) = -Lje{[v,w]),
for any v,w £ g.
A form is called right invariant if R*u = ω for all g Ε G. Of course, if ω
is right invariant, then so is άω. If inv : χ \ч> χ~λ denotes the inversion map
on G, then
inv о Rg = Lg-i о inv,
and so
R*g о inv* = inv* о L*-i.
As a consequence, if ω is left invariant, then
R* о inv*cj = inv* о L*iu = inv*cj,
so inv*cj is right invariant. Similarly, if ω is right invariant, then inv*cj is
left invariant.
Lemma 8,82. Teinv = — id : q ->■ 0.
Proof. Let υ € 0. Then the curve t н+ exp(tv) has tangent v. Thus
(exp(tv))"1 has tangent Teinv · v. But (exp(tv))~ = exp(—tv) so we must
have Teinv -v = —v. Ώ
Proposition 8.83. Let inv : χ ι-> х~г be the inversion map on G.
(i) Ifue ttk(G), then (inv*u;)e = {-l)kue.
(ii) If ω is left and right invariant, then άω = 0.
Proof, (i) It suffices to assume that ω = fun A · · ■ Λ ω4 for certain 1-forms
wn,..., ω** that may be taken to be left invariant. But the result will follow
if we can show that (inv*cj)e = — lue for any 1-form. So let ν £ 0. Then, by
Lemma 8.82,
(inv*o;)e (v) = ше (Teinv · v) = -ue(v).
(ii) From (i) we have
(inv*u;)e = (-1)4-
386
8. Differential Forms
If ω is left and right invariant, then both inv*cj and ω are left invariant and
this continues to hold globally:
inv*td = (-l)fccj;
άω is also left and right invariant, so
inv*du> = (-l)fe+1d£J.
On the other hand,
inv*du) = dinv*u; = (-l)kdu,
so (-l)k+1dw = (-l)kdu and then άω = 0. D
Corollary 8.84. If G is abelian, then g is abelian.
Proof. If G is abelian, then every left invariant form is also right invariant,
so άω = 0 for all left invariant forms ω. But then by equation (8.13), for
any v, w € g we have ωβ([υ, w]) = 0 for all uje € g*. Thus [v, tu] = 0 for any
г;,ги. D
If Xi,..., Xn are left invariant vector fields with Xi(e),..., Xn(e) dual
to the basis u;1(e),... ,cjn(e), then
[Х^е),Ха(е)] = ^4хк(е) forl<i,j<n,
A;
where ώ· are the structure constants from Definition 5.56. We also have
[X„ X3] = J2k °ijXk for 1 < г, j < n, so by the equation above and equation
(8.13) we have for г < j,
dw*^,*,) = -ω*([*,Χ,·]) = -cfc.
Since dwk(Xi,Xj) gives the components of е&Д we have
(8.14) do;* = - Σ 4ω* л ^ = ~\ Σ 4'ω* Λ "j>
which are called the equations of structure, or the structural
equations.
We now use the concept of Lie algebra-valued forms and the
product [ , ]A introduced earlier. Recall the left Maurer-Cartan form uq- If
ω1,..., ωη is a basis of left invariant 1-forms dual to Χι,..., Xn, then
(omitting tensor product signs),
η
uG = ^2Хг(е)ш\
8.8. Invariant Forms
387
Indeed, if Xg = Σ2-ι otXiid) € TaG, then Χ = Σ? ι a% is the unique left
invariant vector field such that X{g) = Xg and we have
(£xt(e)u/W) Σα'ΧΜ
4=1 ' ι 1
= X(e) = TLg-i-X(g)=uG(Xg).
Moreover,
άωα = J2 Хк{е)амк = - £ Хк(е) (у, 4ω' Λ ω')
к 1 fc-1 Ч<7 '
=-
££с*Х*(е)и/ЛиА
fc 1 г<3
On the other hand we have
1^g^g]a =
г
Л 1 г 1
η
,3 1
ij=l fc к i<3
so that an alternative and concise form of the equations of structure is the
single equation
(8.15) άω0 = -^[ω0,ω0]Α.
Using what we learned at the end of Section 8.4, this equation may be
written as
άωα{ΧίΥ) = -[ωα{Χ)1ωα{Υ)]
for any I,7e X(G). If G is a matrix group, then the structural equation
can be written as
Indeed, if we abbreviate ouq to just ω, then we have {άω)% — άω1· and
[uG(X),uG(Y))) = (ω(Χ)ω(Υ)-ω(Υ)ω(Χ)))
-ω{Χ)ίω{Υ))-ω{Υ)ίω{Χγό
= (ωίΛωή{Χ,Υ).
Equation (8.15) is also called the Maurer-Cartan equation.
388
8. Differential Forms
Problems
(1) Let Sn denote the group of permutations of the set {1,2,..., N}. Now
let Gk,e be the subgroup of Sk+t which consists of permutations that
leave the sets {1,..., к} and {к + 1,..., к + £} each invariant. A cross
section of Gk,e is a subset К of Sk+e that contains exactly one element
from each coset in Sk+i/Gk,i- Show that for any such cross section we
have
wAf?(vli...,vfc,vfc+lj...Jvw)
= Σ sgn^Mv^,..., ν^)77(ν^+1,..., νσ^+£).
σβΚ
Also, show that the set of all (&, ^)-shuffle permutations is a cross section
of Gkii.
(2) Show that if vi,..., v* € V, then v\ A · · · Avk φ 0 if and only if v\,..., Vk
axe linearly independent.
(3) Let /i,..., fn be smooth functions on an open set in an η-manifold. Let
ρ be in their common domain. Then df\ A · · - Λ dfn is nonzero at ρ if and
only if /i,..., fn agree with the coordinate functions of a chart whose
domain is a neighborhood of p.
(4) (a) Let ν G f\hV. Show that if υ A w = 0 for all w € /\n~kV, where
η = dim V, then ν = 0.
(b) Using part (a), show that more generally, if ν A w = 0 for all w G
f\m V, where m < η — fc, then ν = 0. [Hints: If ν A w = 0 for
all w Ε Λ™ν> then ν Л (гиЛх) = 0 for all w G f\mV and all
ж G /\п-к-т V. Elements of the form wAxas above span Дп к V.]
(5) Prove Proposition 8.23.
(6) Show that the sphere is orientable.
(7) Prove (i) of Proposition 8.4.
(8) Prove Proposition 8.30.
(9) Prove equation (i) of Corollary 8.58.
(10) Prove Cartan's lemma: Let к < η — dim Μ and let ωχ,...,^ be 1-
forms on Μ which are linearly independent at each point. Suppose
that there are 1-forms θχ,..., 0fc such that
к
y^Oj Αωί = 0 (identically).
г=1
Problems
389
Then there exists a symmetric к х к matrix of smooth functions (Aij)
such that
к
0i — ^ AijUj for i = 1,..., A;.
3=1
(11) Let Μ = R3\{0} and let
xdy Adz + ydz Adx + zdx Ady
Ш= (ж2+у2 + 22)3/2 ·
Find άω and determine whether ω is closed and if so, whether it is exact.
Find the expression for ω in spherical coordinates.
(12) Show that every simply connected manifold is orientable.
(13) (a) Let V and W be vector spaces with dimV = η and dimW =
m. Show that if A G L(V, W), then there is a unique map Лд :
/\kY -* /\kW such that Λλ(ιιιΛ· ■ -Ли*) = Av\A- -AAv^ whenever
νι,.,.,υ* e v.
(b) Show that if ei,..., en is a basis for V and /i,..., fm is a basis for
W and if Aei = ^ai/j> then for 1 < i\ < · · · < ifc < η we have
Лд (^ Л · · - Л eik) = ^ а£"'£/л Л · · · Λ /Λ,
l<ji<-<jfc<m
where
is the fc χ fc minor determinant of the matrix (aj) given by
a€Sk
(14) Finish the proof of Proposition 8.13.
(15) Prove Proposition 8.35.
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Chapter 9
.ntegration and Stckes'
Theorem
Let Μ be a smooth η-manifold possibly with boundary dM and assume that
Μ is oriented and that dM has the induced orientation (Definition 8.75).
Definition 9,1. The support of a differential form α Ε Ω(Μ) is the closure
of the set {p e Μ : a(p) φ 0} and is denoted by supp(a). The set of all
fc-forms that have compact support is denoted Ω*?(Μ), and the set of all
fc-forms with compact support contained in U С Μ is denoted by Ω^(Ϊ7).
Let us consider the case of an η-form a on an open subset U of Rn. Let
(u1,..., un) be standard coordinates on J7. We may write a = a du1 Λ · ■ * Λ
dun for some function a. If α has compact support in C/, we may define the
integral J^ a by
I a= I a du1 Λ ■ ■ · Λ dun
Ju Ju
:= / a(u) (hi1 ■ · · ditn,
Ju
where this latter integral is the Riemann (or Lebesgue) integral of α(·). For
this we extend α(·) by zero to all of Rn and integrate over a sufficiently
large closed η-cube containing the support of α(·) in its interior. We could
have written l^it1 ■ ■ ■ dun\ instead of du1 · · · dun to emphasize that the order
of the du% does not matter as it does for du1 A · · · Λ dun. If U is an open
subset of the half-space Кд>с, then we define fv a by the same formula. If
φ : V -» U is an orientation preserving diffeomorphism of open sets in Rn,
then det D<f> > 0. Let w1,..., un denote standard coordinates on f/, and let
391
392
9. Integration and Stokes7 Theorem
v1,... )Vn denote standard coordinates on V. Then by the classical change
of variable formula,
j a= f a(u)du1---dun= f α ο φ(ν) \detD<f>\ dvl · · -dvn
Ju Ju Jv
= I ao<t)(v)detD(l>dv1---dvn
Jv
= / ao(f)(v)detD<t)dvl A---Advn= f φ*α.
Jv Jv
So
(9.1) / a = f φ*α.
Ju Jv
Next consider an oriented η-manifold Μ without boundary and let a E
Ωη(Μ). If a has compact support inside U for some positively oriented
chart ({7, x), then x"1 : x(C7) —> U and (x-1)*a has compact support in
x(C/) С Rn. We define
!a-S
JU Jx
(CO
The change of variables formula (9.1) shows that this definition is
independent of the positively oriented chart chosen. In fact, if (y, V) is another
chart and a has support inside U Π V, then (x_1)*a has support inside
x(U Π V) and (y-1)* ot has support in y({7 Π У). Thus since χ о у"1 is
orientation preserving, we have
/ (ι>=/ (х-1Г«=/ (xoy-yCx-^a
Jx{U) J%{Uf\V) Jy(UnV)
= [ (y-1)*a=/ (у"1)**.
Λ(^ην) Jv(u)
!y(UnV) Jy(U)
The same definition works fine in case Μ has nonempty boundary, but there
is a small technicality. Namely, suppose we wish to work only with positive
charts taking values in a fixed half-space Кд>с. Then as long as η > 2 there
is no problem, but if η = 1, we are faced with the fact that there may
be no positively oriented Rj^>c-valued atlas at all even if Μ is orientable.
Some authors define manifold with boundary completely in terms of a fixed
half-space and seem unaware of this little glitch. Since we allow multiple
half-spaces, this is not a problem for us, but in any case, we could modify
the definition slightly in a way that works in all dimensions and for any
chart. Let Μ be an oriented manifold with boundary. If α has compact
support inside U for some chart (i/, x), then
(9.2) I a = sgn(x) f (χ-1)*α,
JU Jx(U)
9. Integration and Stokes7 Theorem
393
where
/ \ f 1 if (U, x) is positively oriented,
\ — 1 if (f7,x) is not positively oriented.
This could be taken as a definition. Once again, the standard change of
variables formula shows that this definition is independent of the chart
chosen.
Remark 9.2. Because we have used the sgn(x) factor in the definition, we
can use arbitrary charts. But the manifold still must be oriented so that
sgn(x) makes sense!
If α Ε Ω£(Μ) has compact support but does not have support contained
in some chart domain, then we choose a positively oriented atlas {(х$, Ui)}
for Μ and a smooth partition of unity {(#, Ui)} subordinate to the atlas,
and consider the sum
(9.3) W Α« = Σ/ (ОЧЛ")-
Proposition 9,3. In the sum above, only a finite number of terms are
nonzero. The sum is independent of the choice of atlas and smooth partition
of unity {(ρ», 17»)}.
Proof. First, for any ρ Ε Μ there is an open set О containing ρ such that
only a finite number of pi are nonzero on O. But a has compact support,
and so a finite number of such open sets cover the support. This means that
only a finite number of the pi are nonzero on this support. Now let {(χ*, V%)}
be another positive atlas and p* a partition of unity subordinate to it. Then
we have
ςΧ/'«=ς/κ(«ς^)
3 3
Since the sum (9.3) above is the same independently of the allowed
choices, we make the following definition:
Definition 9.4. Let (M, [w]) be an oriented smooth manifold with or
without boundary. Let α Ε Ωη(Μ) have compact support. Choose a positively
394
9. Integration and Stokes7 Theorem
oriented atlas {(хг, Ui)} and a smooth partition of unity {(pu Ui)}
subordinate to {U{}. Then we define
J(M\M) iM
We usually omit the explicit reference to the orientation [vj] and simply
write fM a.
Remark 9.5. Of course if we take (9.2) as a definition, then we may take
Σ / A<* = ^sgn(xO / (χΓχ)*(Λα)
and LM w,v α = Σζ $Ό рга even for an arbitrary unoriented atlas. Once
again, we still need Μ to be oriented. In the online supplement we
introduce twisted тг-forms, and these may be integrated even on nonorientable
manifolds!
In case Μ is zero-dimensional, and therefore a discrete set of points
Μ = {pi,P2j - · ·}> an orientation [w] is an assignment of +1 or —1 to each
point. Then, if а = / G Ω°(Μ), we have
J (MM)
where we choose the ± according to the orientation at the point.
9.1. Stokes' Theorem
In this section we take up the main theorem of the chapter. It is the
fundamental theorem of exterior calculus known as Stokes' theorem. Our
definition of integration works for any (positive) atlas, but we can use a specific
atlas for theoretical purposes. We will employ R™1<0-valued atlases in this
section. This is Мд>с where с = 0 and λ = — и1. Let us consider two special
cases of integration.
Case 1. This is the case of a compactly supported (n — l)-form on ln*
Let Dj = fdu1 A · · · Λ άνβ Λ ■ · · Λ dun be a smooth (n — l)~form with compact
support in Rn, where the caret symbol over the dv? means that this j-th
9.1. Stokes' Theorem
395
factor is omitted. Then we have
/ du)j = / difdu1 A ■ ■ · Λ dv? Α · - ■ Λ dun)
и1 О их<0
= / {df Adu1 A-· AdvJ A---A dun)
u1 0
= / (y^^dukAdu1A---Adu> A---Adun)
ux<0
Y~ldJ du1--dun = V
} диз
by the fundamental theorem of calculus and the fact that / has compact
support. All compactly supported (n — l)-forms ω on Rn are sums of forms
of this type and so we have
/ άω = / cj.
Case 2. This is the case of a compactly supported (n — l)-form on
R£1<0. Let Uj = fdu1 A · · ■ Λ dvJ A · · ■ Λ dun be a smooth (n — l)-form with
compact support meeting <Ж£1<0 = K£i=0 = {0}xRn l. Then, if j φ 1,
/ duj= d{fdul A · ■ ■ Λ Sui Λ - ■ ■ Λ ditn)
«1<0 ux<0
If j — 1 we have
/ άωχ= l (-1)J_1 ( / 14^1 du2A---Adun
Ун». Л—i \J-oodu1 )
u1 <0
= / /(0,u2,...,un)du2A--.Adun = / ωι.
For this last equality, it is important that we have set things up so that
du2 A * · * Λ dun be positive on 5R™1<0 with the induced orientation. Since
clearly LRn Uj = 0 if j φ 1 or if Uj has support that does not meet
Ж^1<0, we see that in any case /Rn duj = fdRn ω3. All compactly
396
9. Integration and Stokes* Theorem
supported (n — l)-forms ω on K™i<0 are sums of forms of this type, and so
summing we have
/ d»= [
ω.
Let us assume that Μ is an oriented smooth manifold of dimension
η > 2. Then there is a positively oriented atlas {(Ua,x-a)}aeA consisting
of nice charts so either xa : Ua = R" or xa - Ua — R£i<0· Let {pa} be a
smooth partition of unity subordinate to {C/a}. Notice that {ра\ц пдм) *s
a partition of unity for the cover {Ua Π дМ} of дМ. Next we apply the
special cases above. For ω € Ωη_1(Μ) with compact support, we have that
dw = Σά(Ρ<*ω) = Σ ά(Ρ<*ω)
= Σ / №м = Σ / *№)***>)
= Σ / (fe1)ν«ω) = Σ / ρ<*ω= ω>
where we have used the fact that x"1 : xQ (dUa) —> dUa is orientation
preserving. We have now proved Stokes' theorem stated below for the η > 1
case. The η = 1 case is easily proved directly and amounts to the familiar
fundamental theorem of calculus.
Theorem 9.6 (Stokes' theorem)· Let Μ be an oriented smooth manifold
with boundary (possibly empty) and give дМ the induced orientation. Then
for any ω Ε Ω"~Χ(Μ) (i.e. with compact support) we have
f dui= [
Jm Jd
ω,
дМ
Note that fdMu := 0 if дМ = 0. It can be shown that if (17, x) is a
chart for a manifold such that M\U has measure zero, then
JM Jx\
(x-Vu;.
(U)
The definition of integration using partitions of unity is fine for the
theoretical purposes we intend to pursue (such as cohomology), but for
actually calculating integrals it is nearly useless. Consider the task of
integrating the form ω :— zdy Λ dz + xdx Λ dy over the sphere S2 С I3,
Technically speaking, we are integrating the restriction of ω to £2. Let
a(u,v) = (cos и cos v, sin и cost?, sin v) for (u,v) G (0,2ττ) χ (—π, π). This
gives a parametrization of a portion of the sphere. Thus σ plays the role
of x_1. The image of our parametrization is the domain of the chart x,
and the complement of the domain of the chart has measure zero in 52.
9.2. Differentiating Integral Expressions; Divergence
397
Because of this it seems plausible that we would get the correct answer by-
calculating /(0 2тг)х( тгтг)67*^· ^u* σ*ω d°es ηο^ have compact support in
(0,2π) χ (—π, π), and so this is not justified in terms of the theory we have
developed. Let us try it anyway. We pull the form back to the it, ν space and
then integrate just as one normally would in calculus of several variables:
Js2 </(ot
σ ω
2ТГ]Х(-7Г,7Г)
Λθ,27Γ]χ(-π,π) V d(u,v) d{u,v)J
J-π JO
2π
cos2 v cos и sin ν du dv = 0.
о
We are led to a nice integral in this case, but in general, proceeding in
this way may lead to improper Riemann integrals, and anyway, it is not
immediately clear how to connect this with our theory given in terms of
partitions of unity. That the answer we obtained above is indeed the correct
answer can be seen by applying Stokes' theorem:
f ω5^ ί άω= /0 = 0.
Js2 Jb Jb
Here В is the unit ball and we used the fact that άω = 0.
A practical way of calculating integrals of forms is to break up the
manifold in a nice way and add up the integrals over the pieces. In Problem 10
we ask the reader to prove the following theorem:
Theorem 9,7. Let Μ be an oriented η-manifold with possibly nonempty
boundary. Suppose that there are subsets Dx С Шп for г = 1, ...,ϋΓ and
smooth maps φι : Di —> Μ such that the following assertions hold:
(i) Each Di is compact and has a boundary of measure zero.
(ii) Each <f>i restricts to an orientation preserving diffeomorphism of
the interior of Di to the interior of фг (D%).
(iii) φι (Di) Π φ] (Dj) is either empty or contains only boundary points
of φι (Di) and φ] (Dj).
Then, for any smooth η-form ω with support in \}фг (D%), we have
к
9,2. Differentiating Integral Expressions; Divergence
Suppose that S С М is a fc-dimensional regular submanifold with boundary
dS (possibly empty) and Φ* is the flow of some complete vector field X e
398
9. Integration and Stokes3 Theorem
X(M). In this case, Φ^(5) is also a regular submanifold with boundary. We
then consider jg Js Φ%η for some fc-form η with compact support. We have
=a III·'«'-')]
-[/μ£»5«4-,')]-/μ£λ·
But also /5 Φ J 77 — /Φί5 77. Thus we obtain the useful formula
-£ / η= £χη-
As a special case (t = 0), we have jg |t=0 /Φί5 V — Js £xV- We can go further
using Cartan's formula £χ = ίχ od + doix. We get
л / ^ = л / ф* ^ = / £χ7? = / ***ϊ+ / ώχ7?
«t J<&tS at JS J$tS JφtS J<&tS
= / %χάη+ / %χη.
This formula is particularly interesting in the case when S = Ω is an open
submanifold of Μ with compact closure and smooth boundary 5Ω, and
where η = μ is a volume form on M. Let Ω* := ΦίΩ. We have jg /fl μ -
/ant ^^ an(* *ken
— t*= %χμ-
dt\t=oJnt Jdn
Definition 9.8. If μ is a volume form orienting a manifold M, then €χμ =
(div Χ) μ for a unique function div X called the divergence of X with
respect to the volume form μ.
We have
and
d\ f f ·
-d μ= ΐχμ
at\t=oJQt JdQ.
li\ V= Сх^= \ (divX)/i.
«Ч*-0Л14 «/Ω </Ω
Prom this we conclude that
/ (divX)/i = / ϊχμ.
Jci JdQ
This formula helps to give geometric meaning to div X. The above formula is
a version of Gauss' theorem and can be easily proven not just for domains
9.2. Differentiating Integral Expressions; Divergence
399
in a manifold with a volume form, but for a general manifold with boundary
with a volume form:
Theorem 9.9. Let Μ be α manifold with boundary and let μ be a volume
form on M. In particularj Μ is oriented by [μ]. Then we have
/ (άινΧ)μ= / %χμ
JM JdM
for any compactly supported smooth vector field X.
Proof. The proof uses the trick we used above:
/ (divX)/i= / £χμ
JM JM
= / άιχμ+ / τχάμ
JM JM
— / άΐχμ — / %χμ (Stokes' theorem). D
JM JdM
Now let £Ί, ...,£nbea local frame field on U С Μ and 01,..., θη the
dual frame field. Then for some smooth function ρ we have
μ = ρθιΛ···Λθη,
and a simple calculation gives
η
ίχ {ρθ1 л···л 0П) = Y^{-iy-1Pxke1 л · ■ ■ л ек л · · · л θη.
fc=i
Then we have
ίχμ = Cx (ρθ1 Λ · ■ · Л 0я) = Αχ (ρθ1 Λ · · · Λ θη)
П
= άΣ(-ιγ-ιΡχΨ л ■. ■ л вк л ■.. л θη
η
= ^2(-l)^ld{pXk) Λ 01 Λ · ■ ■ Λ 0* Λ · · ■ Λ 0η
η η
= Σί-1^"1 Σ(ΡχΙ,№ Λ 01 Λ · · · Λ 0* Λ · · · Λ 0η
fc=l i=l
= Σ &*>χ")ι) ρ01 Λ ■ ■ ■Λ θη = Σ (Ι^*) *
k-i \ρ ' к ι \ρ '
where (рХк)^ := d(pXk)(Ek). Thus we end up with a nice formula
άΐνΧ = Τ-(ρΧ%·
400
9. Integration and Stokes' Theorem
In particular, if Ek — -^ for some chart (t/,x) = (ж1,..., xn), then
(9.4) Λ,Χ-ti^X»).
If we were to replace the volume form μ by —μ, then divergence with
respect to that volume form would be given locally by Σ£=ι ^~£я(~рХк) =
Υλ-ι pgfi (рХк) ап^ so the orientation seems superfluous! What if Μ is
not even orientable? In fact, since divergence is a local concept and
orientation is a global concept, it seems that we should replace the volume form in
the definition by something else that makes sense even on a nonorientable
manifold. On nonorientable manifolds, pseudoforms can be used in place
of forms for many purposes. Alternatively, many situations can be
handled by going to the two-fold orientation cover. See the online supplement
[Lee, Jeff] for a discussion.
9.3. Stokes' Theorem for Chains
Here we give another version of Stokes' theorem that is very useful in
connection with topology. We say that distinct points po, · · · >Pfc+i Ξ №n are
in general position if they are not contained in any fc-dimensional aifine
subspace of Rn. Equivalently, po, - · · ,Pk are in general position if the vectors
px — po, ■ - · >Pk ~ Po are linearly independent. A set of the form
{ρ ρ Λ
Y^tiPi : 0 < t1 < 1 and ^U = l\
i=0 г 0 J
for po,---iPk hi general position is called a geometric fc-simplex. The
set above is a closed convex set and is the smallest such set containing the
points po,*..,Pk- The integer к is the dimension of the simplex, and a
geometric 0-simplex is just a point. Let us denote the geometric fc-simplex
determined by the points po,... ,Pk by (po,... ,ρ*). If (po,... ,Pk) is such
a geometric fc-simplex and {pn,... ,ptj} С {po,... ,pjb+i}, then рг1,.. · ,Pi
are in general position and the geometric j-simplex (pia,..., ptj) is called a
face of (po,... ,Pfc). In particular, the geometric (k — l)-simplices obtained
by omitting one of po,... ,р& are called boundary faces. The г-th boundary
face of (po,... ,Pfc) is the simplex obtained by omitting the рг and is denoted
di (po, -. - ,Pfc) = (po> - - · jpi, · · · ,Pfc). Geometric simplices can be combined
to create geometric simplicial spaces, but we take a slightly more flexible
approach.
Definition 9.10. For к > 0, the set Ak := {a € Rk : a* > 0 and £a* < 1}
is called the standard fc-simplex in R*. Note that Δ0 = R° consists of
just a single point denoted 0. In other words, Ak is (ео,-..,е*), where
eo = (O5...,0),e1-(l,O,...,0),e2 = (0,l,...,O),..., etc.
9.3. Stokes' Theorem for Chains
401
Definition 9.11. Let Μ be ал η-manifold. A Cr map σ : Ak —> Μ is
called a Cr singular fc-simplex. Let G be an abelian group. A formal sum
с = Σσ οσσ, where the sum is over all singular simplices σ, οσ € G, and
са = О for all but finitely many simplices σ, is called a smooth singular
fc-chain with coefficients in G. A 0-simplex σ : Δ° -> Μ is often identified
with the image σ(0) = ρ Ε Μ.
By definition, οσ — 0 for all but finitely many simplices; we say that a
singular fc-chain has finite support. The set of all Cr fc-chains with
coefficients in G is an abelian group denoted С&(М, G)r, where we use additive
notation for the group operation. Addition is given by
ί Σ c°A + (Σ cH = Σ (c-+c*) σ·
The cases г = 0 and г = oo are the most important. We will restrict to
the r = oo case and drop the superscript in Cfc(M, G)°°. We now define
the notion of the i-th face of a singular simplex and then the boundary
of a simplex. First, if qo,... ,<& are distinct points in Rn, then there is a
unique affine map Шк —» Mn which maps e$ to щ for 0 < г < к. This map
restricts to a map from Ak onto the convex hull of <?о> - - -, Qk> which will be a
geometric fc-simplex if q$, · · · > Qk are *п general position. For given q^ ..., q^
denote this map by α (#о>..., qk). For 0 < г < Л, let fk : Ак~г ->■ Ah be
a(e0,..., ej,..., efe). Explicitly, /J(0) = 1, /i(0) = 0 and for к > 0,
/0^a\...,afc-1)=il-£ai,a1,...,afe-iy
//i(a\...la*-1) = (a1l...la<-1>0,ai,...lafc-1).
Thus /* is a homeomorphism of Δ*"1 onto its image, which is the г-th
boundary face of Δ*\ Thus it parametrizes the face of Ak opposite e$ while
keeping track of the order of the vertices.
The г-th face of a singular fc-simplex σ is the singular (k — l)-simplex σ%
given by
σ«:=σο/*.
Definition 9.12. The boundary operator d^ : Ck(M,G) -> Cfc_i(M,G) is
defined on a simplex σ by
к
дка := Σ("1)ν
г=0
402
9. Integration and Stokes' Theorem
Figure 9.1. A face of σ
and then extended to be a group homomorphism dk Σσ °°σ = Σσ с^сг.
If σ is a 0-simplex, then we define доа = О (the group identity) so that for
a 0-chain Σσ οσσ we have dk Σσ С<?а = 0-
We will denote all the maps dk simply by д for all fc. It can be shown
that
(9.5) do# = 0.
Exercise 9.13. Prove that д о д = 0. Hint: First show that /* ο /* ι
fj о f]£i for к > 1 and i > j. The maps fj* are defined above.
It is convenient to define Ck{M, G) to be the trivial group {0} for all
к < 0 and define дк — О for all к < 0. Then д о 9 = 0 remains true. The
sequence of spaces and maps
·■· Acfc+1(M,G) Acfc(M,G) AC* !(M,G) A ···
is called the singular chain complex with coefficients in G.
Let Zk(M,G) Kerdfc and Bk(M,G) :— Im<9fc_i. Then because of
equation (9.5) we have Bk(M,G) С Zk(M,G). The fc-th singular
homology group Hk(M; G) with coefficients in G is defined by
^(М'С)-Щё)'
If с G Zk(M, G), then the equivalence class of с is denoted [c]. If ci and C2
are ^-chains in the same class, then c\ — c<i + dc for some с € Bk(M,G ,
and in this case we say that c\ and C2 are homologous (or in the same
homology class). The most important choices for G are R and Z.
Exercise 9.14. Check that if G = R, then Cfc(M,R),Zfc(M,R),£fc(M,l
and Нк{М\ R) are all vector spaces in a natural way and the boundary maps
д extend to linear maps.
We define the integral of a fc-form α over a smooth singular fc-simplex σ
as
/ а := / ф*а,
9.3. Stokes7 Theorem for Chains
403
and then for a chain с = Σα οσσ G Cfc(M, R) we can define
\α = Υ^οσ α.
Jc σ Ja
If σ is a 0-simplex and if / € Ω°(Μ) = (7°°(Μ), then /σ / = /(0). We state
without proof the following version of Stokes' theorem:
Theorem 9.15 (Stokes' theorem for chains)· Let Μ be α smooth manifold.
For с G Cfc+i(M,R)? and a a k-form on M, we have
da — a
Jc Jdc
dc
For a proof see [War]. Notice that in this version, Μ is not assumed to be
orientable. Also, a need not have compact support since the chain с has
finite support. If σ is a 1-simplex and / G C°°(M), then the above reduces
to
'# = /(*(!))-/(σ(0)).
L
Now we define the de Rham map. First, if a G Zk(M) С iifc(M), then
we can define an element la of Hk{M\R), the dual space of Η^(Μ\R): For
[c] G tffc(M;R) represented by с G Zk(M]M) we define
Ia{[c]) = ja.
This is well-defined since if c+ dd G [c], then
Ia{c + dc') = f a
Jc+dd
— a+ a= a+ da (Stokes)
Jc Jdd Jc Jd
= I a (since da = 0).
This gives a linear map Zk(M) -» #fe(M;R). Now if a G Bk(M) (image of
d), then a — άβ and
/a([c]) = [a= [άβ - f β = 0 (since 5c = 0).
Л Ус */0C
lap called the de Rham
tf*eR(M)^tf*(M;R),
Idc
Thus we obtain a linear map called the de Rham map
Г& ( λ/Γ\ ν. Zlk
where #<ieR(M) = Zk(M)/Bk(M) is the de Rham cohomology defined
earlier. The content of the celebrated de Rham theorem is in part that this
map is an isomorphism. The theorem is fairly difficult to prove.
404
9. Integration ала Stokes7 Theorem
Theorem 9.16 (de Rham). The de Rham map defined above is an isomor-
phism,
#lR(M)^tffc(M;R).
Proof. See [Bo-Tu] or [War]. D
We have defined Hk(M;№) using smooth singular chains, but we could
have used continuous chains. The result is isomorphic to Нк(М]Ж) as we
have defined it (see [War] or [Bo-Tu]).
9.4. Differential Forms and Metrics
Let (V,p) be a real scalar product space (not necessarily positive definite).
We wish to induce a scalar product on L^t(V) = Д (V*). Even though
elements of L*lt(V) Ξ /\k(V*) can be thought of as tensors of type (0, k)
that just happen to be alternating, we will give a scalar product to this
space in such a way that the basis
(9-6) {eilA..-Aei"}il<...<ifc = {ef}
is orthonormal if e1,... , en is orthonormal. Recall that if α, β G V*, then
by definition (α,β) = g(cfi,/?**). Now suppose that α = α1Λα2Λ···Λα|!
and β = β1 Λ β2 Λ · · · Λ βΗ, where the a1 and β1 are 1-forms. Then we want
to have
(α\β) = (α1 Α α2 Λ · - · Λ ak \ β1 Λ β2 Λ ■ ■ ■ Λ β1*)
= det[(a\/3%
where [{αι,β^)] is an η χ η matrix. Notice that we use {α\β) rather than
(α, β) since the latter could be taken to be the natural inner product of α
and β as tensors—the latter differs from the first by a factor. We want to
extend this bilinearly to all fe-forms. We could just declare the basis (9.6
above to be orthonormal and thus define a scalar product. Of course, one
must then show that this scalar product does not depend on the choice of
basis. For completeness, we now show how to arrive at the appropriate scalar
product using universal mapping properties and obtain some formulas. Fix
β1, β2)..., fik ε V* and consider the map
given by
/^i,/^,...,/3* '· (α1,α2, ■ - ■ ,«*) н> det [<а*,0*>] .
Since this is an alternating multilinear map, we can use the universal
property of Д (V*) to see that this map defines a unique linear map
9.4. Differential Forms and Metrics
405
such that
PfiijP fi (α1 Λ α2 Λ · · · Λ afe) = det [(a\ ?)] .
Similarly, for fixed a € /\k(V*), the map ma: (β1,β2,...,β*) ι-)·
ββΐ 02 βΐι{α) is alternating multilinear and so gives rise to a linear map
m* ■ /\\v*) -> R
such that
πια{βχ Л /З2 Л ■ · · Λ /?*) = μβ1ίβ2 ^ (α).
Lemma 9.17. The map
Μ·): Λ V) x AV)-^
defined by
(α\β) := πια{β)
is symmetric and bilinear. We have
{α1 Λ α2 Λ ■ · · Λ ak \ β1 Α β2 Λ ■ ■ ■ Λ /3fc)
= άέί[(α\β*)]
for Informs a1, a2,... ,afc,/?\/?2,... ,/?*.
Proof. By construction, the map is linear in the second slot for each fixed
a. Fix β and write β as a sum β = Σ &u...ifc/?11 Λ · · · Λ /3lfc in any way. Then
™*{β) = Σ ^ι-*™" О3*1 л ■ ■ ■л Pik)
But each map α н->- 'ββΐβ2_βΗ (α) is linear and socuH τηα(β) is also linear.
Now let p(','):WxW4Rbe any bilinear map on a real vector space
W. If S С W spans W and if <p(si, S2) = ¥>(52, si) for all si, $2 ε 5, then <p is
symmetric. The set of all elements β £ /\k (V*) of the form /? = /31 Λ · · · Λ /3fc
for 1-forms β1,..., /3fc, is a spanning set. Since
= det[(ai,^)]=det[{^',ai)]
= {β1 Λ β2 Λ ■ ■ ■ Λ β* Ι α1 Λ α2 Λ · - ■ Λ afe),
we conclude that (·|·) is symmetric. D
The bilinear map defined in the previous lemma is a scalar product for
the vector space Д (V*). Notice the vertical bar rather than a comma in
406
9. Integration and Stokes' Theorem
the notation. If e1,..., en is an orthonormal basis for V*, then {en A
Л е1к}^<...<1к is orthonormal since
(eilA'--Aeifc|eJ'1A---AeJ'fc) =
ei2,ei2
which is zero if ii,..., г* is not a permutation of ji,..., jk> while on the
other hand, if (ji, ...,jk) — (σ (ii),..., σ (i*))» then every column of the
above determinant has exactly one nonzero entry, which is either 1 or —1.
Abbreviating e4 A · - · A elfe to e1 and so on, we have in the orthonormal case
So, if а = Σ ctfe1, then (e7 | a) = (e7 | eIsjaf and
ar=(e?\e?)(eT\a) = ±(e?\a).
We see that the scalar product need not be positive definite. Unless we
give a specific linear order to the basis {e7}, the signature is perhaps best
thought of as the indexed set {e(ij}j so that
a = J2e(i)(er\a)er.
I
We are very much interested in Д (V*), but note that Д (V) is also a scalar
product space in the analogous way so that
{v\ Л V2 A ■ · · A Vk | w\ A гиг A ■ · · A Wk) — det [(υ», Wj)\.
Exercise 9.18. Let R| denote Minkowski space. Determine the index of
the scalar product on Д (Rf) described above.
We already have a scalar product denoted {·, ·) for tensors. We wish to
compare it with the scalar product (·|·) just defined on forms. For simplicity
we consider the positive definite case. If a = J2aieI and β = Σβ/β/ are
Ar-forms considered as covariant tensor fields, then by Problem 18 of Chapter
7 we have
Now let e1,..., en be an orthonormal basis for V*. Then
9.4. Differential Forms and Metrics
407
and we have
(α β) = Σ,(<*ϊ*Γ\βΐή = Ea^/<eV>
ϊ,Ι ij
We conclude that
(α\β) = ±(α,β),
so the two scalar products differ by a factor of k\. Now let 01,..., θη be any
basis for V* (not necessarily orthonormal). Fot a given a E Д (V*), we can
also write a — ^ Σ aje1 and then as a tensor
a = £[ Σ^··-^1 л · ·' л Рк = Σ^ι-^^1 ® """ ® **·
But when α = ^a^fl7 and β = Y^bfO1 are viewed as fc-forms, we must
have
(α\β) = ^β) = ^α^ = ^α/.
Definition 9.19. We defined the scalar product on /\k V* = L^t(V) by first
using the above formula for exterior products of 1-forms and then extending
(bi)linearly to all of /\kY*. We can also extend to the whole Grassmann
algebra Д V* = φ Д V* by declaring forms of different degree to be
orthogonal. We also have the obvious similar definition for Д V and Д V.
If we have an orthonormal basis e1,..., en for V*, then e1 Λ · · ■ Л еп 6
Дп V*. But Д71 V* is one-dimensional, and if t : V —> V is any isometry of
(V*,3*), then the dual map £* : V* —> V* is an isometry of (V,#). Thus
£*el Л · · · Л ten = ie1 Л · - · Л en. In particular, for any permutation σ of
the letters {1,2,..., n} we have e1 Λ ■ ■ ■ Λ en = sgn^e^1) Λ · ■ · Л е*<4
For a given ordered basis (ei,..., en) for V, with dual basis (e1,..., en),
we have
(e1 Л · ■ ■ Л en\el Л--Леп> = eie2 · ■ · en = ±1.
Exercise 9.20. Let (ei,...,en) be orthonormal. Show that the only
elements ω of Дп V* with | (ω} ω) | = 1 are e1 Λ · · · Λ en and —e1 Λ ·· · Λ en.
Given a fixed orthonormal basis (ei,.. .,en), all ordered orthonormal
bases for V fall into two classes. Namely, those bases (/i,..., /n) for which
/^•••Λ/71 = ехЛ---Леп and those for which fl Л---Л/71 = -е1Л-"Леп.
Thus for each orientation of V there is a corresponding element of Дп V*
called a metric volume element for (V,g). The metric volume element
corresponding to a basis (ei,..., en) is just e1 Л · · · Л е71. On the other hand,
408
9. Integration and Stokes7 Theorem
we have seen that any nonzero top form ω determines an orientation. If
we have an orientation given by a top form ω, then obviously, (ei,... ,en)
determines the same orientation if and only if u;(ei,..., en) > 0 since this
means that ω = cel A - · · Λ en for some с > 0.
Proposition 9.21. Let an orientation be chosen on the scalar product
space (V, g) and let € =(ei,... , en) be an oriented orthonormal frame so
that vol := e1 Л - · Л еп is the corresponding volume element Then if
Τ — (/ι,..., fn) is a positively oriented basis for V with dual basis J"* -
{f\...,F),then
yo\ = yf\d^)\f1A---Afn,
where дц = (/i,/j). If g is positive definite, then det(gij) > 0 and so
vol-^det(^)/1A...A/n.
Proof. Let e% = J^a^p. Then we have
=Σ 44><л η=Σ а>1дкт,
fc,m fc,m
so that ±1 = det([<4])2det([gkm]) = (det([a|.]))2(det(5i_;))-1. Thus,
±VW(</u)l = det([4]).
But since £ and Τ are both positively oriented, we must have det([a£]) > 0
and so
^|det(i/0)| = det([4]).
On the other hand,
vol - el Λ · ■ · Л еп
= (Σ</"0Λ · ■ ■Λ (Σ^/*1) = ^(М)/1 л... л л
and the result follows. If g is positive definite, then det([ajj )2 det( [flr*"1]) = 1
so det(^j) > 0. □
Fix an orientation and let (ei,..., en) be an orthonormal basis in that
orientation class. Then we have the corresponding volume element vol =
е1Л- · -Леп. Now we define the Hodge star operator * : Д* V* -► f\n к V*
for 1 < к < η, where η — dim(V).
Theorem 9.22. Let (V, 5) be a scalar product space with dim(V) = η a d
the corresponding volume element vol. For each к, there is a unique linear
9.4. Differential Forms and Metrics
409
isomorphism * : /\k V* -l· /\n k V* such that а A */3 = (a|/3)vol /or oft
Proof. Given 7 £ /\n~kV*, define a linear map L7 : /\k V* ->> R by
requiring that L7(a) vol = а A 7. By Problem 3, if L7(a) = 0 for all a, then
7 = 0 and 7^L7 gives a one-to-one linear map Д V* -> (Д* V*)*. But
since dim Д* V* = άίπι(Λ*ν*)*, this must be a linear isomorphism. Thus,
for each β G Д V*, there is a unique element *β such that
L^(a) = (a|/5>
for aU a e /\kV*. This defines a map * : /\k V* -> /\n~k V* such that
α Λ */3 = L*^(a)vol = (a|/3) vol.
This map is easily seen to be linear. The equation above also shows that it
is one-to-one and hence an isomorphism. D
Proposition 9.23. Let (V,g) be a scalar product space with dim(V) =
η and corresponding volume element vol. Let (ei,...,en) be a positively
oriented orthonormal basis with dual (e1,..., en). Let σ be a permutation of
(1,2,..., n). On the basis elements εσ^ Λ · · ■ Λ εσ№ for /\k V* we have
(9.7) *(c'W Λ ■ · ■ Λ βσ<*>) = €σιβσ2 · · · βσ, sgn(a)e^fc+1) Λ · · · Λ βσ^}.
/η οί/ьег words, if we let {ik+ъ · · ■ > *n} = {1 j 2,..., n}\{ii,..., г&}, йеп
♦(e*1 Λ ■ ■ ■ Λ eifc) = ±(ehei2 · · · 6ije<fc+l Λ ■ · · Λ eS
wfrere we ia&e ifee + sign if and onft/ if e4 A · · · Λ elfe Λ elfc+L Λ · ■ ■ Л е%п =
е1Л---Леп.
Proof. Formula (9.7) above actually defines a map Λ*ν*->Λη"*ν*. For
this proof we denote this map by * and show it satisfies the same defining
equations as the Hodge star. It is enough to check that the defining formula
α Λ *β = {ο\β) ν°1 holds for typical (orthonormal) basis elements a = en A
- · Λ e** and β = emi A · · · Λ em*. We have
(9.8) (emi Λ - - - Λ emfc) Λ *(е41 Λ ■ ■ · Λ eik)
= emi Λ · - · Λ emk A (±е*к+г Λ · ■ ■ Λ einl).
The last expression is zero unless {mi,..., m^} U {ί*+ι,..., in} = {1, 2,
...,n}, or in other words, unless {ii,..., i*} = {mi,..., m^}. But this is
also true for
(9.9) (emi Λ - - · Λ emk | eh A · ■ ■ Λ eifc) vol.
On the other hand if {ή,..., ifc} = {mi,..., m^}, then both (9.8) and (9.9)
give ± vol. So the lemma is proved up to a sign. We leave it to the reader
to show that the definitions are such that the signs match. D
410
9. Integration and Stokes' Theorem
Remark 9.24. In case the scalar product is positive definite, 6i = 62 = · · ■
= en = 1 and so the formulas are a bit less cluttered.
The Hodge star operators for each к can be combined to give a Hodge
star operator /\Y* —> Д^*· We also note that the scalar product on
до γ* _ jj jg jug^. ^a|^ _ a^ ^hen, we g^-щ have the formula α Л */3 =
{θί\β) vol for any α, β in the algebra Д V*.
Proposition 9.25. Let V be as above with dim(V) = n. The following
identities hold for the star operator:
(1) *1 = vol;
(2) *vol = (-l)indte);
(3) **a = (-l)^W(-l)*(r»-*)a jora € д*у*/
(4) (*a|*^) = (-l)ind^){a|/3).
Proof. (1) and (2) follow directly from the definitions. For (3), it suffices
to let a = ea^ Л · ■ · Λ βσ^ for some permutation σ G £п. We first compute
*(6σ(*+ΐ)Λ.. .Α6σ(η)γ Wemust have *(е<т(*+1)л· · -Ле^71)) = се^Л· · -Ле^*
for some constant с. On the other hand,
Mfc+i) · ■ ■ 4")vo1 = (ea{k+l) л ''' л βσ(η) |е'(Л+1) Л ■ · - Л eff<n>) vol
= (e***1) Л ... Л e*<n>) Λ *(βσ(*+1> Λ ■ · · Λ e*<n>)
= (εσ^+1) Λ ..· Λ е'<п>) Л ce'W Λ · ■ ■ Λ βσ^
= (-l)^n-^csgn(a)vol
so that с = £a(k+i)''" ea(n){~l)k^n~k^ 8δη(σ). Using this and equation (9.7),
we have
* * (e'W Λ ... Λ e*<*>) = *€σ(1)6σ(2) · - · ea{k) pgn(a)e*<*+1> л ... Λ e'M
= 6σ(1)^σ(2) * " * €σ(Αϊ)6σ(Α.+ι) ■ · ■ €σ(η) (sgn(ff))
χ(-1)*(η-*)βσ(1)Λββ.Λβσ(*)
= (-l)indto)(-l)*("-*)e^(l) л ... Λ βσ(*\
which implies the result. For (4) we compute as follows:
<*a| * β) vol = *α Λ * * β = (-l)ind(s) (-!)*("-*) * α Λ /3
= (-1)Μ(9)β Λ *α = (_l)ind(p) ^ γο1 в Q
We obtain a formula for the star operator that uses a basis that is not
necessarily orthonormal. Recall ej*'"* defined by equation (8.2). If we let
Cti...tn := 6i\2.'.'in' ^en 6»i-*n is just the sign of the permutation (J^'£). Using
this notation, we have the following theorem.
9.4. Differential Forms and Metrics
411
Theorem 9.26. Let (V, g) be an oriented scalar product space with
corresponding volume element vol. Let e1,..., en be a positively oriented basis of
V*. If ω = ЬХшп...1кег'Л---Ле*>° (οτω = Ei1<i2...<.ifc ^...^^■••Ле^,
then
(9.10) *ω = yj\det[9ij}\ £ ^^к^...мк+1...^к+1 Л · ■ ■ Л е?».
jk+l<~'<jn
Proof. Let us use the common abbreviation g := detf^]. Let ^ denote
the operator defined so that *м) is given by the right hand side of equation
(9.10) above. Our task is to show that A — *. For this we need to show that
α Λ ίβ = (α\β) vol for all α, β e /\к V*. First note that for fixed ju ..., jn
we have eji...jn^*.'.^ = £ii...%n (no sum). We compute the coefficients of
а Л 4/3 using formula (8.3):
(«л ^)u...in = fci(n-fc)i Σ ал ..л (Щк^^п 4::£
= _ —S^\a\l/2a- · /3mi-mfc6 ■ β*1--?*
fc!(n — A;)! jfc! ^ "ji-j*^ c"21...mfcjfc+1...jntil_tn
— Irrl1/2 - Y^rv- ■ Rh-hf. · . · >-**
= ΙίΙ1/2έΣαΛ··Λ^"Λί*ι-ί- = ((«l/3)vol)n...in
and so α Λ 4*β = (α\β) vol; thus by uniqueness we have $ — *. D
The definitions we gave for volume element, star operator, and the
musical isomorphisms all apply to each tangent space TpM of a semi-Riemannian
manifold and these concepts globalize to the whole of TM accordingly.
Let Μ be oriented. The metric volume form induced by the metric
tensor g is defined to be the η-form vol such that volp is the metric volume
form on TPM matching the orientation. If (t/, x) is a positively oriented
chart on M, then we have
νο1Ιΐ7 — γ Idet^ijJIcte1 Λ ■ ■ · Λ dxn.
If / is a smooth function with compact support, then we may integrate /
over Μ by using the volume form:
/ /vol.
Jm
The volume of a compact oriented Riemannian manifold is
э1(М) := /
JM
vol(M):= / vol.
Im
412
9. Integration and Stokes' Theorem
The volume of an open set D С М is defined as
vol(L>) := sup it /vol: supp(/) С D with / e CC°°(M) and 0 < / < 1J ,
where C£°(M) denotes the space of smooth functions with compact support.
Example 9.27. The metric volume form vol^n on Rn is dx1 Л ■ · · Λ ώ?η,
where χ1,... ,xn are standard coordinates. Also,
volUn(vlp,..., vnp) = det(vb ..., vn),
where vip = (ρ, ν,) Ε TpRn = {ρ} χ Rn.
Once again assume that Μ is oriented by a metric volume form "vol".
The star operator gives two types of maps, which are both denoted by *.
Namely, the bundle maps * : /\k T*M -► f\n~k Γ*Μ, which have the obvious
definition in terms of the star operators on each fiber, and the maps on
forms * : Ω*(Μ) —> fin_fe(M), which are also induced in the obvious way.
The definitions are set up so that * is natural with respect to restriction
and so that for any oriented orthonormal frame field {Ει,..., En} with dual
{01,..., 0n} we have *(^ Л- · -A0,fc) = ±{ehei2 ■ ■ ■ eik)9jlA· · -Λ^*-*, where,
as before, we use the + sign if and only if θ%1 A · · ■ Λ θ4 А ДО1 Л · · ■ Λ θ3η к =
θ1 Λ ■ ■ ■ Λ θη = vol. As expected we have
*1 = vol,
*vol=(-l)ind4
**a = (-l)ind^(-l)fe(ri-/c)a
for а е Ω*(Μ).
Example 9.28. On open sets in R3 the star operator associated with the
standard metric and volume is given by
/ H> fdx A dy A dz,
fidx + f2dy + f3dz h-y fidyAdz + f2dzAdx +fcdxf\dy,
g\dyt\dz + g2dzAdx + gsdxAdy H> g\dx + g2dy + gsdz,
fdx Ady Adz H> /.
For an oriented Riemannian manifold, div will always be the divergence
with respect to the metric volume form (giving the orientation). From
equation (9.4) we see that the local coordinate expression for the divergence of
where detg = det(gij).
9As Differential Forms and Metrics
413
Definition 9.29. The gradient of a smooth function /ona Riemannian
manifold (M, g) is defined to be
grad(/) := Uf,
and the Laplacian Δ is defined by
A/:=-div(grad(/))
if Μ is oriented. If Δ/ = 0, then / is called a harmonic function.
The minus sign in the above definition is a matter of convention, and
this choice of sign is popular with differential geometers. Ironically, this
choice of sign makes the Laplacian a positive operator.
Lemma 9.30. If Μ is an oriented semi-Riemannian manifold with metric
volume element vol, then %χ vol — *bX for Χ G Χ(Μ).
Proof. We show that for fixed ρ G Μ we have ivi vol = *\>v\ for all v\ G
TPM. First consider the case where ν is not a null vector and (v±, υχ) — e± =
±1. Since the maps (г>2,..., vn) \-ь iVl vol(i>2,..., vn) and (t^,..., vn) t-l·
(*bvi) (v2, -. -, vn) are both alternating multilinear maps, it suffices to check
that they are equal on some basis. We extend to an orthonormal
basis (г>1, г>2,..., vn) with dual basis e1,..., en. Then we wish to show that
iVlvol(viiy...,Vin_1) = (♦bv!)^,...,^^) for 1 < h < -· < in-i < n-
But if Vix = vi, then
biVolivi!,...,^^) = νόΙ{ν1}νι>...) = 0
and
(*bui) Ki, ·-, vin^) = ex (*ex) (vil}..., ^n_J
= φ2Λ···Λβη(νι,...) = 0.
On the other hand, if v^ φ νχ, then (г^,..., Vin_x) = (г>2,..., νη) and
iVl vol(v2,..., vn) = volOi, v2,..., vn) = 1
while
(*bvi) (v2,..., vn) = ex (*ex) (v2, ...}vn)
= ^е2Л...Ле>2..^п) = 1.
Thus ίυι vol = *b^i for all nonnull vectors v\. But since both sides are
linear in v\, this establishes the result. □
Proposition 9.31. If Μ is an oriented semi-Riemannian manifold with
metric volume element vol, then ά\νΧ = (—l)md5 * d * bX for X G X(M).
414
9. Integration and Stokes' Theorem
Proof. By Lemma 9.30 and Cartan's formula (8.56), we have £χνο1 =
d ίχ vol = d * bX so that * div X vol = *d * bX or
div X * vol = *d * \>X
or
(-l)ind5divX = *d*bX D
Proposition 9.32. If Μ is an oriented semi-Riemannian manifold with
metric volume element vol, then
div fX = (grad /, X) + f div X.
Proof. Write vol = μ and compute as follows:
(div fX) μ = ΙΙ^χμ = di/χμ + ί/χάμ
= d (fixμ) = d/ Λ ίχ/χ + /<Κχμ
= df Λ ζχ/i + / (div X) μ
= -ix (df Α /χ) + ίχ# Λ μ + / (div Χ) μ
= d/(X)/z + /(divX)/z
= «X,grad/> + /(divX))Mf
where in the third line we use the fact that ίχ is a graded derivation and
df Α μ = 0. D
Exercise 9.33. Show that the local expression for Δ/ is
9.5. Integral Formulas
Definition 9.34. Let S be a regular codimension one submanifold of an
η-dimensional Riemannian manifold (Myg). A smooth vector field N along
5 is called a unit normal field if (iV, N) = 1 and if iV(p) ± Tp5 for every
peS.
Let (M, 5) be oriented by a metric volume form vol and suppose that a
regular (n — 1)-dimensional submanifold 5 of Μ has a (global) unit normal
field N along 5. If N is extended to a field N on a neighborhood of S,
then iff vol is an (n — l)-form on this neighborhood. Clearly, the restriction
of i^ vol to S is independent of this extension, and so we can denote this
restriction by ijyvoll^. Then we have
(iNvol\s)p (vi,..., vn-i) = vo]p(JV(p)s vi,... ,vn_i)
for vi,..., vn-i e T]t>£. This is a volume form on 5 which orients S and is
clearly the metric volume form for the induced metric t*g on S. Note that
9.5. Integral Formulas
415
inyoI\s is sometimes denoted by i^vol, but this is an abuse of notation
since in vol(p)(vi,..., νη-ι) makes sense for vi,..., vn_i € TpM, but this is
not what we want. On the other hand, Js ζ^νο1|5 is fs%* (z^vol), where
ι : S ·4 Μ is the inclusion map. In turn, this means the same thing as
Js ijv vol by convention.
Proposition 9.35. Let S be α regular codimension one submanifold of an
η-dimensional oriented Riemannian manifold (M, g) and let ι : S ^-ϊ Μ be
the inclusion. Let N be a smooth unit normal vector field along S. Let vol^
be the volume form of Μ and vols the volume form of S with respect to the
orientation induced by N and the metric L*g. If X is a smooth vector field
along S, then
ix volM = {X, N) vols.
Proof. Let XT := X - (X, N) N\
ix voIm = ixr voIm +i(x,N)N voIm
= ixr volM + {X, N) iN voIm,
and since i;v voIm\$m = vols, it remains to show that гхт voIm = 0. But
for υι,...,νη-ι € TpS,
ίχΎ VOW(Vi, . . . , Vn-l) = V01m(^T, Vl, - . . , Vn-l) = 0
since XT, vi,..., vn-i cannot be linearly independent. D
Now suppose that an oriented (M, g) has a boundary 9M. Since dM
is basically a codimension one regular submanifold of M, the above
constructions can be applied to dM so that dM is Riemannian with induced
metric. Since dM has a global outward pointing vector field, we may
normalize to obtain a global outward pointing unit normal field N along 5M.
The orientation induced on dM by the metric volume form in^oIm^m ls
exactly the induced orientation on dM described previously. We will denote
ijv voWldiw by vol^M when the orientation and normal field are understood.
Exercise 9.36. Let χ : V —^ i/ С М be a parametrization of an open
subset of a surface S in R3 and Ν :~ (χ^ι χ jcu2) / \\χ^ι χ х^гЦ. Show
that if dS := %n vol|5, where vol is the usual volume form on R3, then
x*dS = ||χ„ι χ х^гЦ du1 A du2.
We now introduce a special (n — l)-form σ on Rn (we use the same
symbol for all η if no confusion arises). This form is used in many constructions.
If ж1,..., xn denote standard coordinates, then
η
(9.11) σ := ^(-1)*_1ж* ώ'Λ-ΑώΆ-Λ dxn,
i=l
416
9. Integration and Stokes7 Theorem
where, as usual, the caret denotes omission. For example, on R3
σ = χ dy Λ dz — у dx Л dz + ζ dx Λ dy.
Proposition 9.37. // we use the outward pointing normal field N on Sn~l
and the metric induced from Rn, then the induced volume form on 5n_1 (the
area form) is given by
volsn-i = tV,
where ι: Sn~l «->■ Rn is inclusion.
Proof. Let νι = Σν{ d/dxj\x e TxSn~l. Identify elements of TxSn~l with
vectors in W1 so that v* is the column vector (vj,..., v™)T. Then we use
expansion along the first column by cofactors to obtain
vol5n ι(ϋι,...,νη-ι) =det(x,vi,...,t;n-i)
= ^(^ir1xidet(Ml)
2=1
η
= ^(-l)*"1^ (ώ1 Λ - -. Л dxi Л ■ - ■ Л dxn\ (wi,..., νη-ι),
i-l
where M» is the submatrix obtained by deleting the first column and i-th
row. D
Exercise 9.38. Show that the volume form on Sn~1{R) = {Y%=i(x%)2 =
R2} is jtV, where &: 5П_1(Я) М- Rn is the inclusion map.
Theorem 9.39. Let Μ be an oriented Riemannian η-manifold with
boundary. For any X 6 X(M) with compact support,
/ div X volM = / <X, N) voldM,
Jm JdM
where N is the outward pointing unit normal field on 8M.
Proof. We have fM divX voIm = fM d{%x voIm) = JdM^x уо^м w^c^ ls
equal to fdM (X, JV) νοΙ&Λ/ by Proposition 9.35. D
Corollary 9.40. Let Μ be a compact oriented Riemannian η-manifold with
boundary and X € X(M). Then for f € C°°(M) we have
[ <X, grad /) voW + / / div X volM = f f {X, N) voW.
J μ Jm JdM
Proof. We have
/ ({grad/, X) + f div X) voW = / div fX = / / (X, N) voW. □
Jm Jm JdM
9.5. Integral Formulas
417
Theorem 9.41. Let Μ be an oriented Riemannian η-manifold with
boundary and f,g e C°°(M). Then
/ (grad /, grad g) volM - / fAg volM = f (grad g, N) voW
JM Jm JdM
and
I (-fAg + gAf) volM = /[/ (grad g,N)-g (grad /, N)] vo\dM
JM JdM
where N is the outward unit normal field.
Proof.
- / fAgvolM= / /div(gradg) volM
Jm Jm
- f <grad 5> N) volaAf - / (grad g, grad /) volM
JdM Jm
by Corollary 9.40. The second formula follows from the first by interchanging
the roles of / and g and subtracting. □
The integral equations of the previous theorem are called Green's first
and second formulas respectively (when comparing with other versions in
the literature, do not forget our convention: Δ/ :— — divgrad/). If the
boundary of Μ is empty, then the boundary terms are zero, so that for
instance fM fAg vo\m = Jm 9&f voIm = 0. We close this section with an
application of Green's formulas.
Theorem 9.42 (Hopf). Let (M,g) be a compact, connected and oriented
Riemannian η-manifold without boundary. If f Ε C°°(M) and Δ/ > 0 (or
Af < 0), then f is a constant function.
Proof. Suppose Δ/ > 0. Then integration gives
0 < / 1Δ/ vo\M = / /Δ1 volM = 0,
Jm Jm
which means that Δ/ = 0 on M. Now use Green's second formula with
/ = g to get
/ ||grad/||2volM= / /Δ/νο1Μ = 0,
Jm Jm
so that ||grad/||2 = 0 on M. Thus / is constant since Μ is connected. G
418
9. Integration ала Stokes' Theorem
9.6. The Hodge Decomposition
In this section we follow [War]. Let (M, g) be an oriented semi-Riemannian
η-manifold. (However, we soon restrict to the compact Riemannian case.)
Let vol denote the metric volume form. For each &, the scalar products
induced by gp on Д* Τ* Μ for each ρ combine to give a symmetric C°°(M)-
bilinear map
(■|-> : Пк(М) χ Ω*(Μ) -> C°°(M)
with (α\β)(ρ) := {α(ρ)\β{ρ))ρ = 9ρ{μ{ρ),β{ρ)). We see that (α\β) vol =
α Λ *β. We can put a scalar product (·|·) on the space Ω(Μ) — Y^k Qh(M)
by letting (α\β) = 0 for α e ttkl(M) and β Ε Ω*2 (Μ) with fa φ к2 ала
letting
(α\β)= ί αΛ*/3= f <a|/?)vol if α, β Ε Ω*(Μ).
Jm JM
Definition 9.43. Let (Μ, g) be an oriented semi-Riemannian n-manifold.
For each к with 0 < к < η = dim Μ, the codifferential δ : Ω* (Μ) -►
Ω*-1(Μ) is defined by
S := (_i)bd(e)(_1jn(fc+i)+i # fo
and 5 = 0 on Ω°(Μ). These operators combine to give a linear map Ω(Μ) ->
Ω(Μ), which is also denoted 5.
Remark 9.44. Notice that if Μ is nonorientable, then * is still defined
locally by choosing an orientation valid on a chart domain. But * occurs
twice in the formula for <S, so any sign ambiguities cancel and thus δ is
globally well-defined even if Μ is nonorientable!
Proposition 9.45. The codifferential δ is the formal adjoint of the exterior
derivative on Ω(Μ). That is,
(άα\β) = (α\δβ)
for all α, β in Ω(Μ) with compact support in the interior of M.
Proof. It suffices to check that (άα\β) = (α\δβ) for α а (к — l)-form and β
a fc-form. In this case, d*β is an (n — k + l)-form and so
= (_l)ind(ff) (_!)η(*+1)+1 „, „, d „, β
= (_i)n(fc+i)+i(_1)(n-fc+i)(fc-i)d „, β = (_i)fe2d * β
9.6. The Hodge Decomposition
419
where we have used that n(k + l) + l + (n — k + 1) {k-1) = 2k + 2kn — k2 =
k2 mod 2. Then we have
d (α Λ *β) = άαΛ*β + (-l)fe_1a Λ d * /3
= da Λ */3 + (-l^-^-l^a Λ * δβ
= da Λ *β — а Л * δ β
since fc — 1 + к2 = fc(fc + 1) — 1, which is always odd. Now we integrate and
use Stokes' theorem to get the desired result:
/ άα/\*β= αΛ*δβ.
Jm Jm
Obviously, δ is defined on fifc(i/) for open sets U in Μ and δ is natural
with respect to restriction.
Definition 9.46. For 0 < к < η, the differential operator Δ : Ω(Μ) ->
Ω(Μ) defined by Δ = δά+ άδ is called the Laplace-Beltrami operator.
For each k, the restriction of Δ to fifc(M) is also called the Laplace-Beltrami
operator (on fc-forms). If Δω = 0, we call ω a harmonic form.
Notice that by Remark 9.44 the Laplace-Beltrami operator is defined
whether or not Μ is orientable. On Ω°(Μ) = C°°(M) the operator Δ
reduces to the Laplacian defined earlier.
Proposition 9.47. The following assertions hold:
(i) For all α, β Ε Ω(Μ), we have
(Δα\β) = (άα\άβ) + (δα\δβ).
(η) Δ is formally self-adjoint That is (Αα\β) = (α\Αβ) for all α, β G
Ω(Μ) with compact support in the interior of M.
Proof. By Proposition 9.45 we have
(Αα\β) = {δάα + άδα\β) = {δάα\β) + (άδα\β) = (άα\άβ) + {δα\δβ)
= {α\δάβ + άδβ) = (α\Αβ),
which proves (i). But by symmetry (Δα\β) = (άα\άβ) + (δα\δβ) = (α|Δ/3),
which proves (ii). D
In the remainder of this section, we restrict attention to an oriented
compact Riemannian тг-manifold (M,g) without boundary. In this case,
(α\β) — fMa Λ*β defines a positive definite inner product on Ω(Μ). We
write ||α|| := {μ\α)Ύί2.
Proposition 9.48. // (M,p) is compact, oriented and Riemannian, then
Δω = 0 if and only if άω = 0 and δω = 0.
420
9. Integration and Stokes' Theorem
Proof. This follows from (Αω\ω) = (άω\άω) + (δω\δω) since (·|·) is positive
definite in the Riemannian case. D
Now we wish to consider the equation Δα; = α for α G Ω*(Μ). First
notice that if Δα; = α holds, then for any β e iik(M) we have (Αω\β) =
(ω\Αβ) = (α\β). Then since |(ω|Δ/?)| = \{α\β)\ < \\α\\ \\β\\, the map έω :
β ι-» (ω\Αβ) is a bounded linear functional Employing a common idea from
the theory of differential equations, we make the following definition:
Definition 9.49. A bounded Unear functional £ : Ω*(Μ) -> R is called a
weak solution of the equation Δα; = α if
£(Αβ) = (α\β)
for all β Ε Пк(М).
Notice that if a weak solution t is represented by u>, then it must be an
ordinary solution since in that case we have
(Αω\β) = (ω\Αβ) = £(Αβ) = (α\β)
for all β, which means that Δα; = α. We will use the following powerful
regularity result (see [War] for a proof):
Theorem 9.50. Let α Ε Ω*(Μ). If I is α weak solution of Αω = a, then
there exists ω e ilk(M) such that
for all β Ε flk(M) and hence
Αω = a.
We will also need a rather technical result whose proof can also be found
in [War].
Proposition 9.51. Let {оц}™ be a sequence in tlk(M). Suppose that for
some С > 0 we have
\\ai\\< С and ||Δα<||<σ
for all i. Then {oti}f* has a Cauchy subsequence.
Let Hk = {ω e Qk(M) : Δα; = 0} and let (Uk) X denote {β G Ω*(Μ):
(β\ω) = 0 for all ω e Hk}>
Lemma 9.52. There exists a constant С > 0 such that \\β\\ < 0\\Αβ for
al^e{Hk)L.
9.6. The Hodge Decomposition
421
Proof. Suppose there is no such constant С Then we may find a sequence
{A}f С {Hk) X such that || A|| = 1 while lim^oo ||Δβ|| = 0. By
Proposition 9.51, {/34}i° has a Cauchy subsequence, which we may as well assume
to be {/?i}i°· Hence we have that for any fixed θ G Ω*(Μ), the sequence
{{βΐ\θ)} is Cauchy in R and so converges to some number. Now we define
i : Пк{М) -+ R by
1(θ):- lim(A|0).
г—too
Then,
ί(ΔΘ) = lim (β|Δβ) = Urn (Δ&|0) = 0,
г—Уоо г—>оо
and since £ is clearly bounded, it is a weak solution to Δω = 0. By Theorem
9.50, there must exist β e Qk(M) such that £{θ) = (β\θ) for all θ e Ω*(Αί).
It follows that (β\θ) = lim^oo(/?i|0) for all 0 and so /% —► /3 since
HA--/?ll2 = (A-/?l&-/?)
= (А|&)-2(/ЗД + (/?|/?)->о.
Since ЦАЦ = 1, we must have \\β\\ — 1 and of course β e (%k) . But by
Theorem 9.50, Αβ = 0 so β € Uk Π (Ή*) Χ = 0, which contradicts ||0|| - 1.
We conclude that С exists after all. D
Theorem 9.53 (Hodge decomposition). Let (M, g) be compact and oriented
(and without boundary). For each к with 0 < к < η = dim Μ, the space
of harmonic k-forms %k is finite-dimensional. Furthermore, we have an
orthogonal decomposition ofilk(M),
Qk(M) = A(nk{MJ\®rik
= άδ (pk(M)) θ δά Ulk(M)\ θ Пк
= d (Ω*"1 (Μ)) θ δ (θ*+1(Μ)) θ Пк.
Proof. If Пк were infinite-dimensional, then it would contain an infinite
orthonormal sequence {tt>i}f°. In this case, we would have
lk-u;;||2 = 2
for all г, j with г ф j. By Proposition 9.51, this sequence would contain a
Cauchy subsequence. But this contradicts the above equation. Thus %k
must be finite-dimensional.
We now prove the orthogonal decomposition ilk(M) = Δ (Ω*(Μ)) ®%k.
The other two decompositions can be derived from the first, and we leave
422
9. Integration and Stokes7 Theorem
this as a problem for the reader (Problem 9). Choose an orthonormal basis
cji, ..., ωά for Hk. If ot G ΩΛ(Μ), then we may write
d
α = β + ^2(ωί\α)ωί
i=l
where β б (Нк) . It is easy to show that this decomposition is unique,
so we have the orthogonal decomposition Ω*(Μ) = (Wk) θ Hk and our
task is to show that {Uk)± = Δ(Ω*(Μ)). Since (Δα|ω) = (α|Δω) - О
whenever ω 6 Hfc, we see that Δ (Пк{М)) С (Uk) ±. Now let α € (ft*)1
and define a linear functional £ on Δ (Ω*(Μ)) by
*(Δ0) := (α|0).
This is well-defined since if Αθ± = Δ02, then ίι-feE 4k and so (a|0i) -
(α\θ2) = (α|βι - 02) = 0. We show that £ is bounded. Let φ := θ - #(0),
where #& : Ω* (Μ) —> ftfc is the orthogonal projection. Then using Lemma
9.52 we have
|ί(Δ*)| = \£(Αφ)\ = \(α\φ)\ < \\α\\ \\φ\\
<С||а||||Д*|| = С||а||||Д0||.
By the Hahn-Banach theorem, the functional £ extends to a bounded
functional I defined on all of Ω* (Μ), which is then a weak solution of Δα; = α. By
Theorem 9.50, there is an ω 6 Qk(M) with Δα; = α so (Hk) ± С Δ (Ω*(Μ))
and so (Нк)± = А{Пк(М)). D
In order to take full advantage of the Hodge decomposition, we now
introduce a so-called "Green's operator" Gk : Ω*(Μ) —> {V,k) . We simply
define бг*(а) to be the unique solution of Δα; = α — Я* (α) where Hk :
Ω*(Μ) —► 7ife is the orthogonal projection as above. The Gk combine to
give a map G : Ω(Μ) -► фд. (ftfc) also called the Green's operator.
Lemma 9.54. Let Gk : Ω*(Μ) -> (Нк) be the Green's operator defined
above for each fc. Then
(i) Gk is formally self-adjoint;
(ii) if L : Ω*(Μ) -ν ΩΓ(Μ) is Кпеаг and commutes with A, then G
commutes with L (that is, L о Gk = Gr о L). In particular, G
commutes with d and δ.
Proof. We have
(Gfc(a)|/9) = (Gk (α) \β - ΗΗ{β)) = (Gk{a)\AGk(fi)) = (AGk(a)\Gk (β))
= (a-Hh(a)\Gk{fi)) = i<*\Gk(P)),
9.6. The Hodge Decomposition
423
so G is self-adjoint. For each j, let 7Tj : Ω^Μ) —► (KJ) denote orthogonal
projection (thus π3 + H3 = id^). Now suppose that L : ΩΛ(Μ) —> ΩΓ(Μ)
is linear and commutes with Δ. Notice that by definition we have Gk =
(A\ (Hk) L) ovTfc. The fact that LA = AL implies that ^(Ή*) С Ur. Also,
since Δ (Пк{М)) = (Kfe)X, we have ЩН^) С (ТГ)^ Thus
Ζ/ О 7Tfc = 7ГГ О L,
Lo(A|(«*)-L) = (A|(7f)J-)oL|(KP)xl
and
(A|(^)-L)-1°b=L|(Wr)xo(A|(^)-L)-1.
It follows that G commutes with L. D
We note in passing that G maps bounded sequences into sequences that
have Cauchy subsequences. Indeed, suppose that {щ}™ С Qfe(M) is a
sequence with ||с^|| < С. If βχ := <3(αζ·), then using Lemma 9.52 we have
ЦАЦ < ||ΔΑ|| = \\щ - H(ai)\\ < \\oi\\ < C,
and so by Proposition 9.51, {/%}i° has a Cauchy subsequence.
Theorem 9.55. Let (M,#) be a compact Riemannian manifold (without
boundary). Then each de Rham cohomology class contains a unique
harmonic representative.
Proof. First assume that Μ is orientable and fix an orientation. Let α Ε
0,к(М) and use the Hodge decomposition and the definition of G to obtain
a = AGk(a) + Hk(a) = dSGk(a) + 5dGk{a) + Hk(a).
Then since G commutes with cf, we have
a = dSGk(a) + SGk+1{da) + Hk{a).
So if da = 0, then a — Hk(a) = d5Gk{a), and so Hk(a) represents the same
cohomology class as a. To show uniqueness, suppose that αχ and a<i are
both harmonic and in the same class so that a2 — ot\ = άβ for some /3.
Then we have 0 = άβ + (αϊ — a2). But αϊ — a2 is orthogonal to άβ since by
Proposition 9.48
(άβ\α!-α2) = (β\δα1-δα2) = (/510) = 0-
Next suppose that Μ is nonorientable. If π : Mor —> Μ is the 2-
fold orientation cover of Μ (Section 8.7), then there is a unique metric on
Mor such that π restricts to an isometry on sufficiently small open sets (π
is a Riemannian covering). Now suppose that с Ε Hk(M) is a de Rham
cohomology class and consider the class π*с € Hk(MOT) (see the discussion
after Definition 8.41). Since Mor is orientable, π*с is uniquely represented
424
9. Integration and Stokes' Theorem
by a harmonic form η on Mor. If τ : Μοτ -> Μ01 is the involution which
transposes the two points in each fiber, then it is easy to see that r is an
isometry so that 7**77 is also harmonic. It follows from the identity ποτ π
that τ*η is also a representative of the cohomology class π*α Indeed, [τ*η\ -
τ* [η\ = т*7г*с = 7г*с. So by uniqueness, we must have τ*η = η, which is
exactly the condition that guarantees that rj — π*η for some η G Q,k(M).
But π is a local isometry and so by Problem 5, η must be harmonic. We
show that η represents c. Suppose that с = [μ] for μ G Ω*(Μ). Then both
π*μ and π*η represent the class 7r*c, so π* (η — μ) = άβ for some β. But
then, as for τ*η and η above, τ*β is in the same class as β, which means
thatd/3 = dr*/3. Then
π* (η - μ) = άβ = ± (d/3 + dτ*/3) = <* Q(0 + r*/?)) .
Since r is an involution, /3 + τ*β is τ-invariant, that is, r* (/3 + r*/3) =
(β + τ*/3), so there exists α G Ω*(Μ) with ττ*α = ±(/? + r*/3). Thus
π* (τ/ — /χ) = άπ*α = 7T*d!a,
and since π* is a local diffeomorphism, we must have η — μ — da, which
means that [η] = [μ] = с. The uniqueness of the harmonic representative 7/
is clear. D
Notice that if τ is the involution of Mor introduced in the above proof,
then π*α = τ*π*α for any a G Q,k(M).
Corollary 9.56. If Μ is compact, then its cohomology spaces Hk(M) a e
finite-dimensional
Proof. Any manifold can be given a Riemannian metric, and so if Μ is
compact and oriented, then Theorems 9.53 and 9.55 combine to give the
result. Now suppose that Μ is not orientable. Let a G ΩΛ(Μ) and suppose
that π*α = άβ so that π* [α] = 0. Then π*α = τ*π*α = τ*άβ = άτ*β. Now
π*θ = \{β + τ*β) for some θ and so
π* α = ά(^(β + τ*β)\ = άπ*θ = π*άθ.
Since π is a local diffeomorphism, we have a = άθ от [а] = 0. Thus the
map 7Г* : Hk(M) -> Hk(MOT) has trivial kernel and is injective. The finite-
dimensionality of Hk{M) now follows from that of Hk(MOT). D
The results above allow us to give a quick proof of Poincare duality for
de Rham cohomology. Choose an orientation for M. We define a bilinear
pairing Hk(M) χ Hn~k(M) -> К as follows:
((M.M))=/
Jn
ω Α η.
Μ
9.7. Vector Analysis on R3
425
This is well-defined since if ω\ = ω + da and ^ = 77 + g?/3 (with ω, η closed),
then by Stokes7 theorem
/ ωχ Ληχ= Ι ω Αη + da Αη + ω Αάβ + άαΛάβ
JM JM JM JM JM
= ωΑη + ά(αΑη)- ά(ωΑβ)+ ά(αΑάβ)
J Μ J Μ J Μ JM
= ω Α η.
JM
We wish to show that the pairing defined above is nondegenerate. Thus,
given any nonzero [ω] G Hk(M) we wish to produce a [77] G Hn~k(M)
such that (([ω], Ι7?])) Φ 0- Choose a Riemannian metric and metric volume
element for M. We may assume that ω is the harmonic representative for
[ω]. But it is easily checked that Δ commutes with * and so *cj is also
harmonic. In particular, *ω is closed and so represents a cohomology class
[*ω]. Then [*cj] is the desired class since
(([α;], [*ω])) = ω Α *ω = / {ω\ * ω) vol — ||α;||2 > 0.
JM JM
For each fixed [ω] Ε Hk(M), we have the linear map i^ G (Hn~k(M))*
given by i[w](M) ·= ((M> fa]))· Since the pairing is nondegenerate, the map
[ω] *-*> £[ω] defines an isomorphism from Hk(M) to (#n-fc(M))*. Thus we
have proved the following:
Theorem 9.57 (Poincare duality). If Μ is an orientable compact n-mani-
fold without boundary, then we have an isomorphism
нк(м)^(нп-к(м)У
coming from the pairing defined above.
We will take up Poincare duality again in Chapter 10.
9.7. Vector Analysis on R3
In R3, the 1-forms may all be written (even globally) in the form θ = f\dx +
fzdy+fedz for some smooth functions /1 , /2 and /3 and all 2-forms β may be
written β = g\dyAdz+g2dzAdx+gzdxAdy. The forms dyAdz, dzAdx, dxAdy
constitute a basis (in the module sense) for the space of 2-forms on R3 just as
dz, dy, dz form a basis for the 1-forms. The single form dxAdyAdz provides
a module basis for the 3-forms in M3. Suppose that x(u,v) parametrizes a
surface 5 С R3 so that we have a map χ : U —> R3. Then the surface is
426
9. Integration and Stokes' Theorem
oriented by this parametrization, and the integral of β over S is
/ β = / 9idy Λ dz + g<idz Adx + g^dx A dy
Js Js
f ( / / \\9(y,z) . . чч d(z,x)
■'■•««••йгШ)*'*·
Here and in the following we disregard technical issues about integration
(but recall Theorem 9.7).
Exercise 9.58. Find the integral of β = χ dy A dz + dz A dx + xz dx Λ dy
over the sphere oriented by the parametrization given by the usual spherical
coordinates φ, Θ, p.
If ω = h dxAdyAdz has support in a bounded open subset ί/cl3 which
we may take to be given the usual orientation implied by the rectangular
coordinates ж, у, ζ, then
/ ω = I hdx Ady Adz = / hdxdydz.
Ju Ju Ju
In order to relate differential forms on R3 to vector calculus on R3, we
will need some ways to relate forms to vector fields. To a 1-form θ = f\dx +
Hdy + hdz, we can obviously associate the vector field |}0 = f\\ + /2J + /зк.
But recall that this association depends on the notion of orthonormality
provided by the dot product. If θ is expressed in say spherical coordinates
0 = fidp+f2dd + fzd(f>, then it is not true that J|0 = /ιΐ + Αί + Λ^- Neither
is it generally true that Ц0 = /iP + /2$ 4- /з<£, where p,0, φ are unit vector
fields in the coordinate directions1 and where the ft are just the ft expressed
in polar coordinates. Rather, our general formulas give
p psmv
In rectangular coordinates ж,у, ζ, we have
Ц: dx -» i,
Ih dy -+ j,
Й: dz -> k,
while in spherical coordinates we have
jj: dp н> /э,
(J : ρδίηθάφ h-¥ φ.
Here 0 is the polar angle ranging from 0 to π.
9 J. Vector Analysis on R3
427
As an example, we can derive the familiar formula for the gradient in
spherical coordinates by first just writing / in the new coordinates /(ρ, #, φ) :=
f(x(p, 0,0), j/(p, 0, φ), z(p, 0, φ)) and then sharping the differential
to get
At m/ df д idf a i a/a
dp dp ρδθδθ рвшвдфдф'
where we have used
(9ij) =
1 0
0 ρ
0 0
0
0
ρ sin θ
-1 ι
=
1 О О
о J о
о о г
ρ sin θ -Ι
In order to proceed to the point of including the curl and divergence
of traditional vector calculus, we need a way to relate 2-forms with vector
fields. This part definitely depends on the fact that we are talking about
forms in R3. We associate to a 2-form η the vector fields Й(*т/). Thus
g\dy Adz + g^dz Adx + g$dx A dy gives the vector field X = gii + paj + Ззк.
Now we can see how the usual divergence of a vector field comes about.
First flat the vector field, say X = Л1+/а1+/зк, to obtain \>X = f\dx +
hdy + fsdz and then apply the star operator to obtain f\dy Adz + fedz Л
dx + fedx A dy. Finally, we apply exterior differentiation to obtain
d{f\dy Adz + f2dz Adx + fedx A dy)
= dfx A dy A dz + dj^ Adz Adx + dfy A dx A dy
= ( -r— dx + -r—dy + -z—dz ] Л dy A dz + the other two terms
V ox dy ο ζ J
— -£-dx A dy A dz + ~^—dx A dy A dz +
dx
Oh
\ dx dx
dx dx dx
dx
dx Ady A dz.
dx
dx Ady Adz
Now we see the divergence appearing. In fact, if we apply the star operator
one more time, we get the function divX = Щ~ + ^ + Ц^. We are thus
led to *d* (\>X) = div X which agrees with the definition of divergence given
earlier for a general semi-Riemannian manifold.
428
9. Integration and Stokes' Theorem
What about the curl? For this, we just take d (bX) to get
d (fidx + f2dy + fodz) = df\ Adx + df2 Ady + dfs A dz
/ p\ £ О £ Q £ \
= ( -^—dx + -z—dy + -z—dz ] Л dx + the obvious other two terms
\ ox dy dz J
and then apply the star operator and sharping to get back to vector fields
obtaining
(dh dh\.^(bh dfA.fdh df2\, lv
In short, we have
fl*d(bX) = curlX.
Exercise 9.59. Show that the fact that dd = 0 leads to both of the following
familiar facts:
curl(grad/) = 0,
div(curlX)=0.
The 3-form dx Ady Adz is the (oriented) volume element of R3, Every
3-form is a function times this volume form, and integration of a 3-form
over a sufficiently nice subset (say a compact region) is given by fDu) =
JD fdx A dy A dz = JD f dx dy dz (usual Riemann integral). Let us denote
dx Ady A dz by dV. Of course, dV is not to be considered as the exterior
derivative of some object V. Let it1, it2, it3 be curvilinear coordinates on an
open set U С R3 and let д denote the determinant of the matrix [дц] where
gij = (^-, g^j). Then dV = yfgdu1 A du2 A duz. A familiar example is the
case when (it1, it2, u3) are spherical coordinates ρ,θ,φ, in which case
dV = p2sinOdp Α άθ Α άφ,
and if D С R3 is parametrized by these coordinates, then
/ /dV- / /(ρ,θ,φ)ρ2 sin0 dp Α άθ Α άφ
JD JD
= Ι /(ρ,θ,φ)ρ23\τιθάράθάφ.
JD
If we go to the trouble of writing Stokes' theorem for curves, surfaces
and domains in R3 in terms of vector fields associated to the forms in an
appropriate way, we obtain the following familiar theorems (using standard
9.8. Electromagnetism
429
notation):
jfv/.cir = /(r(b))-/(r(a)),
/ / curl(X) x dS = * X · dr (Stokes' theorem),
/// div(X) dV = // X · dS (Divergence theorem).
Similar and simpler things can be done in R2 leading for example to the
following version of Green's theorem for a planar domain D with (oriented)
boundary с = dD:
/ ( — тр J dxAdy^ / d{Mdx + Ndy)= Mdx + Ndy.
All of the standard integral theorems from vector calculus are special
cases of the general Stokes' theorem.
9.8· Electromagnetism
In this subsection we take a short trip into physics. Consider Maxwell's
equations2:
VB = 0,
V χ Ε + — = 0,
от
VxB-er=J·
Here E and В, the electric and magnetic fields, are functions of space
and time. We write Ε = E(i,x), В = B(i,x). The notation suggests that
we have conceptually separated space and time as if we were stuck in
the conceptual framework of the Galilean spacetime. Our purpose is to
slowly discover how much better the theory becomes when we combine
space and time in Minkowski spacetime R|. Recall that R| is treated
as a semi-Riemannian manifold, which is R4 endowed with the indefinite
metric (x,y)v = —x°y° + Σϊ=ιχ1ντ· Here the standard coordinates are
conventionally denoted by (я°,жг,а;2,х3), and x° is to be thought of as
a time coordinate and is also denoted by t (we take units so that the
speed of light с is unity). In what follows, we let r = (x1,^2,^3), so that
(x°,x1,x2,x3) = {t,T).
2Actually, this is the form of Maxwell's equations after a certain convenient choice of units,
and we are ignoring the somewhat subtle distinction between the two types of electric fields Ε and
D and the two types of magnetic fields В and Η and their relation in terms of dielectric constants.
430
9. Integration and Stokes' Theorem
The electric field Ε is produced by the presence of charged particles.
Under normal conditions a generic material is composed of a large number
of atoms. To simplify matters, we will think of the atoms as being composed
of just three types of particle; electrons, protons and neutrons. Protons carry
a positive charge, electrons carry a negative charge and neutrons carry no
charge. Normally, each atom will have a zero net charge since it will have an
equal number of electrons and protons. If a relatively small percent of the
electrons in a material body are stripped from their atoms and conducted
away, then there will be a net positive charge on the body. In the vicinity
of the body, there will be an electric field which exerts a force on charged
bodies. Let us assume for simplicity that the charged body which has the
larger, positive charge, is a point particle and stationary at ro with respect
to a rigid rectangular coordinate system that is stationary with respect to
the laboratory. We must assume that our test particle carries a sufficiently
small charge, so that the electric field that it creates contributes negligibly
to the field we are trying to detect (think of a single electron). Let the test
particle be located at r. Careful experiments show that when both charges
are positive, the force experienced by the test particle is directly away from
the charged body located at ro and has magnitude proportional to qe r2,
where r = |r — ro| is the distance between the charged body and the test
particle, and where q and e are positive numbers which represent the amount
of charge carried by the stationary body and the test particle respectively.
If the units are chosen in an appropriate way, we can say that the force F
is given by
zr r~ro
F = qe- -j.
|r-r0|
By definition, the electric field at the location r of the test particle is
(9.12) E = g7^5.
|r-r0|
If the test particle has charge opposite to that of the source body, then
one of q or e will be negative and the force is directed toward the source.
The test particle could have been placed anywhere in space, and so the
electric field is implicitly defined at each point in space and so gives a vector
field on R3. If the charge is modeled as a smoothly distributed charge
density ρ which is nonzero in some region U С R3, then the total charge is
given by integration Q = J^ p(t, r) dVr and the field at r is now given by
E(t,r) = J^ p(t) y). Γ_^3 dVy. Since the source particle is stationary at ro,
the electric field will be independent of time i. A magnetic field is produced
by circulating charge (a current). If charge e is located at r and moving
with velocity ν in a magnetic field B(t,r), then the force felt by the charge
is F = eE + |v χ Β, where ν is the velocity of the test particle. The test
9.8. Electromagnetism
431
particle has to be moving to feel the magnetic part of the field! At this point
it is worth pointing out that from the point of view of spacetime, we are not
staying true to the spirit of differential geometry since a vector field should
have a geometric reality that is independent of its expression in a coordinate
system. But a change of inertial frame can make В zero. Only by treating
Ε and В together as aspects of a single field can we obtain the proper view.
Our next task is to write Maxwell's equations in terms of differential
forms. We already have a way to convert (time dependent) vector fields Ε
and В on R3 into (time dependent) differential forms on R3. Namely, we
use the flatting operation with respect to the standard metric on R3. For
the electric field we have
Ε = Ехдх + ЕРду + Ezdz н> S = Exdx + Eydy + Ez dz.
For the magnetic field we do something a bit different. Namely, we flat and
then apply the star operator. In rectangular coordinates, we have
В = Вхдх + Вуду + Bzdz н-> В = Bxdy Л dz + Ву dz Л dx + Bzdx Л dy.
If we stick to rectangular coordinates (as we have been), the matrix of the
standard metric is just I = (£jj), and so we see that the above operations
do not numerically change the components of the fields. Thus in any
rectangular coordinate system we have
EX = EX, EP = Ey, EZ = EZ
and similarly for the B's. It is not hard to check that in the static case where
8 and В are time independent, the first pair of (static) Maxwell's equations
are equivalent to
dE = 0 and dB = 0.
This is nice, but if we put time dependence back into the picture, we need
to do a couple more things to get a nice viewpoint. So assume now that Ε
and В and hence the forms £ and В are time dependent, and let us view
these as differential forms on spacetime Rf. In fact, let us combine £ and В
into a single 2-form on Rf by setting
τ = в + ε л dt.
Since T is a 2-form, it can be written in the form Τ — ^Εμι/άχμ Adxv', where
Ρμν — —Ενμ and where the Greek indices are summed over {0,1,2,3}. It
is traditional in physics to let the Greek indices run over this set and to let
Latin indices run over just the "space indices" 1,2,3. We will follow this
convention for a while. If we compare Τ = Β + £ Adt with \Εμνάχμ Λ dxv',
432
9. Integration and Stokes7 Theorem
we see that the Γμν form an antisymmetric matrix which is none other than
0
Ex
Ey
Ez
—Ex
0
-bz
By
-Ey
Bz
0
—Bx
-ΕΖΛ
-By
Βχ
0 J
Our goal now is to show that the first pair of Maxwell's equations are
equivalent to the single differential form equation
Let N be an n-manifold and let Μ = (a,6)xJV for some interval (a, b).
Let the coordinate on (a, 6) be t = x° (time). Let (ж1,..., xn) be a
coordinate system on N. With the usual abuse of notation, (x0,^1,... ,xn) is a
coordinate system on (a,b)x7V. One can easily show that the local
expression du = θμ/μι„.μΗ Α άχμ Α άχμι Λ · ■ · Λ άχμΗ for the exterior derivative of a
form ω = /μι...μ^χμι Λ * ■ · Λ άχμ* can be written as
3
(9.13) du> = Σ $ωμι...μΛ Λ dxl Α άχμι Λ ■ ■ · Λ άχμ*
+ 9οωμι„.μ}ζ Λ dx° Λ άχμι Λ · · ■ Λ άχμ*,
where the μι sum over {0,l,2,...,n}. Thus we may consider the spatial
part ds of the exterior derivative operator d on (a, b)xS = M. That is, we
think of a given form ω on (a, b) χ 5 as a time dependent form on N so that
d^o; is exactly the first term in the expression (9.13) above. Then we may
write άω = dsoj + dt A dtu as a compact version of the expression (9.13).
The part dsui contains no di's. By definition J = B+£ Adt on RχR3 = Rf,
and so
dT = dB + d{£ A dt)
= dsB + dtA dtB + {dsS + dtA dtS) A dt
= dsB + (dtB + dsS) A dt.
The part dsB is the spatial part and contains no eft's. It follows that dl
is zero if and only if both dsB and dtB + ds£ are zero. Unraveling the
definitions shows that the pair of equations d$B — 0 and dtB + dsS = 0
(which we just showed to be equivalent to dT = 0) are Maxwell's first two
equations disguised in a new notation. In summary, we have
dsB = 0 V · В = 0
dtB + dse = o ^ VxE + f = o
Below we rewrite the last pair of Maxwell's equations, where the
advantage of combining time and space together manifests itself to an even
greater degree. Let us first pause to notice an interesting aspect of the
<£F = 0
9.8. Electromagnetism
433
first pair. Suppose that the electric and magnetic fields were really all
along most properly thought of as differential forms. Then we see that
the equation dT — 0 has nothing to do with the metric on Minkowski
space at all. In fact, if φ : Rf —^ Rf is any diffeomorphism at all, we
have dT = 0 if and only if ά(φ*Τ) = 0, and so the truth of the
equation dT = 0 is really a differential topological fact; a certain form Τ is
closed. The metric structure of Minkowski space is irrelevant. The same
will not be true for the second pair. Even if we start out with the form
Τ on spacetime it will turn out that the metric will necessarily be
implicit in the differential forms version of the second pair of Maxwell's
equations. In fact, what we will show is that if we use the star operator for
the Minkowski metric, then the second pair can be rewritten as the
single equation *d * Τ = J, where J is formed from j = (j1, j2, j3) and ρ
as follows: First we form the 4-vector field J = pdt + jldx+j2dy + j3dz
(called the 4-current) and then using the flatting operation we produce
J — —pdt + j1 dx + j2 dy + j3 dz = J$dt + J\ dx + fady + J$ dz, which
is the covariant form of the 4-current.
We will only outline the passage from *d * Τ — J to the pair V - Ε = ρ
and VxB-^ = j. Let *s be the operator one gets by viewing differential
forms on Rf as time dependent forms on R3 and then acting by the star
operator with respect to the standard metric on R3. The first step is to
verify that the pair V ■ Ε = ρ and VxB — ^=jis equivalent to the pair
*sds *s £ = Q and — dtS + *sds *s В — 3, where j :— jldx+j2dy + j3dz and
В and S are as before. Next we verify that
* Τ = *s<? — *s& A dt.
So the next goal is to get from *d * Τ = *J to the pair *sds *5 £ = Q and
-<?t£ + *sds *5 В = j. The following exercise finishes things off.
Exercise 9.60. Show that *d * Τ — -dtS — *sds *s £ Λ dt + *s<ts *s В and
then use this and what we did above to show that *d * Τ = J is equivalent
to the pair *sd>s *5 S = Q and — dt£ + *sds *5 В = j.
We have arrived at the following formulation of Maxwell's equations:
*d * Τ = J.
If we just think of this as a pair of equations to be satisfied by a 2-form
Τ where the 1-form J is given, then this last version of Maxwell's
equations makes sense on any semi-Riemannian manifold. In fact, on a Lorentz
manifold that can be written as (a, b)xS — Μ with the product metric
—dt®dt χ #, for some Riemannian metric д on £, we can write Τ = B+£Adt,
434
9. Integration and Stokes' Theorem
which allows us to identify the electric and magnetic fields in their covariant
form.
9.9. Surface Theory Redux
Consider a surface Μ с R3. In what follows, we take advantage of the
natural identifications of the tangent spaces of R3 with R3 itself. Let ei,e2
be an oriented orthonormal frame field defined on some open subset U of
Μ and let ез — βι χ β2· We can think of each ei, e<x and ез as vector fields
along U. Using the identifications mentioned above, we also think of them
as R3-valued 0-forms.
Note that the identity map id^3 may be considered as an Revalued 0-
form on R3. Let I := t* (idR3) where t : U С Μ «-► R3 is inclusion. Then
dl := L*d (id]R3) = 0 since idK3 is constant. Thus, I is an Revalued 0-form,
and we may write
I = βχθ1 + e202,
where (01, Θ2) is the frame field dual to (ei, ег). Note that volj^f — θ1 Α Θ2.
We have the Revalued 1-forms dej and
3
de3 = Σ ekUi
к ι
for some matrix of 1-forms (аЛ). Note that if г; Ε TpM, then dej (ν) = Vvej,
where V is the flat Levi-Civita derivative on R3. (In fact, we should point
out that if i, j, к is the standard basis on R3, then any field X along U may
be written as X = /ii+/2j+/3k for some smooth functions /i, /2, /3 defined
on U and may be considered as an R3-valued 1-form. Then dX(v) = VVX =
dfi(v)i + df2{v)j + б?/з(г;)к.) For an arbitrary tangent vector ν we have
ω)(ν) = {de0(v),ei) = - {e^de^v)) = -ω?(ϋ),
and so it follows that the matrix (a;*·) is antisymmetric. We write Revalued
forms on U as Σλ-ι ekVk where ^ G Ω(17). This is to conform with the
order of matrix multiplication.
Theorem 9.61. Let Μ С R3, βχ, e<i, θ1, θ2 and (cjj) be as above. Then the
following structure equations hold:
2
fc=l
ω\ Α Θ1 + ωΐ Α θ2 = О,
з
άω) = -Σω\Αω].
9.9. Surface Theory Redux
435
Proof. We calculate as follows:
2 2
0 = dJ = d^2ek9k -^аекАвк + екАάθκ
к 1
fc-1
2/3
fc=l ^ 1 / k=l
2 2 2
j=l fc=l j=l fc-1
= ][> (cW + JVfc Λ β») + e3 ($>| Λ **) >
J=l \ Л 1 / \fc 1 /
and it follows that dQ% = - Σ* ι <4 л °к and ωι л ^ + ω1 л *2 = °-
Also,
з
О = ddtj = d ^2 ekUj
к-l
= J^rfefc AcjJ + Y^ek Л <Ц*
fc 1 fc-l
3.3 ν 3
= Σ(Σβ*α;*)Λα;*+Σ^ΛΛ;5
fc-1 4 1 / г-1
3 3
г-1 Ч 1 '
3 ,.,г л /,,fc
and so dcjj = — Σ*=ι ^fc Λ ω:
D
Notice that for ν G TpM we have de3(v) = У^ез = -£(г>) and so
II(v,w) = (—йез(г;),гу) (recall Definition 4.20). Therefore,
II{euej) = (-des(ei),e3) = / - £%Λω£(е;),еЛ = -о^(е»)
and so
7/(ebei) JJ(ei,e2)
iT(e2,ei,) i7(e2,e2)
^(ei) ^f(ei)
^з(е2) ^f(e2)
Since (ei, e2) is an orthonormal frame field, the matrix of the first
fundamental form is the identity matrix. It follows that the matrix which represents
436
9. Integration and Stokes' Theorem
the shape operator is the same as that which represents the second fundar
mental form. Thus
J7(ei,ei) II{e\e2)
II{e2,ex}) II(e2,e2) J
Notice that de^ = Σ&=ι екшз reduces to de^ — βιω^ + ζ2ω\ since ω\ = 0.
We have
tf = det
= -det
u>l(ei) w|(ei)
^з(е2) ω|(β2) J
so that
ωΐΛωΙ = -ΚθιΛθ2,
ω\ί\ωΙ = Κθι Α θ2,
άω\ = Κ θ1 Α θ2.
Recall that for ν, w G TXS2 we have vol^v, w) = (ν χ ω, ж). If we consider
ез as a map ез : f/ С Μ —> 52, then it is called the Gauss map. If Μ is
assumed oriented, then we may take ез to be equal to a global normal field.
If υ, w Ε ΤρΜ, then we have
el vol52 (г>, w) = <de3 (ν) χ de3 (it;), e3)
= ((ωΙ(ν)βχ + L)l(y)e2) x (ω£(ιυ)βι + ω|(w)e2) ,e3)
= cj3Acj2(v,tt;) = Κθι Αθ2(ν,ιν) = KvolM(v,w).
Thus we obtain
ез V0I52 = К vo\m ·
This shows that the Gauss curvature is a measure of distortion of the signed
volume under the Gauss map. In particular, if ез : U С Μ —> S2 is
orientation reversing at p, then the curvature at ρ is negative. Let ρ Ε Μ with ρ
in the domain U of ез. If A is a nice domain in C/, then ез maps A to a set
in S2. Without worrying about measure-theoretic technicalities, we have
vol(e3 (A)) = / V0I52
Jez(A)
= ез V0I52 = / К voIm
Ja Ja
One can get an idea of the curvature near a point on a surface by visualizing
the Gauss map (see Figure 9.2). For example, it is clear that the right
circular cylinder has Gauss curvature zero since it maps every region of the
cylinder onto a set with zero area. It is also fairly clear that the saddle
surface in the diagram has negative curvature.
Problems
437
Figure 9.2. Gauss map
Problems
(1) Let l : S2 «-> R3\{0} be the inclusion map. Let r = £*ω where
χ dy A dz + у dz A dx + ζ dx A dy
ω= {x2 + y2 + z2)*/2 '
Compute fs2 τ where S2 is given the orientation induced by τ itself.
(2) Let Μ be an oriented smooth compact manifold with boundary dM
and suppose that dM has two connected components JVo and N\. Let
%i : Ni ^ Μ be the inclusion map for г = 0,1. Suppose that α is a
p-form with ifa = 0 and β an (n — ρ — l)-form with %\β = 0. Prove
that in this case
/ da Α β - (-l)p+1 / α Α άβ.
Jm Jm
(3) Consider the set up in the proof of Theorem 9.22 where 7 G /\n~k and
L7 : Д V* —> R is defined so that L7(a) vol = a A 7. Show that
7 i-> L7 G (Λ* V*)* is linear. Show that if L7(a) = 0 for aU a e /\kV*,
then 7 = 0. Thus 7 t-> L7 is injective (and hence an isomorphism). See
the related Problem 4.
438
9. Integration and Stokes7 Theorem
(4) Recall the notion of a manifold with corners as in Problem 21 in
Chapter 1. Define orientation on manifolds with corners. Let Μ be an n-
manifold with corners. Show that if CM is the set of corner points of
M, then M\CM is a manifold with boundary. Develop integration
theory and Stokes' theorem for manifolds with corners. [Hint: If ω is an
(n — l)-form with compact support in the domain of a chart (f/, x), then
define
/ W:=£/(x-i)V
JdM J^[ JF%
where
Fi := {x € Щ : xl = 0}
is given the induced orientation as a subset of the boundary of {x 6 Rn :
xi > 0}.]
(5) Show that if / : (M, g) —> (TV, h) is an isometry or a local isometry and
ω is a fc-form on JV, then /*cj is harmonic if and only if ω is harmonic,
(6) Let Μ be a connected oriented compact Riemannian manifold with
Laplace operator Δ. A smooth nonzero function / is called an eigen-
function for Δ with eigenvalue λ if Δ/ = λ/.
(a) Show that zero is an eigenvalue and that all other eigenvalues are
strictly positive.
(b) Show that if Δ/ι = X\f\ and Δ/2 = Mfz for λι Φ λ2, then
(/i|/2) = /m/i/2^M = 0.
(a) Show that if ρ : Μ —> Μ is a smooth covering space of multiplicity
m, then for any compactly supported ω G Ωη(Μ) we have
/ ρ*ω — ml ω.
Jm Jm
(7) Let Μ С Rn+1 be an oriented hypersurface. If JV is a positively oriented
normal field along Μ with Ν = Σ N%d/dx%, then the following formula
gives the volume form corresponding to the induced metric on M:
volM := ^2{-ΐγ-ιΝ4χι Λ ■ ■ ■ Λ efe* Λ ■ · · Λ <tan+\
г
where χ1,..., ζη+1 are the standard coordinates on Rn+1 and the dxl
are restricted to M.
(8) Let U be a starshaped open set in Rn and let a:1,..., xn be standard
coordinates. Given
ω= ^ Uiu_jkdx4 A ·· ■ Λ dx%k,
h<--<ik
Problems
439
define
Ιω:= Σ Σ(-1)α"1(Γ**1ί*1 ^ά1)
ii<-<ika-l \J° '
χ xiadxh Λ · · · Λ dxio£ Λ · ■ · Λ azifc,
where the caret means omission. Show that if άω = 0, then d/α; = 0.
(9) Derive the decompositions
Ω*(Μ) = <W (ω*(Μ)) θ δά (ω*(Μ )) θ Ή*
and
Ω* (Μ) = d (Ω*"1 (Μ)) θ 5 (Ω*+1(Μ)) θ Ή*
from
ttk(M) = A(ttk(M)) ®Hk.
(10) Prove Theorem 9.7.
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Chapter 10
De Rham Cohorrclogy
We have already given the definition of de Rham cohomology (8.41) and the
de Rham theorem which says that for a compact oriented smooth manifold
M, the de Rham cohomology is isomorphic with the singular cohomology.
In this chapter we introduce just a few of the most basic tools proper to the
subject such as the Mayer-Vietoris sequence. We shall also introduce the
compactly supported de Rham cohomology and revisit Poincare duality. In
this chapter all manifolds will be without boundary.
For a given η-manifold M, we have the sequence of maps called the
de Rham complex
о 4 ω°(μ) Λ η\Μ) 4... 4 Ωη(Μ) 4 о,
and we have defined the de Rham cohomology groups (actually vector
spaces) as the quotients
Hk(M) =
Bk{M)'
where Zk{M) := Ker(d : Ω*(Μ) ->· Uk+1{M)) and Bk(M) := Im(d :
Ω* X(M) -» Uk(M)). The elements of Zk(M) are called closed fc-forms
or fc-cocycles, and the elements of Bk(M) are called exact fc-forms or k-
coboundaries. If / : Μ ->■ N is smooth and \β] G Hk(N), then since d
commutes with pull-back, it is easy to see that /*/J is closed because β is
closed. Thus we obtain a cohomology class [ί*β\· Also, [/*β] depends only
on the equivalence class of β. Indeed, Ίίβ — β' = άη, then /*β — /*β' = ά/*η.
Thus we may define a linear map /* : Hk(N) -► Hk(M) by /* [β] := [Γβ].
Let us immediately review a simple situation from Section 2.10, which
will help the reader better see what the de Rham cohomologies are all about.
441
442
10. De Rham Cobomology
(10.1) θ = {
Let Μ = R2\{0} and consider the 1-form
xdy - ydx
^:= —2.2'
We got this 1-form by taking the exterior derivative of θ = arctan(y ж .
This function is not defined as a single-valued smooth function on all of
R2\{0}, but ΰ is well-defined on all of R2\{0}. One can extend θ to be
defined on the plane minus the ray {x < 0, у — 0} as follows:
arctan(y/x) if χ > 0,
arccos(z/\/x2 + y2) if χ < 0 and у > 0,
— arccos(z/γ^χ2 + y2) if ж < 0 and 2/ < 0.
See Problem 1. However, this is still not defined on all of R2\{0}. One
may also check that άϋ 0 and so ϋ is closed. Using Proposition 2.125 and
Example 2.126 of Section 2.10, it is easy to see that we have the following
situation:
(1) # := xdJ^x is a smooth 1-form on R2\{0} with du 0.
(2) There is no function / defined on (all of) R2\{0} such that ΰ = df.
(3) Prom Section 2.10 we know that for any ball S(p, ε) in R2\{0} there
is a function / € C°°(B(p,e)) such that #|β(ρε) = df.
Assertion (1) says that $ is globally well-defined and closed, while (2
says that ΰ is not exact. Assertion (3) says that ϋ is what we might call
locally exact or locally conservative. What prevents us from finding a (global
function with ϋ — df? Could the same kind of situation occur if the
manifold is R2? The answer is no, and the difference between R2 and R2\{0} is
that Hl(R2) = 0 while tfx(R2\{0}) φ 0.
Exercise 10.1. Verify (1) and (3) above.
We recall from Chapter 2 that this example has something to do with
path independence. In fact, if we could show that for a given 1-form a, the
path integral J a only depended on the beginning and ending points of the
curve c, then we could define a function f(x) := f* a, where f£ a is just
the path integral for any path beginning at a fixed x$ and ending at x. With
this definition one can show that df = α and so a would be exact. In our
example, the form i? is not exact, and so there must be a failure of path
independence.
Exercise 10.2. A smooth fixed endpoint homotopy between a path со :
[a, b] —l· Μ and c\ : [a, 6] —> Μ is a one parameter family of paths hs such
that ho — cq and h\ = c\ and such that the map H(t, s) := h8(t) is smooth
on [a, 6] x [0,1]. Show that if a is an exact 1-form, then -^ Jh a = 0.
10. De Rham Cohomology
443
The de Rham complex and its cohomology can be viewed in terms of
differential equations. For example, the task of finding a closed 1-form
f dx + gdy on an open set U С R2 amounts to finding a solution of the
differential equation
*-«=0.
дх ду
The "trivial" solutions are the exact forms since they are automatically
closed. Thus the de Rham cohomology Ηλ(υ) is a space of solutions
modulo the "uninteresting" solutions. Similar statements apply to the higher
cohomology groups.
Since we have found a closed 1-form on R2\{0} that is not exact, we
know that Ях(М2\{0}) ф 0. We are not yet in a position to determine
#X(R2\{0}) completely. We will start out with even simpler spaces and
eventually develop the machinery to bootstrap our way up to more
complicated situations.
First, let Μ — {ρ}. That is, Μ consists of a single point and is hence a
0-dimensional manifold. In this case,
" R for к = 0,
пк(Ы) = zk№) = {
0 for к > 0.
Furthermore, Bk({p}) = 0 and so
Hk({p}) = {
R for& = 0,
0 for к > 0.
Next we consider the case Μ = R. Here, Z°(R) is clearly just the
constant functions and so is (isomorphic to) R. On the other hand, B°(R) =
0 and so
Н°(Ш)=Ж.
Now since d : ΩΧ(Ε) -> П2(Ш) = 0, we see that ZX(R) = nx(R). If g(x)dx G
Ω1(Ε), then letting
f(x) := / g(x) dx
Jo
we get df = g(x)dx. Thus, every D}{R) is exact; £X(R) = Ω1^). We are
led to
Hl{R) = 0.
From this modest beginning we will be able to compute the de Rham
cohomology for a large class of manifolds. Our first goal is to compute Hk(Rn)
for all fc. In order to accomplish this, we will need some preparation. The
methods are largely algebraic, and so we will need to introduce a bit of
"homological algebra".
444
10. De Rham Cohomology
Definition 10.3. Let R be a commutative ring. A differential i?-complex
is a direct sum of modules С = ®feGg Ck together with a linear map d:C ->
С such that dod = 0 and such that d{Ck) С Cfe+1. Thus we have a sequence
of linear maps
> С*"1 A Ck A C*+1 -► · · ·
where we have denoted the restrictions df~ — d\ Ck all simply by the single
letter d.
Let A = ®ke%Ak and Б = φ^ζ-Β* ^e differential complexes. A map
/ : A -► Б is called a chain map if / is a (degree 0) graded map such that
d о f = f о d. In other words, if we let f\Ak := Д, then we require that
fk(Ak) С Вк and that the following diagram commutes for all k:
Ak+i
Ak
вк
-i-^A
/fc-1
_1—*-в
h d
[к ^
r
k d
/fc+1
Bk+l
Notice that if / : A —> В is a chain map, then Ker(/) and Im(/) are
complexes with Ker(/) = фЛ€ЖКег(Д) and Im(/) - φ*€ΖΙιη(Λ)· Thus
the notion of exact sequence of chain maps may be defined in the obvious
way.
Definition 10.4. The fc-th cohomology of the complex С = 0*.6Ζ Ск is
Kei{d\Ck)
Η* (С) :=
1m(d\Ck-lY
The elements of Ker(d\Ck) (also denoted Zk{C)) are called Ar-cocycles,
while the elements of lm{d\Ck x) (also denoted Bk(C)) are called
/c-coboundaries.
If / : A —> В is a chain map, then it is easy to see that there is a natural
(degree 0) graded map Д : Η (A) -> H(B) defined by Д([ж]) := [f(x)\ for
Definition 10.5. An exact sequence of chain maps of the form
o^aAbAc^o
is called a short exact sequence.
10. De Rham Cohomology
445
Associated to every short exact sequence of chain maps is a long exact
sequence of cohomology groups:
d*
/·
Hk+\A) — Hk^(B) -?U Hk+1(C)
Hk{A)
Hk(B)
9*
Hk(C)
The maps /* and <?* are the maps induced by / and g, and the "coboundary
map" or connecting homomorphism d* : Hk(C) —> Hk+l(A) is defined
as follows: Let с G Zk(C) С Ck represent the class [c] e Hk(C) so that
dc — 0. Starting with this we hope to end up with a well-defined element
of Hk+l(A), but we must refer to the following diagram with exact rows to
explain how we arrive at a choice of representative of our class d*([c]):
-^ Ak+l
/ 9
±+ Bk+i _%. (jk+i >. 0
0
d\
*Ak
d\
• Bk
d\
ck
-*o
By the surjectivity of д there is a b G Bk with д{Ь) = с. Also, since g{db) =
^(fl'(b)) — d° — 0, it must be that db = /(a) for some a G Ak+l. The scheme
of the process is
с —-> b —-► a.
Certainly f(da) = d(f(a)) — ddb = 0, and so since / is 1-1, we must have
da = 0, which means that a E Zk+l(A). We would like to define d*([c]) to
be [a], but we must show that this is well-defined. Suppose that we repeat
this process starting with d = с + da for some a E Ck 1. In our first step,
we find У e Bk with g{ti) = d and then a! with f(o!) = db'. We wish to
show that [a] = [af]. We have g[b -bf) = c — d — da. But there must be an
element β G Bk~l such that g(P) = a. Now we have
g{b-V-dfi) = g{b)-g(1/)-gW)
= g(b)-g&)-dg(fi)
= с — d — da — 0.
446
10. De Rham Cohomology
By exactness at Bky there must be a 7 G Ak such that /(7) = b — b' - άβ.
Now we have
Ка-а'-*у) = /(а)-/{а')-Ш
= db- dti - d(b - У - άβ) = 0,
and since / is infective, we have а — о! — άη = 0, which means that [α] = [α'].
Thus our definition d*([c]) :— [a] is independent of the choices. We leave it
to the reader to check (if there is any doubt) that d*, so defined, is linear.
Let us review the situation we have with de Rham cohomology noticing
how it fits the abstract homological algebra above. We let Ω*(Μ) := 0 for
к < 0 and we have a differential complex d : Ω(Μ) —> Ω (Μ), where d is
the exterior derivative and Ω(Μ) is the direct sum of the Ω*(Μ). In this
case, Яfe(Ω(M)) = Hk(M) by definition. If / : Μ -ν Ν is a C°° map,
then we have /* : Q(N) —> Ω(Μ). Since pull-back commutes with exterior
differentiation d and preserves the degree of differential forms, /* is a chain
map. Thus we have the induced map on cohomology denoted by /* so that
/*:tf*(iV)->#*(M),
where we have used H*(M) to denote the direct sum 0^ίίι(Μ). Notice
that / h-» /* together with Μ \-+ H*(M) is a contravariant functor, which
means that for / : Μ —> N and д : Ν -¥ Ρ we have
(<7°/)* = Г°<Л
(This is why we have now put the stars in as superscripts.) In particular, if
iu : U —>■ Μ is the inclusion of an open set U in M, then L^a is the same as
the restriction of the form a to U. If [a] € H*(M), then iu*([a]) 6 H*{U);
lu* : H*(M) -» H*(U).
Remark 10.6. If a e Zk{M) and 0 e £г(М), then for any a' € fife г(М
and any /3' € Ωί_1(Μ) we have
(a + da') A^ = aA/3 + da'A/3
= α Α β + d(a' Α β)- (-1)*" V Λ άβ
= αΑβ + d(a' Α β)
and similarly αΑ(β+άβ') = αΑβ+ά(αΑβ'). Thus we may define a product
Hk{M) χ H\M) ->· Hk+l{M) by [α] Λ [/3] := [α Λ /3]. This gives #*(M) а
graded ring structure.
ЮЛ. The Mayer-Vietoris Sequence
447
10.1. The Mayer-Vietoris Sequence
Suppose that Μ = U U V for open sets U and V. We have the following
commutative diagram of inclusions:
U
У Ч
(10.2) U nV Μ
А
ν
This gives rise to the Mayer-Vietoris short exact sequence
0 -> Ω(Μ) 4 Ω(ί7) Θ il(V) дЛ Sl(U Π V) -> 0,
where
and
дг>, /3) :- *5(0) - *ί(α) = β\υην - α^ .
Note that j^u G Ω(Ϊ7) while jjo; G O(V). Note also that ij(a) = a|i/nv аП(^
i*()S) = /31^ у live in Ω([/ Π V). Let us write j* suggestively as (jj, jj).
Let us show that this sequence is exact. First if (jj, jj)^) :~ 0"ιω» ·?2ω)
= (0,0), then ω\Ό = ω\ν = 0 and so ω = 0 on Μ = U U V. Thus (jf, Л)
is 1-1 and the exactness at Ω(Μ) is demonstrated. Next, if η G Ω(ί7 Π У),
then we take a smooth partition of unity {ρυ,Ρν} subordinate to the cover
{Uy V} and let ξ := {—{ρνν)υΛΡυν)ν), where (pvv)11 is the extension of
the form pvlunv7! ^У zero to t/, and {puv)V likewise extends pulunv*! *°
V\ This may look backwards at first, so note carefully that we use pv,
which has support in V, to get a function on 17, and similarly p\j is used to
define {puv)V on V\ Figure 10.1 shows the circle as the union of two open
sets U and V and serves as a schematic of what we have in mind. In the
figure η G Ω0 is 1 on one connected component of U Π V and 0 on the other
component. Now we have
di*(-(pw)uAPuv)v)
= (puV)V\UnV+(pvv)U\UnV
= Puvlunv + Pvvlunv
= (ρυ + ρν)η = η·
Perhaps the notation is too pedantic. If we let the restrictions and extensions
by zero take care of themselves, so to speak, then the idea is expressed by
saying that di* maps the element (—ρνν,Ρυη) G il(U) Θ Ω(ν) to ρυη —
(—pvv) = V £ ^(f7 П V). Thus we see that <9г* is surjective.
ггХ^
448
10. De Rham Cohomology
It is easy to see that di*o(jl,j%) = 0 so that Im(jJ\ ft) c Ker(di*). Now
let (α,/З) e Ω(Ϊ7) 0fi(V) and suppose that 8%*{α,β) = 0. This translates to
alt/nV = ^lt/nv» which means that there is a form ω Ε Ω(?7 U V) = Ω(Μ)
such that cj coincides with α on U and with /3 on V. Thus
(Jli J2V = C7l4 JaM = (αι£)»
so that Кег(дг*) С Im(jJ,j2), which, together with the reverse inclusion,
gives Ker(9i*) = Im(jf, jj).
Following the general algebraic pattern, the Mayer-Vietoris short exact
sequence gives rise to the Mayer-Vietoris long exact sequence:
Hk+l(M) ^ Hk+l{U) Θ Hk+1(V) -^ Hk+1(U Π V)
Hk(M)
Hk(UHV)
Since our description of the coboundary map in the algebraic case was
rather abstract, we will do well to take a closer look at the connecting
homomorphism d* in the present context. Referring to the diagram below,
η Ε ΩΛ([7 Π V) represents a cohomology class [η] Ε Hk(U Π V) so that in
particular άη = 0:
0 ■ * Ω*+1(Μ) -^ nk+1{U) θ nk+1{V) -^ Qk+l{U Π V) —^ 0
A
d®d\
0 *· Пк(М) —^^ Ω*([7) θ nk(V) ^^ Ω*([7 Π F) 0
We will abbreviate the map d Θ d to just d so that ^(771,772) — {ащуащ .
By the exactness of the rows we can find a form ξ € ΩΛ(ί7) Θ Ω*(ν) which
maps to 77. In fact, we may take ξ — (—ργη, ρυη) as before. Since άη = 0
and the diagram commutes, we see that άξ must map to 0 in Ω^+1(ί7 Л V).
This just tells us that — d {pvv) an<i d (puv) agree °n the intersection 17 Π V.
(Refer again to Figure 10.1.) Thus there is a well-defined form in Ω^+1(Μ)
10.2. Homotopy Invariance
449
(^J))'
( -dipyifl Λ
=1
Figure 10.1. Scheme for d* on the circle
which maps to άξ. This global form 7 is given by
( —ά(ρνη) on f/,
7= <
[ ά(ρυη) on K,
and then by definition ά*[η] = [7] e Я*+1(М).
Exercise 10.7. Let the circle S1 be parametrized by the angle θ in the
usual way. Let U be the part of a circle with —π/6 < θ < 7π/6 and let V
be given by 5π/6 < 0 < 13тг/6.
(a) Show that H°{U) ^ Я°(У) ** R.
(b) Show that the "difference map" H°{U)®H°{V) 4 Я°([/П V) has
1-dimensional image.
(c) What is the cohomology of S1?
10.2. Homotopy Invariance
Now we come to a result about the relation between H*(M) and H*(M χ R)
which provides the leverage needed to compute the cohomology of some
higher-dimensional manifolds based on that of lower-dimensional manifolds.
One of our main goals in this section is to prove the homotopy invariance
of de Rham cohomology and also the Poincare lemma. We start with a
theorem which is of interest in its own right. Let Μ be an η-manifold and
also consider the product manifold Μ x R. We then have the projection
450
10. De Rham Cohomology
π : Μ χ R ->M, and for a fixed α G Μ we have the section sfl:M4Mxl
given Ьужн (ж,о). We can apply the cohomology functor to this pair of
maps to obtain the maps π* and s*:
MxR H*(MxR)
Μ H*{M)
Theorem 10.8. Given Μ and the maps defined above, we have that π* :
H*(M)^H*(M χ R) and β J :H*{Mx R) ->#*(M) are mutual inverses for
each a. In particular,
#*(MxR)^tf*(M).
Proof. In what follows we let id denote the identity map on Ω(Μ χ R),
Η* (Μ) and Η* (Μ χ R) as determined by the context. We already know that
id = 5* ο π*. The main idea of the proof is the use of a so-called homotopy
operator, which in the present case is a degree —1 map Κ : Ω (Μ χ R) ->
Ω (Μ χ R) with the property that
(10.3) id-π* ο s* = ±(do К - К о d).
The point is that d о К — К о d must send closed forms to exact forms. Thub
on the level of cohomology ±(dо К — Коd) =0, and hence id —π* οs* must
be the zero map. Thus we also have id = π* ο s* on H*(M), which gives the
result.
So let us then prove equation (10.3) above. Let dt denote the obvious
vector field that is related to the standard coordinate field on R under the
projection МхКчК, and let %Qt denote interior product with respect to
ft. The map К is defined on Ω*(Μ χ R) by
Κ(ω) = (-Ι)*"1 ΓπΧ(*Λω) dr = (-l)k~l f (sT о тг)* (щи;) dr.
J a J a
More explicitly, for vi, >..,Vk-i £ ^(g,t)^ x R>
К(и)\Ш) (vi,... ,vk-x) = f ω{Τ8τοΤπ (Vl),... ,TsToTn {vk ι), dt T)dr.
This operator and d are both local and linear over R, and so if id —π* ο s*a =
±(d о К — Ко d) is true on charts, then it is true in general. Thus we may
as well assume that Μ is an open set U in Rn. For any ω € Q,k(U x R), we
can find a pair of functions /i(x,i) and f2(x,t) such that
ω = Д(Ж) ί)π*α + /2(x, ήπ*β Λ dt
for some forms a e Ω^(ί7) and β Ε Vtk~x(U). This decomposition is unique
in the sense that if fi(x, ΐ)π*α + /2(2:, £)7Г*/? Л dt — 0, then /i(#, t)ir*a = 0
10.2. Homotopy In variance
451
and /2{χ,ί)π*β Λ dt = 0. In what follows we will abuse notation a bit.
The standard coordinates on U will be denoted by Jul I · · · J X , and we write
7г*жг simply as x1 so that with t the standard coordinate of R1, we have
coordinates (ж1,... ,xn,t) on U x R. Using the decomposition above, one
can see that Κ (/ι(χ,ί)π*α) is zero and in general
if : ω Η-
(jf/2(i,r)dr)x^
This map is our proposed homotopy operator.
Let us now check that if has the required properties. It is clear from
what we have said that we may check the action of К separately on forms
of the types fi(x, ί)π*α and fz(x, ί)π*β Λ dt.
Case I (type /(χ,ί)π*α). If ω — f(x,t)n*a, then Κω = 0 and so (do
Κ-Κοά)ω = -Κ{άω). Then
K(du) = K(d(f(x,t)Tr*a))
= K(df(x,t) ΛΛ+ {-l)kf(x,t)n*da)
- К fa ψϊάχί Λ π*α + Ydt Λ π*α + (-!)*/(*, t)n*da\
= (-l)kK(jt(x,t)K*aAdt^
= (-1)" Г %MdT x π*α = (-1)*(/(χ,ί) - /(ζ,α))π*α.
On the other hand, since
πΧ/(ζ, ί) π*α = π* [/(χ, α) (β£ о π*) α]
- π* [(/ о sa)a] - (/ о 50 о π*) π*α = /(χ, α)π*α,
we have
(id-π* ο s*)u; = (id -π* о 5*)/(ж, ί)π*α
= f{x, ί)π*α - /(ж, α)π*α
= (/ОМ)-/(*,<0)я-*а
as above. So in this case we get (do К — К о d)w = ±(id —π* ο 5*)ω.
452
10. De Rhaxa Cohomology
Case II (type ω = f(x,t)π*β Λ dt). We have
d ο Κ (ω) = d о К {f (χ, ί)π*β Adt) = d (π* β ί f{x, τ) άτ)
= π*άβ ( ί f(x, τ) dr\ + (-1)*" V0 Λ f{x, t) dt
+ (-Ι)*" V/З Λ £ Qf ^(x, τ) dr) dj
and
Κ°άω = Κά{ίτ:*βί\ dt) = Kd(π*/? Λ fdt)
= Κ (π*άβ Λ fdt + (-l)fe-V/3 Λ df Λ di)
= Α" (π*άβ Λ /Λ + (-l)fe-V*^ Λ Σ §^ϊάχί Λ di )
= Qf /(x, r) dr) ir-d/3 + (-l)fc-V/? л Σ (jT ^(*. Ό*') <***■
Thus (d о if - Κ ο ά)ω = (-1)*-V)9 Л /(i,t) di = (-l)fc"1u;. On the
other hand, we also have (id —π* о s*)u/ = ω since s* di = 0 and so (d о
К — К о d) = ±(id— π* ο s*), which is equation (10.3). As explained at
the beginning of the proof, this implies that π* and s* are inverses and so
Я*(МхМ)^Я*(М). D
Corollary 10.9 (Poincare lemma).
Hk(Rn) = Hk(point) = ( J* V
Jfc = 0,
otherwise.
Proof. One first verifies that the statement is true for Hk(point). Then the
remainder of the proof is a simple induction:
Я* (point) S Hk(point χ R) = Я*(К)
*Hk(RxR)=Hk(R2)
—
* Я^М71"1 xR) = Hk(Rn). □
Corollary 10.10 (Homotopy axiom). /// : Μ —► N and д : Μ -ϊ Ν are
homotopic, then the induced maps f* : H*(N) —> H*(M) and g* : H*(N) -4
H*(M) are equal
10.2. Homotopy in variance
453
Proof. By extending the homotopy as in Exercise 1.77 we may assume that
we have a map F : Μ χ R —> N such that
F{x,t) = f(x) fori>l,
F{x,t)-g(x) fori <0.
If s\(x) :— (x, 1) and sq(x) := (x,0), then f = Fosi and g = Fоs0> and so
f*-4°F*,
9* = s*0oF*.
It is easy to check that s* and sj$ are one-sided inverses of π*, where Mxi-^
Μ is the projection as before. But we have shown that π* is an isomorphism.
It follows that s± = Sq in cohomology, and then from the above we have
/*=5*. □
Homotopy plays a central role in algebraic topology, and so the last
corollary is very important. Recall that if there exist maps / : Μ —>■ N and
g : N —>> Μ such that both fog and g о f are defined and homotopic to
idjv and idjtf respectively, then / (or g) is called a homotopy equivalence,
and Μ and JV are said to have the same homotopy type. In particular, if a
topological space has the same homotopy type as a single point, then we say
that the space is contractible. If we are dealing with smooth manifolds, we
may take the maps to be smooth. In fact, any continuous map between two
smooth manifolds is continuously homotopic to a smooth map. We shall use
this fact often without comment. The following corollaries follow easily.
Corollary 10.11 (Homotopy invariance). If Μ andN are smooth manifolds
which are of the same homotopy type, then H*(M) = H*(N).
Corollary 10.12. If Μ is a contractible η-manifold, then
tf°(M)^R,
Hk(M)=0for0<k<n.
Next consider the situation where A is a subset of Μ and г: A *->· Μ is
the inclusion map. If there exists a map r : Μ —>■ A such that r oi = id^,
then we say that r is a retraction of Μ onto A. If A is a regular submanifold
of a smooth manifold M, then in case there is a retraction r of Μ onto A,
we may assume that r is smooth. If we can find a smooth retraction r such
that tor is smoothly homotopic to the identity id^, then we say that r
is a (smooth) deformation retraction, and this homotopy itself is also
called a deformation retraction. In this case, A is said to be a (smooth)
deformation retract of M.
Corollary 10.13. If A is a smooth deformation retract of M} then A and
Μ have isomorphic cohomologies.
454 10. De Rham Cohomology
Exercise 10.14* Let (7+ and U~ be open subsets of the sphere Sn С Rn+1
given by
C7+ := {{x*) eSn:-e< xn+1 < 1},
U- := {(ж*) € Sn : -1 < xn+1 < ε},
where 0 < ε < 1/2. Show that there is a deformation retraction of {7+ Π17
onto the equator xn+1 = 0 in Sn. Notice that the equator is a two point set
in case η = 0. Show that Sn is not contractible. (See also Problem 6.)
Theorem 10.15 (Hairy sphere theorem). A nowhere vanishing smooth
vector field exists on the sphere Sn if and only if η is odd.
Proof. If X is a nowhere vanishing vector field on £n, then define ν :
X/ \\X\\. We introduce a map [0, l]xSM Sn by
F(x, t) := (cos πί) χ + (sin nt) v{x).
This is a smooth homotopy between the identity map and the antipodal
map a : χ ь->> — χ. Now if vols™ is the volume form on Sn, then [vols ] к
not zero in Hn(Sn). Indeed, suppose that vol^n = du. Then 0 φ vol(5n)
JSn vol = JqU = 0, which is a contradiction. Recall the special n-forra on
Rn+1 given in Cartesian coordinates by
n+l ^
σ := J^(-l)71 Veto1 Λ · · - Λ dx* Λ · · - Λ ώ?η+1.
t-l
It was shown previously that volsn is the restriction of σ to 5η. But if η is
even, then n+l is odd, and it is clear from the above formula that α*σ — -σ.
Since vol^n is obtained from σ taking restriction, we have a* vol^n = vols ,
and so also on the level of cohomology:
a* [volsn] = — [volsn].
But this is impossible since if the homotopy exists, we must have a* id*
on Hn(Sn). We conclude that a nonvanishing vector field does not exist on
Sn if η is even.
If η is odd, then we can easily construct a nowhere vanishing vector field
on 5™. For example, if η = 3, then let
д д д д
Χ - Х2Т, Я1- h X±-z Я3-—,
σχχ ΟΧ2 дхз Οχι
where Ж1,Ж2?#з,#4 are the standard coordinates on R4, and restrict X to
S3. Notice that X is indeed tangent to S3. The generalization to higher
odd dimensions should be clear. D
10.2. Homotopy Invariance
455
A trivial consequence of the Poincare lemma is that the cohomology
spaces of a Euclidean space are finite-dimensional. Below we use an
induction that shows that this is true for a large class of manifolds which include
all compact manifolds. For this and later purposes, we introduce a technical
condition. An open cover {Ua}aeA of an η-manifold Μ is called a good
cover if for every choice ao,..., α& the set Uao П · · · П Uak is diffeomorphic
toRn (or empty).
Proposition 10.16. // {Οβ)β^β is any open cover of an η-manifold M,
then there exists a good cover {Ua}aeA such that each Ua is contained in
some Ο β.
Proof. The proof requires a result from Riemannian geometry (see Problem
2 of Chapter 13). We know that every manifold can be given a Riemannian
metric. Each point of a Riemannian manifold has a neighborhood system
made up of small geodesically convex sets. What matters to us now is that
the intersection of any finite number of such sets is geodesically convex and
hence diffeomorphic to Rn. Actually, the assertion that an open geodesically
convex set in a Riemannian manifold is diffeomorphic to Rn is common in
the literature, but it is a more subtle issue than it may seem, and references
to a complete proof are hard to find (but see [Grom]). Granted this claim,
the result follows. D
Many of the results we develop below will be proved for orientable
manifolds that possess a finite good cover. This includes all orientable compact
manifolds.
Theorem 10.17. If an η-manifold has a finite good cover, then its de Rham
cohomology spaces are all finite-dimensional
Proof. The proof is an induction argument that uses the Mayer-Vietoris
sequence. Our fc-th induction hypothesis is the following:
P(jfc): Every smooth manifold that has a good cover consisting of к open
sets has finite-dimensional de Rham cohomologies.
Since we know that an open set diffeomorphic to Rn has
finite-dimensional de Rham cohomology spaces, the statement P(l) is true by Corollary
10.9. Suppose that P(k) is true. We wish to show that this implies P(k +1).
So suppose that Μ has a good cover {f/i,..., f4+i} and let Mk := U\ U
•••Ut/fc.
Since we are assuming P(fc), Mk has finite-dimensional de Rham
cohomology spaces. Note that Mk Π ί/^+ι has a good cover, which is just
{Ι7ι Π E/fc+i,. ·., CTjb (Ί t/fc+i}. From the long Mayer-Vietoris sequence we
456
10. De Rham Cohomology
have for a given q
—► H«(Mk П Uk+1) A H*(Mk+1) ^$ H*+1(Mk) Θ H*+l(Uk+1) —►,
which gives the exact sequence
0 —> Ker d* A Hq(MM) L^$ Im (ij + tj) —> 0.
Since Hq(Mk Π i/fc+i) and #9+1(Mfe) Θ Ηί+1(1/*+ι) are finite-dimensionaJ
by hypothesis, the same is true of Kerd* and Iui(6q + ij). It follows that
ifg(Mjfe+i) is finite-dimensional and the induction is complete. D
10.3. Compactly Supported Cohomology
Let ilc(M) denote the algebra of compactly supported differential forms on
a manifold M. Obviously, if Μ is compact, then ilc(M) = Ω(Μ), and so
our main interest here is the case where Μ is not compact. We now have a
slightly different complex
.••Αω*(μ) Αω*+1(μ)Λ·..,
which has a corresponding cohomology H*(M) called the de Rham
cohomology with compact support. By definition
нЧм\ - Z^M)
НЛМ)-Щм~у
where Z%(M) is the vector space of closed fc-forms with compact support
and Βς(Μ) is the space of all &-forms du) where ω has compact support.
Note carefully that B^(M) is not the set of exact fc-forms with compact
support. To drive the point home, consider / G C°°(Rn) with compact
support and with / > 0 and / > 0 at some point. Then
ω ~ f dx1 Λ · · · Λ dxn
is exact since every closed form on Rn is exact. However, ω cannot be da
for an a with compact support, since then we would have
/ ω — j da= a — 0,
which contradicts the assumption that / is nonnegative with / > 0 at some
point. This already shows that #™(Rn) φ 0. Using a bump function with
support inside a chart, one can similarly show that Щ{М) Ф 0 for any
orient able manifold M.
Exercise 10.18. Let Μ be an oriented η-manifold. If ω Ε Ω"(Μ) and
/м£^0, then [ω] ^0 in Я?(М).
Exercise 10.19. Show that ЯСХ(М) = К and that H°(M) = 0 whenever
dim Μ > 0 and Μ is connected.
10.3. Compactly Supported Cohomology
457
If we look at the behavior of differential forms under the operation of
pull-back, we immediately realize that the pull-back of a differential form
with compact support may not have compact support. In order to get
desired functorial properties, we consider the class of smooth proper maps.
Recall that a smooth map / : Ρ —> Μ is called a proper map if f~l(K)
is compact whenever К is compact. It is easy to verify that the set of all
smooth manifolds together with proper smooth maps is a category and the
assignments Μ н->- Ω0(Μ) and / ■->· {a «->■ /*α} give a contravariant functor.
In plain language, this means that if / : Ρ —> Μ is a proper map, then
/* : Clc(M) —> Ω0(Ρ) and for two such maps we have (/ о д)* = д* о /* as
before, but now the assumption that / and д are proper maps is essential.
We will use a different functorial behavior associated with forms of
compact support. The first thing we need is a new category (which is fortunately
easy to describe). The category we have in mind has as objects the set of
all open subsets of a fixed manifold M. The morphisms are the inclusion
maps jy,u : V ^ Uy which are only defined in case V С U. For any such
inclusion jvtUi we define a map {jv,u)+ · Ω0(ν) —> Ω0([7) according to the
following simple prescription: For a E Ω<.(ν), let {jvp)*** be the form in
QC(U) which is equal to a at all points in V and equal to zero otherwise
(extension by zero). Since the support of a is neatly inside the open set V,
the extension (jV,*/)* α *s smooth. In what follows, we take this category
whenever we employ the functor fic.
Let U and V be open subsets which together cover M. Recall the
commutative diagram of inclusion maps (10.2). Now for each fc, let us define
a map (-iu , *2*) : Ω*(Ι7 Π V) -► Ω*(17) θ Ω*(ν) by а и- (-ii*a, г2*а).
Now we also have the map ju + 32* : Ω* (V) Θ Ω0(ί7) -* Ω*(Μ) given by
(θίΐ,αί2) ^ ju&i + j2*Qi2· Notice that if {φυ^Φν} is a partition of unity
subordinate to {[/, V}, then for any ω G Ω*(Μ) we can define ωυ := φυω\υ
and ωγ := φνω\ν so that we have
(ju + 32*) (ωυ,ωγ) = h*uu + h*uy = φυω + φγω = ω.
Notice that ωυ and ωγ have compact support. For example,
Supple/) = Supp(0(/a;) С Supp(0iy) Π Supp(cj).
Since Supp(0[/) Π Supp(cj) is compact, Supp(u;£/) is also compact. Thus
ju + J2* is surjective. We can associate to the diagram (10.2), the new
sequence
(10.4) 0 -> nkc{V П U) (-^2+) η*(7) Θ Ω*(Ζ7) л^?* Ω*(Μ) -> 0.
This is the short Mayer-Vietoris sequence for differential forms with compact
support.
Theorem 10.20. The sequence (10.4) is exact
458 10. De Rham Cohomology
We have shown the surjectivity of j\* + J2* above. The rest of the proof
is also easy and left as Problem 4.
Corollary 10.21. There is α long exact sequence
#*+1(t/ η ν) —> Η^λ(υ) θ Η^(ν) —^ н^+1{м)
#*([/ Π V) * H*(U) θ H£{V) > #*(Μ)
which is called the (long) Mayer-Vietoris sequence for cohomology
with compact supports.
Notice that we have denoted the connecting homomorphism by d*. We
will need to have a more explicit understanding of d*: If [ω] € Η*{Μ), then
using a partition of unity {pu>Pv} as above we write ω — ju^u + J2*wv
and then djuuu = —dJ2*ouy on U Π V. Then
(Ю.5) d*[w] = - djuu>u\unv = dJ2*ufy\UnV .
Next we prove a version of the Poincare lemma for compactly supported
cohomology. For a given η-manifold Af, we consider the projection π :
Μ χ R -»■ Μ. We immediately notice that π* does not map Ω*(Μ) into
ilJ(MxR). What we need is a map π* : Ω£(Μ χ R) —> Ω*? Χ(Μ) called
integration along the fiber. Before giving the definition of π* we first note
that every element of Ω£(Μ χ R) is locally a sum of forms of the following
types:
Type Ι: /π*λ,
Type Π: /π^Λώ,
where λ € Ω* (Μ), φ € Ω£_1(Μ) and / is a smooth function with compact
support on Μ χ R. By definition π* sends all forms of Type I to zero, and
for Type II forms, we define
/oo
f(;t)dt.
-oo
By linearity this defines π* on all forms. In Problem 2 we ask the reader to
show that π* ο d = d ο π* so that π* is a chain map. Thus we get a map on
10.3. Compactly Supported Cohomology
459
cohomology:
π* : ff*(M xR)4 H*~l(M).
Next choose e G Ω^№ w^h /e — 1 and introduce the map e* : Ω£(Μ) -4
tt£+1(^fxR)by
e* : a; (-» π*ω Λ 7г£е,
where π2 : Μ χ R -* R is the projection on the second factor. It is easy to
check that e* commutes with cf, so once again we get a map on the level of
cohomology:
e* : tf*(M) -> tf*+1(M χ R) for all k.
Our immediate goal is to show that e* ο π* and π* о е* are both identity
operators on the level of cohomology. In fact, it is not hard to see that
π* о е* = id already on Ω* (Μ). We need to construct a homotopy operator
К between e* ο π* and id. The map Κ : Ω£(Μ χ R) -> Ω*-1 (Μ χ R)
is given by requiring that К is linear, maps Type I forms to zero, and if
ω = π*φ - f Adt is Type II, then
Κ {ω) = Κ(π*φ -fAdt) := π*φ ( I /(я, и) du - T(t) i /(ж, и) du\ ,
where T(t) = J e. In Problem 3, we ask the reader to show that
(10.6) id -e* ο π* = (-l)*"1^ - Kd)
on Ω* (Μ χ R). It follows that id = e* ο π* on #* (Μ χ R), and so finally we
have the following result:
Theorem 10.22. With notation as above the following maps are
isomorphisms and mutual inverses:
тг* : tfcfc(M χ R) -> Η*-λ{Μ),
e* : #*_1(Μ) -> H£{M χ R).
Corollary 10.23 (Poincare lemma for compactly supported cohomology).
We have
tfcn(Rn)=R,
#*(Rn) = 0 t/fe^n.
Proof. By Exercise 10.19 we have #*(R) = R. But from the previous
theorem tf?(Rn) S Щ"1^1) * - · · * #*(»)· Also, trivially, tfcfe(Rn) =
0 for к > η, and if к < η, then
#*(Rn) * Я*"1^""1) * - - - ^ Я°^п"*) - 0
by Exercise 10.19. D
460
10. De Rham Cohomology
The isomorphism H^(Rn) = R is given by repeated application of π*,
which is just integration over the last coordinate of Rn.
Remark: Notice that W1 is homotopy equivalent to Rn^1 and yet,
Щ{Жп) ф tf^R71-1). This contrasts the compactly supported
cohomology with the ordinary cohomology, since the latter is a homotopy invariant.
10.4. Poincare Duality
Looking back at our results for Rn we notice that Hk(Rn) = H^k{Rn)\
This is a special case of Poincare duality which is proved in this section.
Recall that we have already met Poincare duality for compact manifolds
in Section 9.6. The version we develop here will be valid for noncompact
manifolds also. So let Μ be an oriented η-manifold which is not necessarily
compact. For each A;, we have a bilinear pairing
fi*(M)xtt?-fe(M)->R,
(ωι,ω2) н> /
Jm
and as in Section 9.6 this defines a pairing on cohomology:
tf*(M)xtf?-*(M)->R,
flwibM) H> / ωχΑω2.
Jm
In turn, this provides a linear map PDk : Hk(M) -» (#™_A;(M))* defined
as
ωι Лш2,
PDk([<Ji])([u>2]):= ! ωιΛω2.
Jm
Our goal is to show that this map is an isomorphism. We prove this for
orientable manifolds with a finite cover, but the result remains valid more
generally (see [Madsen]). Notice also that unless the cohomologies are
finite-dimensional, we may have Нк(Жп)* ф #cn"fc(Rn). The reason is that
for an infinite-dimensional vector spaces, V* = W does not imply V = W*.
In what follows, we will denote all the duality maps PD^ by the single
designation PD. We use a theorem from algebra called the five lemma:
Consider the following diagram of vector spaces (or abelian groups) and
homomorphisms:
10.4. Poincare Duality
461
Lemma 10,24 (Five lemma). Suppose that the above diagram commutes
up to sign (so for example h[ ο α = ±β ο hi). Suppose also that the rows are
exact and that α,β,δ, and ε are isomorphisms. Then the middle map η is
also an isomorphism.
The five lemma is usually stated for the case when the diagram
commutes, but a simple modification of the usual proof (see [L2]) gives the
version above.
If {[/, V} is a cover of the oriented η-manifold Af, then consider the
following diagram:
- Hk-\U)®Hk-\V) — Hkl(Uf\V) -i Hk(M) — Hk(U)®Hk(V) -
Ϊ 1 1 i
Here the bottom row is the sequence of dual maps from the Mayer-Vietoris
long exact sequence and is exact itself (easy check). The vertical maps are
the duality maps PD (or obvious direct sums of such).
Lemma 10.25. The above diagram commutes up to sign.
Proof. First let us consider the diagram
Hh{M) — > Hk{U) Θ Hk(V)
У
Щ-к{М)* ——--гНГк(иу®нгк(У)*
(Л*) +U2*)
In order to show that (PD Θ PD) о (j* + j*) = ((jb)* + (j2*)*) о PD, it is
enough to show that PD о j* = (ju)* о PD and PD о j* = (j2*)* ° PD.
For a given [ω] e Hk(M)y the linear form PD о Jx([u;]) takes an element
[θ] e Hk(U) to
f 3Ϊω Λ θ.
Ju
On the other hand, ((ju)* о PD) ([ω]) maps [Θ] to
ω Λ ,71*0.
Jm
But ω Λ ju9 and ]{ω Λ θ both have support in U and agree on this set.
Therefore the above two integrals are equal. Similarly, PDoj% = (J2*)*oPD.
462
10. De Rham Cohomology
Next we show that the following square commutes up to sign:
Hk~1(U П V) £-»- Hk(M)
V
щ-к+ιφ n vy _^ щ~к{му
Let {pi/,/>y} be a partition of unity subordinate to {[/, V} as before. If
[ω] € Ял_1(17 Π У), then d*[cj] is represented by a form (which we denote
by ά*ω) that has the properties
ά*ω\υ = — ά(ργω) on Ϊ7,
cf ω|^ = d(puuj) on V.
On the other hand, if [τ] £ l^~fc(M), then d* [r] is represented by a form
d*r which has the properties
-ii*d*r = d(pjju) on [/,
i2*d*r — d(pvoj) on V.
Using the fact that ω is closed, we have
PD о d* {[ω]) ([τ]) = / d*w Λ τ = / ^Лг
= / d(pijuj) Λ τ = / dpi/ Α ω Ατ,
Junv Junv
where have used the fact that ά*ω Α τ has support in U П V. Meanwhile,
(d,)* ° PD (M) (M) = / u>Ad*r = - [ ω Ad (pur)
Junv Junv
= — / ω Λ dpi/ Λ τ,
so the two integrals are equal up to sign. We leave the sign commutativity
of the remaining square to the reader. D
Theorem 10.26 (Poincare duality). Let Μ be an oriented η-manifold with
a finite good cover. Then for each к the map PDk : Hk(M) -> (#™~fc(M))*
is an isomorphism.
Proof. We will prove that PDk is an isomorphism by induction. The
inductive statement P(N) is that PD^ is an isomorphism for all orientable
manifolds which have a good cover consisting of at most N open sets. By the
two Poincare lemmas we already know that if U is diffeomorphic to Rn, then
Hk(U) « (Щ-кф))*, where both sides are either zero or isomorphic to R.
It is then an easy exercise to show that in fact PD^ : Hk(U) —>· (Щ~к(и))*
is an isomorphism. This verifies the statement P(l). Now assume that
ЮЛ. Poincare Duality
463
Ρ (Ν) is true and suppose we are given an oriented manifold Μ with a good
cover {[/χ,..., t/jv+i}· Let Mn := U\ U · · ■ U Un and use Lemma 10.25 with
U := Mn, V := 17jv+i and C/ U У — M. Then by assumption PD is an
isomorphism for [/, V and ί/ Π У, so the hypotheses of Lemma 10.24 are
satisfied. But then the middle homomorphism Hk(M) —> Щ~к(М)* is an
isomorphism (k was arbitrary). D
Corollary 10.27. If Μ is α connected oriented η-manifold with finite good
cover, then H™(M) = R. This isomorphism is given by integration over M.
This last corollary allows us to define the important concept of degree
of a proper map. Let Μ and JV be connected oriented η-manifolds and
suppose that / : Μ —¥ N is a proper map. Then a pull-back of a
compactly supported form is compactly supported so that we obtain a map
/* : Щ (N) —У Щ (M) and the following commutative diagram:
Ш
In
I ^R
Deg/
The induced map Deg^· : R —> R must be multiplication by a real number.
This number is called the degree of/ and is denoted deg(/). If [ω] Ε Щ (Ν)
is chosen so that fNu = 1, then
/ /*a; = deg(/).
Such a form is called a generator and can be chosen to have support in an
arbitrary open subset V of JV. To see this, just choose a chart {U,x) in JV
with U С U С V and let φ be a cut-off function with supp φ С U. Then
let ω := φ dx1 Λ · · ■ Λ dxn and then scale φ so that JN ω = 1. Of course, ω
is closed, but also [ω] ^ 0. Indeed, if we had [ω] = 0, then there would be
an (n — l)-form η with compact support and with ω — άη. But then Stokes'
theorem applies, and so JNu = $Νάη = JdNr} = 0 since dN = 0. This
would contradict fNu = 1.
Theorem 10.28. Let Μ and N be connected n-manifolds oriented by
volume forms μΜ and μχ respectively. If f : Μ -ϊ Ν is a proper map and
у € N is a regular value, then
deg(/) = ^ sigaj,
where sign^/ = 1 if (/*μτν) (#) w a positive multiple of Дм(#) and —1
otherwise. In particular, deg(/) is an integer.
464
JO. De Rham Cohomology
Proof. Using Sard's Theorem 2.34 and the fact that / is proper, we may
choose a neighborhood V of у such that f~l (V) — U^-i^i* where Uzr\Uj 0
for г Ф j and such that f\v is a diffeomorphism onto V for each i. Choose
[ω] € Щ (Ν) with fNu — 1, and such that ω has support in V\ We may
arrange that V and all ЭД are connected so that signx/ is constant on each
Ut. In this case, /\ц is orientation preserving or reversing according to
whether signx/ is 1 or —1. Let χι G U% be the inverse image of у in U%. We
have
N N
deg(f) = / Γω = £ / /Ъ = 2 (sign^/) / ω
= ^ sign,,/. D
*€/ Mi/)
We close this chapter with a quick explanation of the Poincare dual of
a submanifold. Let Υ be an oriented regular submanifold of an oriented
η-manifold. Let the dimension of Υ be η — к. Then we obtain a linear map
ί :Ω? *(M)—>K,
ω н^ / ι ω.
where ι: Υ ^ Μ is the inclusion map. Since this map is zero on exact forms
(by Stokes' theorem), it passes to the quotient giving a linear functional
J : НГк(М)
By Poincare duality PD : Hk(M) £ (Я£~*(М))*, there must be a uniqu
class [θγ] 6 Hk(M) such that this linear form is given by
ί [ω] - ί θγ Λ ω for all [ω] Ε Щ~к{М).
This class (or a given representative θγ) is called the Poincare dual of
the submanifold Y.
For more on this and other important topics such as the Thorn isomo -
phism, the Leray-Hirsch theorem, and Cech cohomology see [Bo-Tu].
Problems
465
Problems
(1) Show that formula (10.1) defines a smooth function θ on the open set
U :- M?\{x < 0, у - 0} and show that
xdy-ydx
άθ = —5-—г— on U.
xl + уг
(2) Show that π* : Ω* (Μ χ R) -* Ω*"1 (Μ) as defined in the discussion
leading to Theorem 10.22 is a chain map.
(3) Prove the homotopy formula (10.6) (or see [Bo-Tu], pages 38 39).
(4) Prove exactness of (10.4).
(5) Prove Corollary 10.11.
(6) Use Exercise 10.14 and the long Mayer-Vietoris sequence to show that
if к — 0 or n,
^ 0 otherwise.
(7) Show that ω e Qn(Sn) is exact if and only if fSn ω - 0.
(8) Determine the cohomology spaces for the punctured Euclidean spaces
Rn\{0}. Show that if Б is a ball, then Rn\B has the same cohomology.
(9) Let Bx be a ball centered at χ 6 Rn. Show that a closed (n — l)-form
on Жп\Вх is exact if and only fs l*oj = 0 for some sphere centered at
x. Here t : 5 ^ W1 is the inclusion map. Show that this is true for all
spheres centered at x.
(10) Let η > 2 and suppose that ω is a compactly supported η-form on Rn
such that /Rn ω = 0.
(a) Let Б1 and B2 be open balls centered at the origin of Rn with
B\ С Βχ С В<х and suppu; С Вь By the (first) Poincare lemma
there is a form a such that da = ω. Show that fdB2 a = 0.
(b) Continuing from (a), show that a is exact on Шп\В (use Problem
9 above).
(c) Let ρ G Ωη 2(Rn\B) be such that dp = a where a is as above.
Show that there is a function g such that if β :·— α — d(gp), then
β is smooth, compactly supported and άβ = ω. Deduce that
tfcn(Rn) = 0.
(11) Find H*(M) where Μ is Rn\{p,$} for some distinct points p,q 6 Mn.
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Chapter 11
Distributions and
jYobenhis' Theorem
The theory of distributions is a geometric formulation of the classical
theory of certain systems of partial differential equations. The solutions are
immersed submanifolds called integral manifolds. The Probenius theorem
gives necessary and sufficient conditions for the existence of such integral
manifolds. One of the most important applications is to show that a subal-
gebra of the Lie algebra of a Lie group corresponds to a Lie subgroup. We
give this application near the end of this chapter. We also develop a classic
version of the Probenius theorem and use it to prove the basic existence
theorem for surfaces.
One can view the theory studied in this chapter as a higher-dimensional
analogue of the study of vector fields and integral curves. If X is a smooth
vector field on an η-manifold Μ, then we know that integral curves exist
through each point, and if X never vanishes, then the integral curves are
immersions. Nonvanishing vector fields do not even exist on many manifolds,
and so we consider a somewhat different question. A one-dimensional
distribution assigns to each ρ e Μ a, one-dimensional subspace Ep of the
tangent space TpM. We say that the assignment is smooth if for each ρ Ε Μ
there is a smooth vector field X defined in an open set U containing ρ such
that X{p) spans Ep for each ρ Ε U. It is easy to see that a one-dimensional
distribution is essentially the same thing as a rank one subbundle of the
tangent bundle.
A curve с : (α, 6) —> Μ with с φ 0 is called an integral curve of the
one-dimensional distribution if c(t) — Ttc · J^ is contained in Ec^ for each
467
468
11. Distributions and Frobenius' Theorem
Figure 11.1. No integral manifold through the origin
t G (a,b). The curve is an immersion, and its restrictions to small enough
subintervals are integral curves of vector fields that locally span the
distribution. Since integral curves cannot cross themselves or each other, the curve
is an injective immersion. The images of such curves are called integral
manifolds (they are immersed submanifolds). Of course, every nonvanishing
global vector field defines a one-dimensional distribution, but some one-
dimensional distributions do not arise in this way. For example, the tangent
spaces to the fibers of the Mobius band bundle MB —l· Sl form a
distribution not spanned by any globally defined vector field (see Example 6.9).
The integral manifolds are the fibers themselves.
11.1. Definitions
We wish to generalize the above idea to higher-dimensional distributions.
Definition 11.1. A smooth rank к distribution on an η-manifold Μ is a
(smooth) rank к vector subbundle Ε —> Μ of the tangent bundle.
Sometimes we refer to a distribution as a tangent distribution to
distinguish it from the notion of a distributional function on a manifold. Recalling
the definition of a vector subbundle (Definition 6.26) and the criteria given
in Exercise 6.27, we see that a smooth rank к distribution on an n-manifold
Μ gives a fe-dimensional subspace Ep С TpM for each ρ £ Μ such that for
each fixed ρ € Μ there is a family of smooth vector fields Χχ}..., X^ defined
on some neighborhood Up of ρ and such that Xx(q),..., Хк{ч) are linearly
independent and span Eq for each q € Up. We say that such a set of vector
fields locally spans the distribution. In other words, Χχ,... ,Χ& can be
viewed as a local frame field for the subbundle E.
Consider the punctured 3-space Μ = R3\{0}. The level sets of the
function ε : (я, у, х) ь^ χ2 + у2 +х2 are spheres whose union is all of R3\{0}.
Now define a distribution on Μ by the rule that Ep is the tangent space at
11 Λ. Definitions
469
ρ to the sphere containing p. Dually, we can define this distribution to be
the one given by the rule
Ep = {ν G TpM : de{v) = 0}.
Thus we see that a distribution of rank n—1 can arise as the family of tangent
spaces to the level sets of nondegenerate smooth functions. More generally,
rank к distributions can arise from the level sets of sufficiently nondegenerate
Rn_fe-valued functions. On the other hand, not all distributions arise in this
way, even locally.
Definition 11.2. Let Ε —У Μ be a distribution. An immersed submanifold
S of Μ is called an integral manifold (or integral submanifold) for the
distribution if for each ρ G S we have Ti(TpS) = Ep, where i : 5 ч М is
the inclusion map. The distribution is called integrable if each point of Μ
is contained in an integral manifold of the distribution.1
If ι: S 4 Μ is the inclusion map, then we usually identify Tl(TpS) with
TPS so the condition which makes S an integral manifold is stated simply
as Ep = TpS for all ρ G S.
Consider the distribution in R3 suggested by Figure 11.1. For each p,
Ep is the subspace of TpR3 orthogonal to
a
yYx
d_
dy
д
+ d~z
P
If there were an integral manifold though the origin, then it would clearly be
tangent to the ж, у plane at the origin. But if one tries to imagine following
a closed path on such an integral manifold around the ζ axis, a problem
appears. Namely, the curve would have to be tangent to the distribution,
but this would clearly force the curve to spiral upward and never return to
the same point.
A distribution as defined above is sometimes called a regular
distribution to distinguish the concept from that of a singular distribution which is
defined in a similar manner, but allows for the dimension of the subspaces
to vary from point to point. Singular distributions are studied in the online
supplement [Lee, Jeff].
Definition 11.3. Let Ε —> Μ be a distribution on an η-manifold Μ and
let X be a vector field defined on an open subset U С Μ. We say that X
lies in the distribution if X(p) G Ep for each ρ in the domain of X; that
is, if X takes values in E. In this case, we sometimes write X G Ε (a slight
abuse of notation).
1In other contexts, integral manifold might be defined by the weaker condition Tt(TpS) С Ер.
We will always require the stronger condition Tc(TpS) = Ep.
470
11. Distributions and Frobenius' Theorem
Definition 11.4. If for every pair of locally defined vector fields X and Υ
with common domain that lie in a distribution Ε —► Μ, the bracket [X, Y]
also lies in the distribution, then we say that the distribution is involutive.
It is easy to see that a vector field X Ues in a distribution if and only if
for any spanning local frame field Xl,..., Xk we have X = Σι=χ flX% for
smooth functions fl defined on the common domain of X, Xl, ..., X&.
Locally, finding integral curves of vector fields is the same as solving
a system of ordinary differential equations. The local model for finding
integral manifolds of a distribution is that of solving a system of partial
differential equations. Consider the following system of partial differential
equations defined on some neighborhood U of the origin in R2:
zx = F{x,y,z),
zy = G{x,y,z).
Here the functions F and G are assumed to be smooth and defined on
ί/xi We might attempt to find a solution defined near the origin by
integrating twice. Suppose we want a solution with jz(0, 0) = zq. First, solve
f'(x) = F(x, 0, f(x)) with /(0) = zq. Then for each fixed χ in the domain of
/ solve dyz(x,y) — G(x,y,z(x,y)) withz(;r,0) = /(x). This function ζ (χ,у)
may not be a C2 solution of our system of partial differential equations since,
if it were, we would have zXy = ZyX. But
zxy = {zx)y = Q-Fix> У>z) = Fy + GFy,
and similarly, zyx = Gx + FGX. Thus the equality of mixed partials gives a
necessary integrability condition;
(11.2) Fy + GFy = Gx + FGx.
We shall see that this condition is also sufficient for the local solvability of
our system.
Exercise 11.5.
(a) Show that solving the differential equations above is equivalent to
finding the integral curves of the following pair of vector fields.
(b) Show that the graph of a solution to the system (11.1) is an integral
curve of the distribution spanned by X and У.
(c) Show that [X,Y] = 0 is the same condition as (11.2).
There are no integrability conditions for finding integral curves of a
vector field. But now we see that we should expect integrability conditions
for the existence of integral manifolds of higher rank distributions.
11.2. The Local Frobenius Theorem
471
11.2. The Local Frobenius Theorem
We will relate the notion of involutivity to the existence of integral manifolds.
Lemma 11.6. Let Ε —ϊ Μ be α distribution (tangent subbundle) on an n-
manifold M. Then Ε —ϊ Μ is involutive if and only if for every local frame
field X\,..., Xk for the subbundle Ε —у М we have
к
[Xi, Xj] = 2_^ cfj^s
s=l
for some smooth functions cL ( 1 < i,j, s < k).
Proof. It is clear that such functions exist if Ε is involutive. Now, suppose
that such functions exist. If X and Υ lie in the distribution, then
к
for smooth functions fl and g*. We wish to show that [X, Y] lies in the
distribution. This follows by direct calculation:
L * 3
= Σ /V [Xit X-[ + J2f (Xtf) Xj -Y^gi (X,f) Xt. Π
hj i,3 hj
We will show that a distribution is involutive if and only if it is integrable.
In fact, we shall see that the integral manifolds fit together nicely. In the
following discussion we identify Rk χ Rn~k with Rn.
Definition 11.7. A rank к distribution Ε —» Μ on an η-manifold Μ is
called completely integrable if for each ρ £ Μ there is a chart (f/, x)
centered at ρ such that x(U) = V χ W С Шк x Шп~к and such that each
set of the form x^1(V χ {у}) for each у € W is an integral manifold of the
distribution.
If x(p) = (α\...,αη), then
x-4V x {</}) = {q 6 U : xk+\q) = a*+\ ..., χ«(β) = β"},
where у = (afc+1,..., an). For a chart as in the last definition we have
T(x-i(Vx{y})) = E\I-4Vx{y})
472
11. Distributions and Frobenius' Theorem
for each у e W. We can always arrange that V and W in the above definition
are connected (say open cubes or open balls). In this case, we refer to such a
chart as a distinguished chart for the distribution. The integral manifolds
of the form x~1(Vx {y}) for a distinguished chart are called plaques. Notice
that since V and W are assumed path connected for a distinguished chart,
the plaques are path connected. The reader was asked to prove the following
result in Problem 4 of Chapter 3, but we will state and prove the result here
for the sake of completeness.
Lemma 11.8. Suppose that f : Ν —> Μ is a one-one immersion and that
Υ is a vector field on Μ such that Y(f(p)) € Tpf {TpN) for every ρ G N.
Then there is a unique vector field X on N that is f-related to Y.
Proof. It is clear that we can define a field X such that X(p) is the unique
element of TPN with Y(/(p)) = Tpf · X{p). The task is to show that X so
defined is smooth. For ρ G iV, let (17, x) be a chart centered at ρ and (V,y
a chart centered around f(p) such that у о / ο χ"1 has the form
(а\...,ате)к>(а\...,^0,...,0).
Then it is easy to check that
Tpf-
dxi
So if
dyi
д
/(ρ)
for smooth functions Yl, then
dxi
where we must have Хг = Υ1 ο /. This shows that the Хг are smooth and
hence the field X is smooth. D
We apply the previous lemma to the situation where S is an immersed
submanifold of Μ and ι : S ^y Μ is the inclusion. In this case, if Υ is
a vector field on Μ which is tangent to 5, then there is a unique smooth
vector field on 5, denoted Y\s, such that Y\s (q) = Y(q) for all q G S. The
vector field Y\s is called the restriction of Υ to S and it is ^-related to У.
Theorem 11.9 (Local Probenius theorem). A distribution is involutive J
and only if it is completely integrable and if and only if it is integrable.
Proof. First, completely integrable clearly implies integrable. Now assume
that Ε —> Μ is a rank к distribution which is integrable and let dim(M) = n.
Let X and Υ be smooth vector fields defined on U which both lie in the
J 1.3. Differential Forms and Integrability
473
distribution. Let ρ Ε U and let S be an integral manifold that contains p.
Then Xq, Yq Ε TqS for all q € U Π 5. Thus the vector fields are tangent to
U Π S and hence if ι : U Π 5 Μ- U is the inclusion map, then X and У are
^-related to their restrictions to U Π S as in the previous lemma. This means
that [X\UnS, Y\uns\ is related to [X>Y] and hence [X,Y]q € TqS = Eq
for all q Ε 17 Π S and in particular [-Ϊ, Y]p Ε i?p. Since ρ was an arbitrary-
element of Uy we conclude that the distribution is involutive.
Now suppose that Ε —> Μ is involutive. Pick a point po Ε Μ and let
(i7,x) be a chart centered at po- By shrinking U if necessary we can find
fields -ΧΊ,..., Xk defined on U that span the distribution. Notice that if we
have a chart of the type produced by Theorem 2.113, then we can easily
arrange that the image of the chart is of the form V x W С Rk x Rn~k,
and it is then easy to see that we have exactly the kind of chart needed (see
the comments immediately following the proof of Theorem 2.113). Thus the
strategy is to replace the frame field X\,..., Xk by a commuting frame field.
Construct a chart (17,x) centered at po such that Xj(po) = g|j| for
1 < i < *- Let π : Шп -> Rk be the map (a1,..., an) \-t (a1,..., ak) and let
/ := π ο χ. We have
д
(γλ/)№(ρο)) = γρο/Α
where (ί1,... ,tk) are standard coordinates on Rfc. Then T^f : Γ^Μ -*
ToRfe maps £^ isomorphically onto ToR*. It follows that Tpf maps i?p
isomorphically onto T^R* for all ρ in some neighborhood of po· Hence, for
each such p, there are unique vectors Yi(p),..., lfc(p) in Ep with Γρ/ (lj(p))
= ^|-|,/ у Thus we get vector fields Yi,... ,1^ whose smoothness we leave
to the reader to check. By shrinking U if necessary, we may assume that
these are defined on U. It is clear that Yi,..., Yjt form a local frame for
Ε and Yj is /-related to ^-. Thus [Yi,lj] is /-related to [^, ^-] and in
particular
д д
Τρί([Υ^]Ρ)^
dtv дР
= 0.
о
Since the distribution Ε is involutive, we have [Yi,lj]p Ε Ερ for each p.
Then since Tpf is injective on Ep, we see that [Yi,lj]p — 0 for all p. Thus
we arrive at a local frame field Yi,..., Υ& for Ε with [YJ, Y}] = 0.
By Theorem 2.113 and the comments above, we are done. D
11.3. Differential Forms and Integrability
One can use differential forms to effectively discuss distributions and involu-
tivity. What is interesting is the way the exterior derivative comes into play.
474
11. Distributions and Frobenius' Theorem
The first thing to notice is that a smooth distribution is locally defined by
1-forms. We need a bit of creative terminology:
Definition 11.10. If α1,...,α*" are 1-forms on an η-manifold Μ and S
is a subset of M, then we say that a1,...,^ (defined at least on some
open set containing S) are independent on S if {al(p),..., α?(ρ)} is an
independent set in Τ*Μ for each ρ € S.
Proposition 11.11. Let Ε —> Μ be α rank к distribution on an n-manifold
M. Every ρ £ Μ has an open neighborhood U and 1-forms a1,... ,an k
independent on U such that
Eg = Ker ax(q) Π · · · nKeran"fc(g) for all qeU.
Proof. Let ρ £ Μ and let ΑΊ,..., X^ span Ε on a neighborhood О of p.
We may extend to a frame field ΧΊ,..., X^..., Xn defined on some possibly
smaller neighborhood U of p. Indeed, choose any other frame field ΥΊ,..., 1^
and then rearrange indices so that -X"i(p),..., -Χ*(ρ), lVu(p), · · · > Yn(p) are
linearly independent. The vectors Xi{q),..., Xk{q), Yk+i{q), ■ ■ · ? ϊη(?) mus^
be independent for all g in some neighborhood of p, which we may as well
take to be O. Indeed, q η· -ΧΊ(?) Λ · · · Λ Xk(q) Λ lfc+i(9) Λ · · ■ Λ Fn(g)
is nonzero at ρ and so remains nonzero in a neighborhood. In any case,
let Χχ}..., Λ&,..., Xn be our extended local frame and let 01,..., θη be
the dual frame. Then it is easy to check that ν Ε ТЯМ is in Eq if and
only if 0j+1(v) = .·- = 0J(v) = 0. Thus we just let (a1,...,***1-*) :=
(0*+1,...,0n). □
The forms described in the previous proposition are sometimes called
local defining forms for the distribution.
The set of all differential forms vanishing on a distribution has a
convenient algebraic structure. Recall that Ω(Μ) is a ring under the wedge
product. If J is the ideal in Ω(Μ) and U С Μ is open, then J\v is the
set of restrictions to U of elements of J and is an ideal in Ω (17) since the
wedge product is natural with respect to restrictions (that is, with respect
to pull-back by inclusion maps).
Definition 11.12. Let J be an ideal in Ω(Μ); we say that J is locally
generated by n—к smooth 1-forms if there is an open cover of Μ such that
for each open U from the cover, there are 1-forms a1,..., an~k independent
on U such that J\u is an ideal of Ω(17) generated by a1,..., an~k.
Exercise 11.13. Show that if J is locally generated by η — к smooth 1-
forms as above, then it defines a unique smooth rank к distribution Ε on Μ
such that for each U in the cover, and corresponding a1,..., an"fe, we have
Eq = Ker al(q) ΓΙ ■ · · П Keran~*(g) for all q e U.
11.3. Differential Forms and Integrability
475
Definition 11.14. Let Ε —► Μ be a rank к distribution on an n-manifold
M. Let U be open in M. A j-form ω G Ω*"(17) is said to annihilate the
distribution on U if for each ρ eU
Up(vi,..., Vj) = 0 whenever νχ,..., vj G £p.
An element ω G Ω(17) is said to annihilate the distribution if each
homogeneous component of ω annihilates the distribution. The subset of all
elements of Ω (17) that annihilate the distribution Ε —>> Μ is denoted 1(17).
It is easy to check that X(U) is an ideal in Ω(Ϊ7). Clearly, local defining
forms for a distribution annihilate the distribution.
Lemma 11.15. The following conditions on α distribution Ε —► Μ are
equivalent:
(i) Ε —l· Μ is an involutive distribution.
(ii) For every open U С Μ, and 1-form ω G Ωχ({7) that annihilates the
distribution, the form du also annihilates the distribution on U.
Proof. The proof that these two conditions are equivalent follows easily
from the formula
(11.3) du>(X,Y) = Χ{ω{Υ)) - Υω{Χ) - ω([Χ,Υ}).
Suppose that (i) holds. If ω annihilates Ε and Χ, Υ lie in E, then the above
formula becomes
MX,Y) = -u([x,Y]),
which shows that άω vanishes on Ε since [X, Y] G Ε by condition (i).
Conversely, suppose that (ii) holds and that X>Y G E. Let a1,...,^-* be
local defining 1-forms. Then each form da1 annihilates the distribution and
also a?{X) = a^(Y) = 0 for all j. Using equation (11.3) again, we have
α'([Χ, Υ]) = -άα{(Χ, Υ) = 0 for all t so that [X, Y]eE. Π
Lemma 11.16. Let Ε —> Μ be a rank к distribution on the η-manifold M.
A j·form ω annihilates Ε if and only if whenever a1,..., an~k are local
defining 1-forms on some open U', we have
(11.4) ω\υ = Υ^α^βί
г=1
for some β1,.. .,βη~* G Qj'l(U).
Proof. Recalling the definition of exterior product makes it clear that if ω
satisfies (11.4), then it annihilates Ε on U. If ω satisfies such an equation
near each point for some generating 1-forms, then ω annihilates Ε on all
ofM.
476
11. Distributions and Frobenius' Theorem
Suppose that α ,..., ап к are local defining forms for the distribution
defined on an open set U and let ρ G U. Then on some neighborhood V
of ρ we can extend the coframe field a1,..., otn~k to a1,..., αη in such а
way that a1,..., an is independent on V. Define Xi, ..., Xn as dual to the
frame otl\v ,.. ■, <*п\у so that Xn &+χ,... ,Xn span Ε on V. Now for any
ω G Qi(U)} we have
wlv = Σ Wii'"b ail|vA""Aa<Jlv
ύ<···<ί<7
where a;tl..·^ = u^X^,... ,Хг,) on V. Thus ω annihilates Ε on V if and
only if ω(Χ%ι,..., X* ) = 0опУ whenever η — к + I < ίχ < - · - < i3 < п.
So if ω annihilates Ε, then in the expansion of ω we need only include those
ω%λ„.% such that ii < · · ■ < i3 and such that at least one index is less than
or equal to η - fe. But in the latter case we would have i\ < η — fc. Thus
we can write
ω\ν= Σ Vivt, <?%*·-* <**'\ν
I\ii<n—k
= Σα<ΙνΑ( Σ ^i2-i3at2\vA'--A<xi3\v\
l 1 \Ϊ2<-<1, /
η к
:=Σ>ΊνΛ«-
г 1
This expression holds on У С [/, but there is a cover of U by such V
on which similar expressions hold. Denote this cover by V. By using a
partition of unity {фу : V Ε V} subordinate to this cover we can patch these
expressions together. Notice that for any V € V we have Φν {®г\у) = Φνα%
for 1 < г < η — к, so
η Λ
vev vev ι ι
= ΣΣ(^αίΙν)Λ^
vev г-\
η—к п к
VGV г 1 t=l VeV
η—к
11,3. Differential Forms and Integrability
477
Corollary 11.17. A distribution Ε —> Μ is involutive if and only if for
every local defining set of 1-forms a1,..., an~k we have
η к
з ι
for some l-forms 7J with 1 < i,j <n — к.
Proof· Just use the above theorem together with Lemma 11.15. D
Theorem 11.18. A distribution Ε —l· Μ on Μ is involutive (and hence
completely integrable) if and only if dl(E) С 1(E).
The condition dX(E) С 1(E) is expressed by saying that 1(E) is a
differential ideal.
Proof. Suppose that dX(E) С 1(E). If ω is any 1-form that annihilates Ε
on an open set C/, then pick an arbitrary point ρ EU and a smooth cut-off
function φ with support in U and equal to one on a neighborhood of p. Then
ά(φω) G 1(E). But ά(φω) (ρ) = άω(ρ), and since ρ was arbitrary, we see
that du annihilates Ε on U. The open set U was also arbitrary so we can
apply Lemma 11.15.
Now suppose that Ε —ϊ Μ is involutive and choose η e 1(E). Without
loss of generality we may assume that η is homogeneous of degree j. Notice
that the case j = 0 is trivial, while the case j — 1 is Lemma 11.15. For
j > 2, consider a local defining set of l-forms a1,..., an~k on an open set
U. Since by assumption η annihilates E, we can use Lemma 11.16 and
Corollary 11.17 to write
n—k n—k
= Σάαι Λβ{ ~ХУ Ad?
г-l i=l
n к п к
г—1 j г=1
Now use Lemma 11.16 again to conclude that άη annihilates Ε and so is in
1(E). D
We close this section with a convenient application of vector-valued
forms.
Proposition 11.19. Let ω be a 1-form on an n-manifold with values in a
finite-dimensional vector space V. Then Ex = Keru^ defines a distribution
478
11. Distributions and Frobenius' Theorem
Ε if dim Ex = к is constant for χ G M. This distribution is involutive and
hence integrable if and only if άω(Χ, Υ) = 0 whenever Χ, Υ lie in E.
Proof. The first statement is cleax once we choose a basis ei,..., en for V
for then we can write ω = Σ^№% for some linearly independent smooth
1-forms ω1,... ,uk which clearly define the distribution.
Choose local spanning fields -ΧΊ,..., Xk defined on an open set XJ. Then
Ε is integrable on U if and only if
[Xu Xj] G span{Ai,..., Xk} for 1 < г, j < Л,
which is true if and only if ω([Χ^ Xj]) = 0 for 1 < г, j < к. But as before,
<L· (XuXj) = XMXj)) - XMXi)) - ujUXuXj])
= -u([Xt,Xj]).
Thus Ε is integrable on U if and only if du (X{, Xj) = 0 for 1 < г, j < k.
This implies the result. D
11.4. Global Frobenius Theorem
First notice that if (f/, x) is a distinguished chart for a distribution on an
η-manifold with x(U) = VxW сШкх Шп~к, then the plaques хч(7х {у})
for each у 6 W are embedded integral manifolds. If О С V and у е W, then
x_1(Ox {y}) is an open subset of the plaque x"1 (V x {y}) (in its submanifold
topology). Clearly, all open subsets of a plaque are of this form.
Proposition 11.20. Let Ε -> Μ be an integrable distribution of rank к
on the η-manifold M. Let (Ϊ7,χ) be a fixed distinguished chart for Ε where
x(U) = V x W С Rfc x Rn~k. If S is a connected integral manifold for E}
then S Π U is a countable disjoint union of connected open subsets of the
plaques of (t/,x). These open connected subsets are open in S and embedded
in M.
Proof. Since the inclusion t : 5 4 Μ is an immersion, the sets S Π 17 =
i~l{U) are open in S. The set 5 Π U is a disjoint union of connected
components each of which is open in S. Since S is second countable, this union
must be a countable union. Let С be a connected component of 5 Π U.
We show first that С С х"*1 (У х {у}) for some у G W. Since (17, x) is a
distinguished chart, the 1-forms dx^1,... ,dxn are local defining forms for
the distribution and so dxkJtl = ■ · · = dxn = 0 on S Π U and hence on C.
But С is connected, so xk+1,..., xn are all constant on C, which puts it in
a single plaque, say x~~l(V χ {у}).
We know that C4Mis smooth. Since x-1(F x {y}) is embedded in
Μ, the inclusion С <-+ χ 1(V x {г/}) is also smooth by Corollary 3.15 and is
thus an injective immersion. Since С and x~1(Vr x {y}) are manifolds of the
HA. Global Frobenius Theorem
479
same dimension, the inclusion С «^ χ l(V x {y}) is a local diffeomorphism
and hence an open map. It follows that С <->■ x~l(V x {y}) is an embedding.
Since the inclusion C4Mis the composition of embeddings C^x^fFx
{y}) <->■ M, it is itself an embedding. D
If Ε —> Μ is an integrable distribution of rank к on the η-manifold Μ
and ρ Ε Μ, then we would like to consider the union
N- [J 5,
where ^(p) is the family of all connected integral manifolds containing p. We
would like to say that N is the maximal integral manifold that contains p,
but we must show that JV is indeed an integral manifold. Let {(Ua, xa)}aeA
be an atlas of distinguished charts for the distribution where ха(иа) =
Va x Wa С Rk x Rn"*\ On each plaque x"1^ x {y}) of {Ua,xa) which is
in JV, we define
ρΓι ο χα : x-l(Va х {у}) -> Ve С Kfe,
and this map is taken as a chart on JV whose domain is a plaque. Charts
obtained in this way will be called plaque charts and the family of all
such charts provides JV with an atlas. The question of the smoothness of
overlaps is routine and is left to the reader. Thus we have a smooth atlas
on JV which induces a topology such that each plaque, and indeed, each
member of ·7·"(ρ), is open. The topology induced by this smooth structure is
often called the leaf topology. It is also evident that JV is connected since
it is the union of connected sets containing a fixed point. We can also see
directly that JV is path connected. Indeed, if q is in JV, then it must be in
some Si G ·7·"(ρ). But also ρ € Si by definition, and so since Si is connected
(and hence smoothly path connected), there is a smooth curve connecting
ρ and q. The questions that remain are whether the topology is Hausdorff
and whether it is paracompact. Once again the proof that JV is Hausdorff
is easy and is left to the reader. The harder question is paracompactness.
But this follows from the fact that JV clearly satisfies property W(k) of
Definition 3.18, and so N is a weakly embedded submanifold by Proposition
3.20. On the other hand, the part of the proof of that proposition that
established paracompactness appealed to induced Riemannian metrics and
material about topologies induced by metrics that we still have not covered.
For this reason we offer a direct and traditional proof. First, since N is
connected, the goal is to prove that it is second countable. The inclusion
Ν ^ Μ is continuous, so JV is contained in a connected component of M.
Thus we may assume without loss of generality that Μ is second countable,
and so also that the atlas {(Ua, xa)}aeA of distinguished charts is countable.
This latter fact is one of the key points. Notice that each plaque is a regular
480
11, Distributions and Frobenius' Theorem
Figure 11.2. Slice accessible from point ρ
submanifold, so each such plaque is itself second countable. Observe that
by construction, if a set TV Π Ua intersects a plaque in Ua, then that plaque
is actually contained in ΝΠ Ua. Indeed, if a plaque of Ua intersects iV, then
there is an element S G F{p) such that S meets the plaque in an open set in
the topology of the plaque (Proposition 11.20). But then the union of this
plaque with S is another element of T{p) and so occurs in the union that
defines N. This reduces the problem to that of showing that N contains
only a countable number of such plaques.
We consider all sequences of the form (ί/α(ΐ),-Ρι),. - -, (t/a(m),-Pm)j
where P$ is a plaque for a chart (ί/α(φΧα(ι)) taken from our countable
atlas of distinguished charts, such that Pm — Ρ is a plaque contained in
Ua(m) Π Ν,
реРъРт = Р, Pi С N and Pi П Pi+ι φ 0 for г 1,..., m - 1.
Notice that τη is not fixed, but must be finite. Let us call such a sequence a
finite connection from ρ to the plaque P. We first show that for each plaque
Ρ contained in some UaC\ N} there is at least one such finite connection
from ρ to P. To see this, consider a point q € Ua Π Ν and suppose that
q 6 P. Let с : [0,1] —> N be a smooth curve with c(0) = ρ and c(l) q.
Now e([0,1]) is a compact set, so there is a sequence t\ < t% < * · · < tm
such that for each г, c([£i,ii+i]) is contained in a distinguished chart, say
(^а(Фха(г))· Of course, each c([ti,U+i\) is also connected and so must be
contained in a connected component of Ua^ Π Ν and thus in some plaque
which we call P*. Clearly we have created a finite connection from ρ to P.
Now consider the family С of all finite connections of ρ to the plaques
contained in the various Ua Π N. Each finite connection can be mapped to
the plaque on which it terminates, and we have just seen that this map is
surjective. Thus the set of plaques of the various ϋβ ΠΝ that are finitely
connect able to ρ must have cardinality less than or equal to that of С It
may already seem that since the atlas is countable, С must also be countable.
However, while the set of all finite sequences ί7α(ι),..., C/a(m) is certainly
11.4. Global Frobenius Theorem
481
countable, there is still the question of how many ways there are to choose
the Pi inside each Ua^ to build a connection of ρ to a plaque. However,
since each plaque Pi is certainly an integral manifold, Proposition 11.20 tells
us that it can only intersect Ua^^ in a countable number of plaques. Thus
for a given sequence ί/α(ΐ),..., Ϊ7α(τη) there is only a countable number of
ways to choose the corresponding P^ It follows that N contains only a
countable number of plaque charts and so is a second countable connected
integral manifold for the distribution. It is clearly maximal by construction.
One final thing to notice is that unless the rank of the distribution is
equal to the dimension of the manifold, Μ has an uncountable number of
maximal integral manifolds, and they partition M. We have now proved the
following:
Theorem 11.21. Let Ε -^ Μ be an integrable distribution of rank к on the
η-manifold M. There is a set of maximal integral manifolds for Ε which
partition N. These are weakly embedded submanifolds.
There is another slightly different way of looking at the above theorem.
Namely, the set of all plaque charts puts a new smooth structure on the set
M. We have seen that the overlaps are only nonempty if the plaques are on
the same integral manifold. This smooth structure and topology gives us a
new manifold which we might denote by M(E). It has the same underlying
set as M, but has a finer topology. The connected components of M(E) are
exactly the maximal integral manifolds.
Proposition 11.22. If L\ and £2 are connected integral manifolds of an
integrable distribution, then either C\ Π Li is empty or it is an open subset
of the maximal integral manifold containing any point of £1 Π £2- The set
£1 Π £2 is also open in the integral manifolds L\ and £2.
Proof. Assume that £1Π £2 is not empty and let ρ Ε £χ Π £2. Then there is
a distinguished chart (t/, x) for the distribution containing ρ such that both
C\C\U and £2ΠΪ7 are plaques containing p. Clearly, the plaques must be the
same. Since plaques are open by definition in the maximal integral manifold,
and since ρ was an arbitrary point in £1Π £2, it follows that £1Π £2 is open
in this maximal integral manifold. The last statement is obvious. D
Corollary 11.23. Suppose that Μ and N are smooth manifolds and that
there is an integrable distribution Ε on Μ χ N. Suppose that for some
connected open subset U of Μ there are smooth functions fi:U—>N and
/2 : t/ —>> iV such that the graphs of /1 and /2 are integral manifolds. Then
these graphs are either disjoint or equal. In the latter case, f\ = f%.
482
11. Distributions and Frobenius' Theorem
Proof· Let Гд and Tf2 the graphs and suppose that Γ/χ Π Γ/2 is not empty.
Let (p0j /i(Po)) = (Po, /2CP0)) € Γ/χ Π Γ/2 and consider the set
5 = {p€^:/i(p) = /2(p)}.
Then 5 is clearly closed. We show that it is also open. Pick ρ e 5. Then
Γ/χ and Tf2 intersect at the point (p, fi(p)) = (p>/2(p))· But since Гд
and Г/2 are integral manifolds of an integrable distribution on Μ x iV, the
previous proposition tells us that Гд П Гд is open in Гд, and so, since
prx : U x N —> i/ restricts to a diffeomorphism on Гд (as for all graphs of
smooth functions), we see that S = ргх(Гд ПГд) is an open set in U. Since
U is connected, we have S = U. Thus f\ = /2 and the graphs are equal. D
If we consider what we have discovered about integrable distributions,
we arrive naturally at the following definition (and new terminology).
Definition 11.24. Let Μ be an η-manifold. A fc-dimensional foliation Ты
of Μ (or on M) is a partition of Μ into a family of disjoint connected subsets
{£<х}аеА such that for every ρ Ε Μ, there is a chart (17, x) centered at ρ of
the form x:U^VxWcRkx Rn^k for connected V and W with the
property that for each Ca the connected components (U Π £α)β of U Γ\£α
are given by
φ{(υπ£α)β) Vx{caj},
where ca^ eW С F are constants. These charts are called distinguished
charts for the foliation or foliation charts. The connected sets £a are
called the leaves of the foliation while for a given chart as above, the
connected components (U Π £α)β are called plaques.
It is easy to see that if Тм is a foliation as above, then there is a unique
integrable distribution Ε —> Μ on Μ such that if ρ is in the domain of a
foliation chart (i7,x), then
Ep = Ker dx\kp+l Π ■ ■ ■ Π Ker dx% .
Furthermore the leaves of the foliation are exactly the maximal integral
manifolds of the distribution. We call this the distribution generated by the
foliation. Combining this observation with Theorem 11.21 we obtain
Theorem 11.25. Every integrable rank к distribution gives rise to a unique
к-dimensional foliation whose leaves are the maximal integral mantfolds.
Conversely, a foliation generates an integrable distribution whose maximal
integral manifolds are exactly the leaves of the original foliation.
Thus the theory of foliations is essentially the study of integrable
distributions. Now we see that the leaves are weakly embedded submanifolds.
The connected components {ϋΓ\£α)β οίϋΠ£α are of the form Cx(UC\Ca)
HA. Global Frobenius Theorem
483
for some χ G Ca (recall Definition 3.17). An important point anticipated by
our concern above is that a fixed leaf Ca may intersect a given chart domain
U in many, even a countably infinite number of disjoint connected pieces no
matter how small U is taken to be. In fact, it may be that U Π Ca is dense
in U. On the other hand, each Ca is connected by definition. The usual first
example of this behavior is given by the irrationally foliated torus. Here we
take Μ = Τ2 := S1 χ S1 and let the leaves be given as the image of the
immersions ia:t\-^ (βίαπί, et7rt), where α is a real number. If a is irrational,
then the image ta(M) is a (connected) dense subset of S1 xSl. On the other
hand, even in this case there are infinitely many distinct leaves.
It may seem that a foliated manifold is just a special manifold, but
from one point of view, a foliation is a generalization of a manifold. For
instance, we can think of a manifold Μ as a foliation where the points are
the leaves. This is called the discrete foliation on M. At the other extreme
a manifold may be thought of as a foliation with a single leaf С = Μ (the
trivial foliation). We also have the following special cases:
Example 11.26. On a product manifold, say Μ χ iV, we have two
complementary foliations:
{{p} x N}peM
and
{Μ χ {q}}qeN>
Example 11.27. Given any submersion / : Μ —> JV, the level sets
{f~1{Q)}qeN form the leaves of a foliation.
Example 11.28. The fibers of any vector bundle foliate the total space.
Example 11.29 (Reeb foliation). Consider the strip in the plane given by
{(x,y) : |ж| < 1}. For a € R U {±oo}, we form leaves Ca as follows:
£a := {(ж, a + f(x)) : |x| < 1} for a <E M,
£±00:={(±l,y):|y|<l},
where f(x) :— exp(уз^г) ~ 1· By rotating this symmetric foliation about the
у axis we obtain a foliation of the solid cylinder. This foliation is such that
translation of the solid cylinder С in the у direction maps leaves diffeomor-
phically onto leaves, and so we may let Ζ act on С by (x, y, z) H- (ж, y+n, z)
and then C/Z is a solid torus with a foliation called the Reeb foliation.
Example 11.30. The one point compactification of R3 is homeomorphic to
S3CR4. Thus S3 - {p} £ R3 and so there is a copy of the Reeb foliated
solid torus inside S3. The complement of a solid torus in S3 is another solid
torus. It is possible to put another Reeb foliation on this complement and
thus foliate all of 53. The only compact leaf is the torus that is the common
boundary of the two complementary solid tori.
484
11. Distributions and Frobenius' Theorem
Theorem 11.31. Let Тм be α foliation on Μ and С a leaf of the foliation.
Then for any smooth map f : N —>■ Μ with f(N) С С the map f : N —> С
is smooth.
Proof. Just use Theorem 3.14. D
11.5. Applications to Lie Groups
First we fulfill the promise of proving Theorem 5.66, which we repeat here
for convenience.
Theorem 11.32. Let G be a Lie group with Lie algebra g. If fj is a Lie
subalgebra ofg, then there is a unique connected Lie subgroup HofG whose
Lie algebra is f).
Proof. For α G G, we let Δα denote the subspace of TaG that is the set of
all X{a) for left invariant vector fields X with X{e) G f). Thus va G Δα if
and only if va = TeLa · ν for some ν G f). If -ΧΊ,..., Xk are left invariant
fields such that -Xi(e),..., -У*(е) is a basis for f), then Xl, ..., Xk span the
distribution which is therefore smooth. Since i) is a subalgebra, it follows
that Δ : α ι-ν Δα is an integrable distribution. Let Η be the maximal
(connected) integral manifold containing e. Note that for any b € G, we have
TeLb(Aa) = Aba so that TLb : TG -> TG leaves the distribution invariant.
Thus Lb induces a permutation of the set of maximal integral manifolds and
takes the maximal integral manifold through a diffeomorphically to that
through ba. Thus if h G ii, then L^-ι takes Η to the maximal integral
manifold containing e, which is just H. In other words, Lh-i(H) = H. This
shows that if is a subgroup. It remains to show that the multiplication map
μ : Η χ Η —> Η is smooth. But the multiplication map into G, Η χ Η -> G,
is smooth and so by Theorem 11.31 the map Η χ Η —ϊ Η is smooth. D
Theorem 11.33. Let G and Η be Lie groups with respective Lie algebras
g and f). If h : q —> f) is a Lie algebra homomorphism, then there is a
neighborhood U of the identity e G G and a smooth map f : U —> Η such
that
f(xy) = f(x)f(y)
whenever x, у Ε U and xy G U, and such that
Tef-v = h(v)
for every ν G g.
Proof. Let t С Q x f) be defined by
l:={(v,h(v)):vee}.
11.5. Applications to Lie Groups
485
The fact that h is a homomorphism implies that I is a Lie subalgebra of
gxf). Thus by Theorem 11.32, there is a connected Lie subgroup К of
G χ Η with Lie algebra t. Now let ι: К «->■ G χ Η be inclusion and define
a homomorphism ρ : Κ —> G by
ρ := prx о l.
If υ Ε 3, then
Τρ.(ν,Λ(ν)) = ν,
and this means that Tp : T^e)K -^ TeG is a, linear isomorphism. Thus by
the inverse mapping theorem, there is a neighborhood V of (e, e) Ε Κ such
that p\v is a diffeomorphism onto an open neighborhood U of e 6 G. Define
the homomorphism φ : К —> Η by
φ := pr2 о 6,
where pr2 : G x Я -> Я is the second factor projection. Notice that Теф ·
(v,h(v)) = Λ(ν). Now let
/ — V^pIv1·
A straightforward diagram chase argument shows that f(xy) = f(x)f(y) if
x,y eU and xy Ε С/.
If ν G 0, then Tp(v, /i(t;)) = i; implies that T(p γ ) ■ ν = (ν, h(v)) so
Τ/-ν = ΤψοΤ (ply1) ■ t; - Τφ ■ (ν, h(v)) = h(v). Ώ
Theorem 11.34. If fi : G —> Η and /2 : G —> Η are Lie group homomor-
phisms with df\ = df<i : g —> f) and G is connected, then /1 = /2.
Proof. Let h := dfi = d/2 and define I := {(v, /i(v)) : ν e g} and К cGxH
as in the proof of the previous theorem. Now define θ : G —l· G χ Η Ъу
θ(χ) := (χ, /χ(χ)). The image of 0 is a subgroup Κι <Z G x H. For vej,
we have Ге0 · ν = (ν,Λ(ν)) so the Lie algebra of ATi must be 6. Since G is
connected, we must have К = K\ which implies that /1 = / on f/, where /
and U are constructed from h as in the last theorem. But equally, /2 = h
on [/, and so by Proposition 5.40 we have Д = /2. □
We say that two Lie groups G and Я are locally isomorphic if there
is a diffeomorphism / from a neighborhood U of the identity of G onto a
neighborhood V of the identity of Η such that f(xy) — f(x)f(y) whenever
x,y and xy are contained in U.
Corollary 11.35. The following assertions hold:
(i) Two Lie groups with isomorphic Lie algebras are locally isomorphic.
(ii) A connected Lie group with abelian Lie algebra is abelian.
486
11. Distributions and Frobenius9 Theorem
Proof, (i) Let h : g —> f) be a Lie algebra isomorphism. Then if / is the map
constructed in Theorem 11.33, then / is a diffeomorphism on some possibly
smaller neighborhood of the identity since Tef = h is an isomorphism.
(ii) By (i) a connected Lie group G of dimension η must be locally
isomorphic to the (additive) abelian Lie group Rn. But a neighborhood of
the identity generates the whole group, and so G is abelian. D
11.6. Fundamental Theorem of Surface Theory
In this section we state and outline the proof of a fundamental theorem
concerning the existence of surfaces with prescribed first and second
fundamental form. Our proof of the main theorem follows [Pa2]. To begin with,
we need a few results about certain systems of partial differential equations.
The first is equivalent to the local Frobenius theorem. For an open set U С
Rk χ IRm, denote standard coordinates by (x, z) = (x1,..., xk^ z1,..., zm).
Proposition 11.36. Let U be an open set in Шк χ Rm and let [Α%Λ be an
m χ к matrix of smooth functions on U. Then the following assertions are
equivalent:
(i) For every (#o52o) € U, there is a neighborhood V of xq in Rk and
a unique smooth map f : V —> Rm with f(xo) = %o such that
(11.5) ^(x)=A)(x,f(x)) for alli,j.
(ii) The functions A1, satisfy the following system of equations on U:
Proof. If (i) is true, then we obtain (ii) by equality of mixed partials of the
f% and the chain rule (see the comments following the proof).
Conversely, consider the vector fields on U defined by
д
Xj '- dxi
A bit of linear algebra tells us that these are everywhere independent. Let
Ε —> U be the distribution spanned by these fields. A straightforward check
using (11.6) shows that
[Xi,Xj] = o,
so there is an integral manifold through each p. Let N be the integral
manifold through (хсь^о)· Using the last m coordinate functions of some
distinguished chart, we obtain a map Φ : Uf —> Mm for some connected open
Uf С U so that the level sets of Φ are integral manifolds. The tangent map
11.6. Fundamental Theorem of Surface Theory
487
Ϊ^,Φ has kernel Ep at each ρ G N, and since ^j | never lies in E, we see
that
дФ
~оЧ
φ 0 on N Π U' for all j.
In particular, this holds at (xo, zo), and the implicit mapping theorem tells us
that a neighborhood of (xo, zq) in a plaque of N is the graph of a function
/ : V -)· Rm with /(ж0) = z0. Define a function F : V -> M.k x Rm by
F(x) := (x,f(x)). Writing ρ = -Р(ж), we see that for each г the vector
д ι
dx* K"' dzT
™~k
dxl
is a linear combination of vectors Xj defined above:
dx*
df
д
/(*)
+Σ&ω£
/(*)
=Σ«*(έ| +Σ*ω£| )·
Collecting terms and comparing we see that cf = δ* and
It follows from Corollary 11.23 that / is uniquely determined on a sufficiently
small connected neighborhood of (x, y). D
It is often convenient to be able to come up with the integrability
conditions for a given application without trying to match indexing and notation
with the above theorem. The basis of the procedure is to set mixed partials
equal to each other. We demonstrate this using the notation of the theorem.
We start with
^4(*>/(*)) = ^Ai(x'/(x))·
Apply the chain rule:
д Аг β Я fr
дА
д
^л
= τ^>/ω)+Σ^4(^(*))ώω·
dza
дхз
Finally, substitute back using the original equations (11.5) and replace all
occurrences of f(x) in the arguments with the independent variable z. We
arrive at the integrability conditions:
dAh ^l^лlf ^9A*>, ν дАк, \>^л1, ч94/ ч
488
11, Distributions and Frobenius' Theorem
The convenience of this may not be clear yet, but we shall shortly
demonstrate the usefulness of this method.
Proposition 11.37. Let U be open in Шк х Rm and (Л*·) аптх к matru
of smooth functions on U. Let (xq,zo) 6 U and suppose that for som
connected open set V, both /i : V -> U С Rm and f2 : V -4 U С Rm are
solutions of
dp
—0{x) = A)(xJ(x)) for all ij,
f(x0) = zq.
Then /i = /2.
Proof. This follows from Corollary 11.23 and the considerations in the proof
of the previous proposition. D
Lemma 11.38. Let V be an open set inЖк and let (Alj) be anmxk matrix
of smooth functions on V x Rm that are linear in the second argument and
satisfy the integrability conditions (11.6) onV x Rm. Then for any xq € V
there is a ball Bxo С V such that for any (a, b) G Bxo xRm there is a solution
defined on Bxo with f(a) = b.
Proof. Let fi be the solution with fi(xo) = e*, where e; is the г-th standard
basis vector of Rm. Then Д,..., /m are defined and Unearly independent
on some ball BXo containing x$ and contained in the intersection of the
domains of the /г. Choose (a,b) € Bxo χ Rm and note that b = Y^brfr{
for some uniquely defined numbers b%. Now define / = Σ blfz on BXQ. Then
writing / = (Z1,..., /m), we have for any χ e V,
A){x, f{x)) = A) (*, £brfr(x)) = £brA)(x, fr(x))
■M(,)_v,
d&K ' дхэ
-ς*££μ-Ι£μ
and
/(a) = £V/r(a) = b. □
Corollary 11.39. Lei V be a simply connected open set in Rk and let (A )
be anmxk matrix of smooth functions onU = V x Rm. Suppose that ea h
AtJ is linear in its second argument. If
11.6. Fundamental Theorem of Surface Theory
489
on V χ Rm, then given any (xq, zq) EV χ Rm, there exists a unique smooth
map / : V -> Rm suc/ι ίΛαί
дР
^j(x) = Л}(х, /(ж)) for all ij,
/(so) - 20.
Proof. Let Xj := g|j| + ΣΓ-Α£(ρ) gpr| be the fields that span an in-
tegrable distribution on V x Rm as in Proposition 11.36. Let Ь(хо,го) be
the maximal integral manifold through the point (xo, ^o)· Let ρ denote the
restriction of the projection ртг : V x Rm -> V to -£(X(bao). Let (αχ,δι) €
L(a.0}Z0) and consider the set
Fax = Р~1Ы).
By Lemma 11.38 above, there is a fixed open set U containing a\ such that
for every (αϊ, b) Ε Fai there is a solution Д : U —> Rm with /(αϊ) = 6.
By Corollary 11.23, the graphs of these solutions are all disjoint open sets
in i(XOjJZO) and ρ restricts to a difFeomorphism on each such graph. Thus
ρ : i(XO(^o) —> V is a smooth covering map. The local solutions guaranteed
to exist by Proposition 11.36 are local sections of this covering. Thus since
V is simply connected, we know from Theorem 1.95 that there is a smooth
lift ρ of idy : V —► V such that p(xo) = (^o»^o)> which in this case means
that we have a global section: ρ ο ρ = idy. Now let / := pr2 ο ρ : V —У Rm.
Then p(x) = (x,f(x)) and f(x) must be smooth and for every a EV the
function / must agree with the unique local solution which takes the value
f(a) at a. D
We return to the situation studied in Chapter 4. Consider an immersion
χ : V —> R3, where V is an open set in R2 whose standard coordinates will
be denoted by u1, it2. Let (Д, /2, /3) be the frame fields along χ defined by
/l :=xui, /2 :=х^2,
/з :=N = X^i X Xu2/ ||x„i X X„a || ,
where x^i = дк/ди1, etc. The first fundamental form is given by the matrix
entries
9%3 = (xu4^uj) for 1 < i,3 < 2,
while the second fundamental form is given by the matrix entries
iij = ~ (N^Xyj) = (N,xuiui) = (/з,х„.„у) -
Let us consider f = (/ь/2,/3) as a matrix function of (v},u2) that takes
values in GL(3). We have
3 3
(Л)* - Ep*7r and CA)u2 = £<tf/r
r=l r=l
490
11. Distributions and Frobenius' Theorem
for some matrix functions Ρ and Q. In matrix notation, we have
f„i = tPf
(1L7) f -to
and these axe called the frame equations. For convenience, we define a
3x3 matrix function G by Gij := (/;, fj) for 1 < г, j < 3 so that
(011 912 0
021 522 0
0 0 1
For any given χ = ^ яг/г» we have
and so
6=Σ (<*)»** = Σσ***·
Now let a; = (/i)ui = ΣΛ^· Tnen a1* = *?» so if we define By
(/i> (Λ")ιιΟ> tnen we nave
(/ί,(/ι)ωι)=βϋ = Σ^Ρί,
(11-9) </i, (/2)„i> = Да = Σ G^P2 ,
<Λ,(/3)«ι> = Βί3 = Σσ*^'
or β = GP. Similarly, if С = (C^·) = (£, (/j)u2), then
(/«,(/ι)β») = σα = Σσ<*ί2ϊ.
(11.10) (/if (/2)u2> = Ci2 = Σ G*<&
(/i> (/з)«2) = Ctf = Σ^Φ*.
or С = GQ. We arrive at
Ρ = G~XB,
Q = G~XC.
We denote the entries of G~l by ptJ so that
/s11 g12 0
(11.11) С"1 := </21 p22 0
\ 0 0 1
Proposition 11.40. We have
\ (0ii)„i 5 (flli)«? -'n
(11.12) B=\ (g12)ul - i (5ll)n2 i(p22)ul -i12
/u *и 0
11.6. Fundamental Theorem of Surface Theory
491
and
(\ (011 )«a \ (912)U2 - 5 (Дй)и1 -^12
(№)„!- ί(Λΐ)„» ^(й2)„а ~*22
*12 ^22 О
In particular, Ρ and Q can be written in terms of the matrix entries of the
first and second fundamental forms.
Proof. The proof is just a calculation, and we only do part. For example,
if г is either 1 or 2, then
Вц — {fi, (/i)ui) = \Xu*»xu*uO = о ^xtt*'xtt*)i*1 = о ^"^ul'
Similarly, for г — 1 or 2 we have
Cii = (fi, {fi)u2) = (xul»xul«2) = 2 (лОцЗ ·
Now, \ {gu)ui = <χ«ι,χ«2>μι = (χ^ι,χ^) + i (5n)u2 from above, and so
B21 = (/2, (/i)tti> = (хЛ^хи2) = 2 (Λ2)„ι - 2 (5ii)«a ·
The entries Б12, C12, C21 are calculated similarly. Next consider Β& for г = 1
or 2. We have
ЯгЗ = {/г, (/з)«1> = <Χ„.,ΛΓ„ι) = -£ц
and
О = (/з,Л>„1 = ((/з)„1 ,/i> + {/З, (/гЫ ,
SO
Взг = —5ϊ3·
The entries Сы and Сгз are obtained in the same way. Lastly, since
(/з,/з} = 1,
0=2 ^3' &ик = {(/з)ц* »/з> = J c if fc _ 2' D
We record an observation to be used later:
B + B* = Gui,
(11.14) , u
The frame equations (11.7) are a system to which Proposition 11.36 applies.
Rather than trying to rewrite the equations in a form that matches that
proposition we obtain the integrability conditions by setting
and then
fu2P + fPu2=fulQ + fQul.
492
11. Distributions and Frobenius' Theorem
Substituting from the frame equations we obtain
f(Jk-Qtti-(PQ-QP))=0.
Now f is a nonsingular matrix, so we have the equivalent integrability
equation
(11.15) PU2 - Qui - (PQ - QP) = 0.
At this point we pause to appreciate an important fact. Namely, direct
calculation reveals that these equations are equivalent to the combination
of the Codazzi-Mainardi equation and the Gauss curvature equation, which
we now see are integrability conditions (see Problem 7). We thus refer to
the above integrability equations (11.15) as the Gauss-Codazzi equations
with apologies to Gaspare Mainardi (1800 1879).
We now turn things around. Rather than assuming that we have a
surface, we take the (gi3) and (£i3) as some given symmetric smooth matrix-
valued functions defined on a connected open FcR2 with the assumption
that (дгз) is positive definite. Furthermore, we now assume that G, P, and
Q are actually defined in terms of these by the formulas above, which we
found to be true in the case where we started with a surface. We will show
that we can obtain a surface with these as first and second fundamental
form.
Theorem 11.41 (Fundamental existence theorem for surfaces). The
following assertions hold:
(i) Let V be an open set in R2 diffeomorphic to an open disk and let
χ : V —> R3 be an immersion with the corresponding first and
second fundamental forms given in matrix form as (gij) and (4?)·
Let у : V -¥ R3 be another immersion with the corresponding forms
{9ij) and{£l3). V
y = /ox
for some proper Euclidean motion f : R3 —> R3, then
9ij — 9ij 3
4j = Ч3 ·
Conversely, if the last equations hold, then у = /ox for som
Euclidean motion /.
(ii) Suppose that (дгз) and(ii3) are given symmetric matrix-valued
functions defined on V with (gi3) positive definite and suppose that G,
В and С are defined in terms of the entries of (gij) and (ίψ) as in
11.6. Fundamental Theorem of Surface Theory
493
formulas (11.8), (11.11), (11.12) and (11.13). Then if
Ρ = G-XB,
Q = G-XC,
and if Pu2 — Qui — (PQ — QP) = 0, then there exists an embedding
χ : V —> R3 such that (gij) and (iij) are the corresponding first and
second fundamental forms.
Proof. We leave the proof of the first part of (i) to the reader, but note that
it can be proved using direct calculation or it can be derived from Theorem
4.22.
For the rest of (i), note that by composing with a translation we may
assume that both χ and у map some fixed point и Ε V to the origin in
IR3. Let (Λ,/2,/3) be the natural frame for χ as above and let (/1,/2,/з)
be that of y. By making a rotation we may assume that these two frames
agree at p. But since we are assuming that g^ = g^ and lxj = ί%3, it follows
that both frames satisfy the same frame equations and so by Proposition
11.37 they must agree on the connected set V. In particular, xu% = yu% for
i = 1,2. Thus χ and у only differ by a constant, which must be zero since
x(u)=y(w).
Next we consider (ii) where (g%j) and (iij) are given. We want to
construct a surface, but first we construct the frame for the desired surface.
Since it is assumed that g = (gtJ) is positive definite, g and the extended
matrix G are both invertible and positive definite. Thus Ρ and Q are
well-defined. Since we assume that the integrability equations Pu2 — Qui —
(PQ — QP) = 0 hold, Theorem 11.36 tells us that we can solve the frame
equations locally, near any point и G V and with any initial conditions
f (u) = fo holding as desired. But the system is linear and our domain is
diffeomorphic to a disk so we can use Corollary 11.39 to obtain a solution on
all of V. Since G is positive definite, we may choose these initial conditions
so that
ifi(u),fj(u)) = Gtj(u) (ij-th entry of G at u).
Having obtained the fi near uy we now wish to obtain a surface. This means
solving the system
x^i = /b
V - /2,
and this time the integrability conditions are derived from
(/ι)** = ih)ui -
Using the frame equations, we obtain integrability conditions
494
11. Distributions and Frobenius' Theorem
This just says that the second column of Ρ is equal to the first column of Q,
which is true. Thus we can find χ : V —► Μ3 with χ(ϊζ) — 0 so that (11.16)
holds.
Next we show that (/», f3) = Gij on all of V. We compute as follows:
= Σοι (frjj)+Σ^· </*>/-> = wo+(qg)%
r s
= (GQ + (GQ)%- = (B + B% = (Gui)y = (Gy)„i
by equations (11.14). Similarly for u2. Thus (fufj) — C?y is a constant,
which must be zero since it is zero at u. Prom (/$, /j) = G^· it follows that
(/з> /з) = 1 and that /i, /2 are independent and orthogonal to /3. It remains
to show that {(/3)^ , f3) = —£ij- We compute as follows:
- <(/з).х ,Л) = (gllhi+gl2li2) {Sufi) + {gl2hi+922h2) (fijj)
= fou*n + gl2ii2)gi3 + (gl2hi+g22£i2)g2j
= *и(Лц + g12g*j) + h2(g2lgij + g22g2j)
This shows that ((/3)^1,/j) = ~hj f°r J = 1)2. The computation of
— {{fs)u2, /j) is similar and left for the reader. D
11.7. Local Fundamental Theorem of Calculus
Recall the structure equations (8.15) satisfied by the Maurer-Cartan form
uq for a Lie group G:
If yx = χχ (e),..., vn = Xn(e) is a basis for the Lie algebra q which extends to
left invariant vector fields Xl, ..., Xn, then the above equation is equivalent
to
where the cy are the structure constants associated to ω1, ... ,ωη, which is
the left invariant frame field dual to Χι,..., Xn. If Μ is some m-manifold
and / : Μ —► G is a smooth map, then ω/ = /*ω<3 is a g-valued 1-form on
M. By naturality we have
or equivalently
11.7. Local Fundamental Theorem of Calculus
495
where ω\ = f*ul for г = 1,... , π. The g-valued 1-form w/ is sometimes
called the (left) Darboux derivative of /. The right Darboux derivative is
defined similarly using the right Maurer-Cartan form.
If we think of a g- valued 1-form on a manifold Μ as a map Τ Μ —l· g,
then ujf = f*DQ = ljq ο Τ/. Prom this point of view we can understand
why Df is a kind of derivative of / by considering the special case where
G is a vector space V with its abelian (additive) Lie group structure. In
this case, the Lie algebra is V itself and the Maurer-Cartan form is just
the canonical map pr2 : TV = V x V —► V, and so for a smooth map
/ : Μ —> V, the Darboux derivative is the differential df = pr2 oT/. Just as
for the differential, the Darboux derivative embodies less information than
the tangent map since the values that the map takes are "forgotten" and
only tangential information is retained. Indeed, notice that if Lg : G -> G
is a left translation and F := Lg ο /, then
uF = F*u)q = f*L*uo = /*ωα = ω/
since uq is left invariant. Hence two smooth maps into G that differ by left
translation have the same (left) Darboux derivative. This generalizes the
fact that two functions that differ by an additive constant have the same
differential.
For a smooth 1-form ϋ = gdt on R, we can always find a smooth function
/ with df = ΰ since by the Fundamental theorem of calculus one need only
choose f(t) = J0 g(r)dr. More generally, if Μ is simply connected and G is
a vector space V, then the fact that Hl(M) = 0 means that every V-valued
1-form is the differential of some smooth / : Μ —> V. For a general G, if
г? is a g-valued 1-form on M, then we may ask for an / such that # = Uf.
But there is no reason to expect a general ϋ to satisfy the above structure
equation that Uf satisfies. Now if we choose a basis {vi} for fl, then there
must be 1-forms i?1,..., ϋη G ill(M) such that
η
i=l
Then άϋ = -\ [#,#]л is equivalent to
where c^· are the structure constants. As we said, this may or may not hold.
These equations are the integrability conditions for the existence of an /
such that *& = Df. More precisely, we have the following theorem.
496
11. Distributions and Frobenius' Theorem
Theorem 11.42. Let Μ be an m-manifold and G an η-dimensional Lie
group. If ΰ is a g-valued 1-form on Μ that satisfies the structural equation
d$ = -I[tf,tf]A, then for every po e Μ there is a neighborhood UPo of po
such that given any (a, 6) Ε UVo χ G there is a smooth function f : UXo —> G
with f(a) = b and ΰ ω/.
Proof. Let prx : Μ χ G -> Μ and pr2 : Μ χ G -¥ G be the canonical
projections and define a 5-valued 1-form on Μ χ G by
Ω — pr*# - pr^G.
For each (p,#) Ε Μ χ G, let Е^д^ = Ker Ω(Ρ)9). Now define a vector bundle
homomorphism Τ (Μ χ G) -4 (Μ χ G) χ 0 by г^) ь-> ((ρ, g), fi(plfl)(v(p>s))).
By Proposition 6.28, if this homomorphism has constant rank, then the
kernel is a subbundle which clearly has fiber E^,^ at (p, g). By linear algebra,
this is equivalent to showing that the dimension of E^g) is independent of
(p,g). This will follow if we show that Tpr^ : E^g) —> TPM is an
isomorphism for all (ρ, ρ). If we identify Γ(ρ55) (Μ χ G) with ΓΡΜ χ TgG,
then Грг2 is just the projection (v,w) »-> г; and similarly for Tpr2. Now if
(v,w) Ε -Б(р,5) and Tprx ■ (г>,ги) = 0 then ν = 0. But, since #(v) ω(?(«/),
we have it; — 0 also. Thus TprjL is injective. It is also surjective since
for any ν Ε TPM, we clearly have {v,TeLg{d{v)) Ε Έ(Μ) and this has ν as
its image.
Now we use Proposition 11.19 to show that Ε is integrable:
aft = pr^tf - prSdwo = Pri (-5 [1?, t?]A) - pr^ (-^ [u;G,o;G]A
= -\ [pritf, pri0]A + 1 [pr^G,pr^G]A .
But prji? = Ω + prjwc, so
<*Ω = -- [(Ω + prfact), (Ω + pr^G)]A - - \ρτ*2ωα, pr^£jG]A
= -^[Ω,Ω]* - ^[Ω,ρΓ^σ]Α - *[pr^G^]A,
which makes it clear that d$l(X, Y) = 0 whenever Ω (X) = 0 and Ω (У) 0.
Now we use the leaves of the distribution to construct the solution.
Let xq Ε Μ and fix go Ε С?. Then let L(Xogo} be the maximal integral
manifold through (жо,0о)· The map Tprx| £(p0j5o) : £(p0,50) -> TpM is
an isomorphism so the inverse mapping theorem tells us that prjL^^
restricts to a diffeomorphism on some neighborhood О of (po, <7o) *n ^(ρο,ρο)*
Let Φ : U —> О С £(po>ffo) denote the inverse of this diffeomorphism. Since
ρΓι ο Φ =idtf, there must be a smooth function / such that Φ(ρ) — (ρ, /(ρ))
11.7. Local Fundamental Theorem of Calculus
497
for all ρ G U. Observe that Φ*(Ω) = 0 since the image of ΤΦ lies in the
distribution by construction. Thus we have
0 = Φ*(Ω) = Φ*(ρτΙ<β - pr5wG)
= Ф*рГ** - Ф*рГ2"<?) = * " /*WG,
or ι?!^ = /*ω<3 = cj/. Let (a, b) eUxG. Now we argue that we may modify
/ so that we still have ϋ v — uj while now /(a) = 6. In fact, if /(a) = bi,
then let g = bj"1b and replace / by Lg ο /. Π
Proposition 11.43. iei Μ be an m-manifold and G an η-dimensional Lie
group. Let ΰ be a g-valued 1-form on Μ and suppose that /i : U —> G and
/2 : U —> G are smooth maps such that ύ\ν = /*cj<3 for г = 1,2. Then if U
is connected and /i(po) — /г(ро) /or some po € 17, £ften /1 = /2-
Proof. Let Фг := (idu,fi)· Then
Фг*(П) = Ф*(рг^-рг^)
= Ф*рг^ - ФГрг^с) = ΰ - f*uG = О,
so Фг : U —► Μ х G is an embedding for i — 1,2 whose image is the
graph of fi and an integral manifold of the distribution generated by Ω that
contains (p(b/(po))· Corollary 11.23 applies to show that Φι(!7) = $2(U)
and /1 = /2. Π
Corollary 11.44. Let M, G, ΰ and Ω be as above and suppose that άΰ —
—j [#, i?]A. ГЛеп ifte restriction of the map prx : Μ χ G? —> Μ to any leaf
of the distribution given by ft is a covering map.
Proof. Let L be a leaf and choose po € M. Choose (po>5o) € L and let
Φ : U —> О С L^^) = L be the diffeomorphism constructed as in the proof
of the previous theorem where U is connected. Now Φ = (idf/,/) where
f(po) = 30 · If (po5 31) is any other point in the leaf, then for g = gxg$l the
map Φι := (id[/,L5 о /) is a diffeomorphism U —» Oi where (ρο>#ι) € Οι
and prx (Οχ) = U. In fact, it is easy to see that Φχ ο Φ"1 : О -> Οχ is a
diffeomorphism. If (po>0i) and (po»52) are distinct points in the leaf, then
we construct diffeomorphisms Φχ : Ο —>■ Οχ and Φ2 : О —> О2 as above, and
Corollary 11.23 applies to show that Οχ and O2 are disjoint and each map
diffeomorphically onto О under prx. It is now clear that Μ is evenly covered
bypr^L. D
Corollary 11.45. Let Μ be a simply connected m-manifold and G an n-
dimensional Lie group. If ΰ is a g-valued 1-form on Μ that satisfies the
structural equation du — — \ [i?,#]A, then for every (po>5o) Ε Μ χ G there
is a smooth function f : Μ —l· G with f(po) — go and $ — ωί·
498
11. Distributions and Frobenius' Theorem
Proof. Let L be the leaf of the distribution determined by Ω that contains
(po5 <7o)· Since Μ is simply connected, we can lift idM : Μ —> Μ to a section
Φ : Μ —> L of prx|L such that Φ(ρο) = 3θ· Once again Φ = (id^,/) for
some smooth function / with f(po) = go and we argue as before to conclude
that ϋ = ω/ (this time globally). D
Problems
(1) Show that the following vector fields define a rank 2 distribution on '.
which is not involutive (and hence not integrable):
X =
У =
β
дх
в
ду'
д
+ Уд~г'
Draw a picture of the portion of this distribution that sits at points in
the x, y-plane and try to see geometrically why the distribution is not
integrable.
(2) Show that the distribution on R4 given by X = щ + x-^ and Υ =
Έχ +У^гу> where (x, у, z,w) axe standard coordinates, has no integral
manifolds.
(3) Let θ be a 1-form. Show that a 2-form η is in the ideal generated by θ
if and only if η Α θ = 0.
(4) Consider again the system of partial differential equations:
zx = F(x,y,z),
zy = G(x,y,z).
Show that the graphs of solutions of these equations are integral
manifolds of the distribution defined by the 1-form θ = dz — Fdx — Gdy. Use
Theorem 11.18 to deduce the integrability conditions for this system.
[Hint: Use Problem 3.]
(5) Let Η be the Heisenberg group consisting of all matrices of the form
A =
The Xij give global coordinates and a diffeomorphism with R3. Let
Vi2>Vi3,V23 be the left invariant vector fields on Η that have values
at the identity with components with respect to the coordinate fields
given by (1,0,0), (0,1,0), and (0,0,1), respectively. Let А{у!2,у13} and
1
0
0
Xl2
1
0
^13
Ж23
1
Problems
499
A{y12iy23} be the 2-dimensional distributions generated by the indicated
pairs of vector fields. Show that Δ{νΊ2,ν!3} is integrable and A{y12)y23}
is not.
(6) Prove (i) of Theorem 11.41.
(7) Show that for a given surface, equations (11.15) are equivalent to a
combination of the Codazzi-Mainardi equations and the Gauss curvature
equations defined in Chapter 4.
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Chapter 12
Connections and
Covariant Derivatives
The terms "covariant derivative" and "connection" are sometimes treated
as synonymous. In fact, a covariant derivative is sometimes called a Koszul
connection. Prom one point of view, the central idea is that of measuring
the rate of change of sections of bundles in the direction of a vector or vector
field on the base manifold. Here the derivative viewpoint is prominent. Prom
another related point of view, a connection provides an extra structure that
gives a principled way of lifting curves from the base to the total space. The
lifts are parallel sections along the curve. In this chapter, we will always
take the typical fiber of an F-vector bundle of rank к to be ¥k. We let
(ei,... ,e/c) be the standard basis of ¥k. Thus every vector bundle chart
(ί/, φ) is associated with a frame field (ei,..., e/J where ei(x) := 0_1(x, e^).
12.1. Definitions
Let π : Ε —> Μ be a smooth F-vector bundle of rank к over a manifold M. A
covariant derivative can either be defined as a map V : X(M) χ Γ(Μ, Ε) —>
Γ(Μ, Ε) with certain properties from which one derives a well-defined map
V : TM χ Γ(Μ, E) -» Γ(Μ, Ε) with nice properties or the other way around.
We rather arbitrarily start with the first of these definitions.
Definition 12.1. A covariant derivative or Koszul connection on a
smooth F-vector bundle Ε -» Μ is a map V : X(M) χ Γ(Μ, Ε) -» Γ (Μ, Ε)
(where V(X, s) is written as Vxs) satisfying the following four properties:
(i) Vfx(s) = fVxs for all / e C°°(M), X e X(M) and s e Γ(Μ,Ε);
501
502
12. Connections and Covariant Derivatives
(ii) VXl+x25 = VXl5 +VX25 for all XUX2 e X(M) and 5 G Γ(Μ,£);
(iii) Vx(si + s2) = Vx5i + Vxs2 for all X G X(M) and sbs2 €
Γ(Μ,£);
(iv) Vx(/s) = (X/)s + fVxs for all / G C°°(M;F), X G X(M) and
seT(M,E).
For a fixed X G X(M), the map Vx : Γ(Μ, Ε) -» Γ(Μ, Ε) is called the
covariant derivative with respect to X.
As we will see below, this definition is enough to imply that V induces
maps Vе7 : X(U) χ T(U,E) -» Γ(ί7,£), one for each open ί/cM, that are
naturally related in a sense we make precise below (this is not necessarily
true for infinite-dimensional manifolds). Furthermore, we also prove that
for a fixed ρ G M, the value (Vxs)(p) depends only on the value of X at
ρ and only on the values of 5 along any smooth curve с representing Xp.
Thus we get a well-defined map V : TM χ Τ(Μ,Ε) -> Γ(Μ,Ε) such that
V^s = (Vxs) (p) for any extension of υ G TPM to a vector field X with
Xp = v. The resulting properties are
(i') Vav(s) = aVvs for all α G R, ν G ΓΜ and s G Γ(Μ,Ε);
(ii') for all ρ G Μ we have Vvi+V2s = Vvis + VV2s for all vi,v2 G TPM,
and 5 еГ(М,Я);
(iii') Vv(5i + s2) = Vvsi + Vvs2 for all υ G Τ Μ and 5Ь s2 G Г(М, jE7);
(iv') for all ρ G Μ we have Vv(fs) = (vf)s(p)+f(p)Wvs for all υ G TPM,
5 G T(M,E) and / G C°°(M;F);
(v') p\-> Vxp5 is smooth for all smooth vector fields X.
A map satisfying these properties is also called a covariant derivative (or
Koszul connection). Note that it is easy to obtain a Koszul connection
in the first sense since we just let (Vxs) (p) := Vxps.
Definition 12.2. A covariant derivative on the tangent bundle Τ Μ of an
η-manifold Μ is usually referred to as a covariant derivative on M.
In Chapter 4 have already met the Levi-Civita connection V on Rn,
which is, from the current point of view, a Koszul connection on the tangent
bundle of W1. The definition of this connection takes advantage of the
natural identification of tangent spaces which makes taking the difference
quotient possible:
i-»0 t
In that same chapter we obtained, by a projection, a covariant derivative on
(the tangent bundle of) any hypersurface in Rn. A covariant derivative on
12.1. Definitions
503
ν°χ-=ιί
a submanifold of arbitrary codimension can be obtained in the same way.
Let Μ be a submanifold of Rn and let X e X(M) and ν e TPM. We have
Xoc] e TPM.
10 /
One may easily verify that V, so defined, is a covariant derivative (in the
second sense above).
Returning to the case of a general vector bundle, let us consider how
covariant differentiation behaves with respect to restriction to open subsets
of our manifold. Recall the restriction map Гу : T(U,E) —» T(V,E) given
by ry : σ ь-> σ\ν and where V С U.
Definition 12.3. A natural covariant derivative V on a smooth F-
vector bundle Ε -» Μ is an assignment to each open set U С М of a map
Vе7 : X(U) x T(U,E) -> T(U,E) written as Vе7 : (Χ,σ) ^ νυχσ such that
the following assertions hold:
(i) For every open U С М, the map Vе7 is a Koszul connection on the
restricted bundle E\v —> U.
(ii) For nested open sets V С /7, we have r*y(V-£a) = ^υχτνσ (nat~
urality with respect to restrictions).
(iii) For X e X(U) and σ e T(U,E) the value (ν^σ)(ρ) only depends
on the value of X at ρ G /7.
Here V^a is called the covariant derivative of σ with respect to X.
We will denote all of the maps Vе7 by the single symbol V when there is no
chance of confusion. We have explicitly worked the naturality conditions (ii)
and (iii) into the definition of a natural covariant derivative, so this
definition is also appropriate for infinite-dimensional manifolds. The definition of
Koszul connection did not specifically include these naturality features and
was only defined for global sections. We shall now see that, in the case of
finite-dimensional manifolds, a Koszul connection gives a natural covariant
derivative for free.
Lemma 12.4. Suppose V : X(M) χ Τ (Ε) —> Τ (Ε) is α Koszul connection
for the vector bundle Ε —» Μ. Then if for some open U either X\v = 0 or
σ\υ = 0, then
(ν*σ)(ρ) = 0 for all pe U.
Proof. We prove the case of σ\ν = 0 and leave the case of X\v = 0 to the
reader.
Let q e U. Then there is some relatively compact open set V with
q G V С U and a smooth function / that is identically one on V and zero
504
12. Connections and Covariant Derivatives
outside of U. Thus /σ = 0 on Μ and so since V is linear, we have V(/a) = 0
on M. Thus since (iv) of Definition 12.1 holds for global fields, we have
Vx(/a)(g) = f(p)(Vxa)(q) + (Xqf)a(q)
= (Vxa)(q) = 0.
Since q G U was arbitrary, we have the result. D
We now define a natural covariant derivative derived from a given Koszul
connection V. Given any open set U С Μ, we define Vе7 : X(U) xT(E\u) -»
T(E\u) by
(12.1) №0(ρ):=(ν^σ)(ρ), Ρ € U,
for X e X(M) and σ G T(E) chosen to be any sections which agree with
X and σ on some open V with ρ e V С V С U. By the above lemma this
definition does not depend on the choices of X and σ.
Proposition 12.5. Let Ε —» Μ be α rank к vector bundle and suppose that
V : X(M) χ T(E) —> T(E) is a Koszul connection. If for each open U we
define Vu as in f 12.1^ above, then the assignment U ь-> Vе7 is a natural
covariant derivative as in Definition 12.3.
Proof. We must show that (i), (ii) and (iii) of Definition 12.3 hold. It is
easily checked that (i) holds, that is, that V^ is a Koszul connection for each
U. The demonstration that (ii) holds is also easy and we leave it for the
reader to check. Now since X -» Vχσ is linear over C°°({7), (iii) follows by
familiar arguments (S/χσ is linear over functions in the argument X). D
Because of this last lemma, we may define VVpa for vp G TpM by
ννρσ:=(ν*σ)(ρ),
where X is any vector field with X(p) = vp e TpM. We say that ν^σ is
"tensorial" in the variable X. The result can be seen as a special case of
Proposition 6.55. We now see that it is safe to write expressions not directly
justified by the definition of Koszul connection. For example, if Χ Ε X(U)
and σ e T(V, E), where U Π V φ 0, then Vχσ is taken to be an element of
T(UnV,E) defined by
(Vxa) (ρ) = νΧρσ := V^pa for allpeUnV.
This is a particularly useful convention when U is the domain of a chart
({7, x) and X = -~i and also when σ is a member of a frame field of the
vector bundle defined on some open set.
In the same way that one extends a derivation on vector fields to a tensor
derivation, one may show that a covariant derivative on a vector bundle
induces naturally related covariant derivatives on all the multilinear bundles.
12.1. Definitions
505
In particular, if π* : Ε* -^ Μ denotes the dual bundle to π : Ε -» Μ we
may define connections on π* : Ε* -» Μ and οηπ^π* :£?®£?*->Μ. We
do this in such a way that for 5 € Γ(Μ, Ε) and 5* € Γ (Μ, £Γ) we have
Vf®E* (5 ® 5*) = Vxs ® 5* + 5 ® Vf s*,
and
(VfV)(s) = *(*·(*))-s*(Vxs).
Of course, the second formula follows from the requirement that covariant
differentiation commutes with contraction:
X(s*(s)) = (VxC(s ® s*)) = C(Vf^*(s ® s*))
= с (vxs ® s* + s ® vf V) = 5*(vx5) + (vf 5*)(5),
where С denotes the contraction given by 5 ® а ь-> a(s). All this works like
the tensor derivation extension procedure discussed previously, and we often
write all of these covariant derivatives with the single symbol V.
The bundle Ε ® Ε* -» Μ is naturally isomorphic to End(.E), and by
this isomorphism we get a connection on End(E). If we identify elements
of Τ (End(E)) with End(TE) (see Proposition 6.55), then we may use the
following formula for the definition of the connection on End(E):
(VxA)(s) :=Vx(A(s)) - A(Vxs).
Indeed, since С : s ® Аи A(s) is a contraction, we must have
Vx(A(s)) = С (Vxs <g> A + s <g> VxA)
= A(Vxs) + (VxA)(s).
Notice that if we fix s G Г(.Е), then for each ρ e M, we have an element
(Vs) (p) of L(TPM, EP)^E® T*M given by
(Vs) (p) : vp ь-> VVps for all vp e TpM.
Thus we obtain a section Vs of Ε ® T*M given by ρ ь-> (Vs) (ρ), which is
easily shown to be smooth. In this way, we can also think of a connection
as giving a map
V:T(#) ->Γ(£®Τ*Μ)
with the property that
V/s = /Vs + s ® rf/
for all s g Г(#) and / e C°°(M). Since, by definition, T(E) = Ω°(Ε) and
Γ(£® Γ*Μ) = Ω1 (£7), we really have a map Ω°(#) -> Ω1 (£7). Later we will
extend to a map Qk(E) ->> ΩΑ:+1(^) for all integral к > 0. Now if X is a
smooth vector field, then X ь-> Vxs G Г(.Е), so we may also view Vs as an
element of Нот(Х(М),Г(£)).
506
12. Connections and Covariant Derivatives
12.2. Connection Forms
Let π : Ε —> Μ be a rank r vector bundle with a connection V. Recall tha
a choice of a local frame field over an open set U С Μ is equivalent to a
trivialization of the restriction ttjj : E\v —> U. Namely, if φ = (π, Φ) is such
a trivialization over /7, then defining ei(x) — 0-1(x,ei), where (e^) is the
standard basis of Fn, we have a frame field (ei,..., β/~). We now examine
the expression for a given covariant derivative from the viewpoint of such a
local frame. It is not hard to see that for every such frame field there must
be a matrix of 1-forms ω = (^))i<i,j<r such that for X G Γ({7, Ε) we may
write
к
i=l
The forms оЛ are called connection forms.
Proposition 12.6. If s = J2islei is the local expression of a section s €
T(E) in terms of a local frame field (ei,...,ejfc), then the following local
expression holds:
Г
I
Proof. We simply compute:
Vxs = Vxfes<ei)
^ i '
г г
г г,г
= Σ{Χ8ί + Σ"ί(χν)βί. π
г
So the г-th component of Vxs is
(12.2) (УХЯ)< = ЛУ + 5>;(Л>Г.
Г
We may surely choose U small enough that it is also the domain of a
coordinate frame {θμ} for M. Thus we have
(12.3) У^ = 5>^еь
fe
12.3. Differentiation Along a Map
507
where αΑ = ω?(θμ). We now have the local formula
(12.4) Vxs = £(X>",V + ΣΣΧ4^)^·
г=1 μ=1 μ=1 r=l
Or, using the summation convention,
Vx5=(l^4l^/)e,
Now suppose that we have two moving frames whose domains overlap,
say и = (βχ,..., β/~) and v! = (e'1?..., ejjj. Let us examine how the matrix of
1-forms ω = (ω\) is related to the corresponding α/ = {ω?) defined in terms
of the frame v!. The change of frame is
which in matrix notation is
v! = ид
for some smooth д : UC\Uf —> GL(n). (We treat и as a row vector of fields.)
For a given moving frame, let Vu := [Vei,..., Ve^]. Differentiating both
sides of v! = ид and using matrix notation we have
uf = ид,
Vu' = V(ug),
uujf = (Vu)g + udg,
ufuf = ugg~lug + ugg~ldg,
u'u* — и'д~1шд + u'g~ldg,
and so we obtain the transformation law for connection forms:
J = g~lug + g~ldg.
12.3. Differentiation Along a Map
Once again let π : Ε —> Μ be a vector bundle with a Koszul connection
V. Consider a smooth map / : N —> Μ and a section σ : Ν —ϊ Ε along /.
Let ei,..., e/c be a frame field defined over U С Μ. Since / is continuous,
О = /-1(/7) is open and e\ о /,..., е& о / are fields along / defined on O.
We may write σ = Σα=ι °"а еа ° / f°r some functions σα : О С Ν —> F.
For any ρ e О and ν Ε TP7V, we define
к / к \
(12.5) νζσ := £ (άσα ■ ν) + J>? (Γ/ («)) σ'(ρ) e«(/ (ρ)).
α=1 \ r=l /
508
12. Connections and Covariant Derivatives
A direct albeit tedious calculation shows that this result is independent of
the choice of frame field. Thus we obtain a global map
V/ :TN xTf(E) ->Tf(E).
The map V·^ : TN χ Tf(E) -» Tf(E) satisfies properties that qualify it
as a covariant derivative along /. Namely, we have
Definition 12.7. Let π : Ε -» Μ and / : TV -» Μ be as above. A
covariant derivative along / is a map V·^ : TN χ Tf(E) -» Tf(E) that
satisfies
(i) V·^ : TN χ Tf(E) -» Tf(E) is fiberwise linear in the first argument:
for all σ G Tf(E), scalars a, b, and u, ν G TpN (p is arbitrary),
(ii) ν£σ = νίσι + ν£σ2 for any и G TN and any σι,σ2 G Γ/(β),
(iii) For ν G Tp7V, /ι G C°°(iV;F), and σ G I>(£), we have
Vl(ha) = h(p)Vla + v(h)a(p).
(iv) If {7 G £(JV), then ρ ^ V^(p)a is smooth for all σ G Tf(E).
(v) If g : Ρ -> N and / : iV -» Μ are smooth, then V/o^ is related to
V-f by the following chain rule:
4°9(*og) = (VfTg.ua)
for it € TF; see the following diagram:
Ε
I*
Μ
In the next section we give a more geometric view of covariant
differentiation. Among other things, we obtain a more geometric way of producing V^
that does not appeal to a frame field. Also, we usually omit the superscript
on V-f since it will be clear from context.
As a special case, consider a section σ of Ε along a smooth curve с :
(α, b) —> Μ. Then we can form the operator V_a σ, which is also denoted by
at
щ or ν^. We have the formula
г г г ч г '
where ei о с is abbreviated to e^.
12.4. Ehresmann Connections
509
12.4. Ehresmann Connections
One way to get a natural covariant derivative on a vector bundle is through
the notion of connection in another sense. What we will define in this section
is a special case of what is called an Ehresmann connection. An Ehresmann
connection is a structure that can be defined in the category of general
smooth fiber bundles, but the two most common instances are defined for
vector bundles and for principal bundles, and in each case extra hypotheses
are added to the definition. We work with vector bundles here and give
an explanation of the principal bundle approach in the online supplement
[Lee, Jeff].
We first describe the construction in words since the notation tends to
obscure what is really a simple geometric idea. First, notice that an ordinary
function of one variable is constant on an interval if its graph is horizontal
over that interval. For sections of a vector bundle, there is no a priori notion
of "horizontal", so we have no principled way to decide what sections should
be considered constant. Consider a moving frame field ei,..., e^ on some
vector bundle. If we had a reason to declare these frame fields to be constant,
then we could differentiate a general section s = ^ slei in a direction υ by
the rule Dvs = Σ (vsl) e^. So it seems clear that having a geometrically
motivated way of picking out which sections should be constant will lead to
a way of differentiating sections. The first step is to notice that we have
a natural notion of vertical for a vector bundle π : Ε —> Μ. The tangent
spaces of the fibers of Ε are to be considered vertical. Thus the vertical
subspace of TyE is TyEp С TyE, where n(y) = p.
Now if we have a vector in TyE for some у G E, we would like to project
it onto the vertical direction. However, this entails having a complementary
horizontal space along which we project. The idea is then to assume that
there is a distribution on E, that is, a subbundle of ТЕ, that is everywhere
complementary to the vertical directions. Thus we obtain a subspace
complementary to the vertical, which allows projection onto the vertical. Once
we have this we can say that a section s G T(E) is horizontal (i.e. constant)
along a curve с : I —> Μ if Ts · c(t) has a vertical projection of zero for
all t. Now we can define a covariant derivative as follows. If υ G TpM and
s is a smooth section, then we can project Tps · ν onto the vertical space
Ts(p)Ep obtaining an element Vvs. But Ev is a vector space, and so we
may identify TS^EP with Ep and take Vvs to be an element of Ep. The
resulting map ν \—l· V^s turns out to define a covariant derivative. We will
also consider sections along general smooth maps f : N -* Μ and obtain
covariant differentiation for sections along /.
There is a canonical "parallelism" on Rn. Recall the canonical map
jp : Rn -» TpRn. A vector v\ <E TpRn is parallel to a vector v2 G TqRn exactly
510
12. Connections and Covariant Derivatives
Figure 12.1. Vertical space
when (jq ° Jp1) (vi) = V2- The map jq о j~l is a special example of what is
called "parallel translation" and in this case establishes what is sometimes
called "distant parallelism". In the presence of a connection, we obtain a
map between fibers of a vector bundle which is called parallel translation.
But this map may depend on a choice of smooth curve connecting the base
points. Locally, this path dependence of parallel translation is due to the
curvature of the connection.
We now proceed more formally. We give the definition of vertical bundle
not just for a vector bundle, but also for a general fiber bundle. First a
lemma:
Lemma 12.8. Let Ε Α Μ be α fiber bundle with typical fiber a k-manifold
F. Fix ρ e Μ and let г: Ep ^ Ε be the inclusion. For all у е Ep, we have
Tyi (TyEp) = Ker[7> : TyE -» TpM] = (Γ^π)"1 (0P) С ТуЕ,
where 0P G TpM is the zero vector. If ψ : Ep —> F is a diffeomorphism
and (V, x) a chart on F, then for all у G ψ~ι {V), dx ο Τυφ maps TyEp
isomorphically onto Rk.
Proof, ποιοη is constant for each smooth curve 7 in iV, so Τπ·(Τι · 7(0)) =
0P. Thus Tyi(TyEp) С (Τ»"1 (Op). On the other hand,
dim((T^)_1 (Op)) = dim Ε - dim Μ = dimF = dim TyEp,
so (Tyi) (Ep) = (Τυπ)~λ (Op). The rest is clear since dx is just Tx followed
by the projection Tx(U) = x(U) xKfc-) Rk. D
Note: As usual we identify Tyi (TyEp) with TyEp.
12.4. Ehresmann Connections
511
Definition 12.9. Let π : Ε —> Μ be a fiber bundle with typical fiber F
and dimF = k. Let VyE := (Тутт)~1 (0P) where n(y) = p. The vertical
bundle on π : Ε —> Μ is the real vector bundle πν : VE —> Ε with total
space defined by the disjoint union
VE:= [J VyE С ТЕ.
yeE
The projection map is defined by the restriction π ν := kte\ve· A vector
bundle atlas on VE is given by vector bundle charts of the form
(πν, dx о ГФ) : тгу ^тГ1 (U) Π Φ"1 (V)) -» (π-1 (/7) Π Φ"1 (V)) χ R*,
where φ = (π, Φ) is a bundle chart on Ε over U and (V, x) a chart in F.
VfE >VE
1 7 J
/*£ —£
ι , ι
Exercise 12.10. Prove: Let / : TV —>· Μ be a smooth map and π : Ε -+ Μ
a fiber bundle with typical fiber F. Then Vf*E —> f*E is bundle isomorphic
to f*VE -» /*£ where /:= p^^*^ : /*E -» Ε and pr2 : Μ χ Ε -> Ε.
Now consider the pull-back bundle f*E where / is as above. Since f*E
is the submanifold οι Ν χ Ε defined by the condition that (g, y) is in f*E if
and only if f(q) = π(?/), a curve in /*£^ must be of the form (ci, C2), where
ci is a curve in TV and C2 is a curve in E, and it must also be the case that
/oci = noc2- Now if prx and pr2 are the first and second factor projections
from Ν χ Ε, then (Грг1,Грг2) gives a vector bundle isomorphism of the
bundle T(N χ Ε) -+ Ν χ Ε with the bundle TN xTE -+ Ν χ Ε, and so
we expect that under this isomorphism Τ (f*E) corresponds to a subbundle
of TN xTE -+ Ν χ Ε.
Exercise 12.11. Show that under the bundle isomorphism (Tpr1,Tpr2) :
T(N χ E)^ TNxTE, the tangent bundle Γ (f*E) corresponds to {(v, ги) G
TNxTE : Tf-v = Τπ-w}. Under this isomorphism (Vf*E), ч corresponds
to {0q} χ VyE.
We need to make an observation. If V is a complex vector bundle and
χ e V, then the tangent space TXV has a natural complex structure. Indeed,
the map jx : V —> TXV is used to transfer the complex structure of V to
that of T^V. In particular, if π : Ε —> Μ is a complex vector bundle, then
we may view TyEp = (Tyn)~ (0P) as a complex vector space. Thus VE is a
complex vector bundle.
the following, refer to the diagram:
NxE
\ /*£ —
V
N ►
Μ
512
12. Connections and Covariant Derivatives
The vertical vector bundle VE is isomorphic to the vector bundle π*Ε
over Ε (we say that VE is isomorphic to Ε along π). To see this, note that
if (v, w) G π*Ε, then π(υ + tw) is constant in t. Prom this we see that the
map from π*Ε to ТЕ given by (v,w) н-> d/di|o(t; + tw), maps into VE1. It
is easy to see that this map is a vector bundle isomorphism:
j:n*E^ VE,
j:(y,w)^> Jyw:= — (y + tw) = wy.
The meaning of the symbol j depends on the bundle we have in mind, but
it is consistent with our previous use in the sense that if Ε is the tangent
bundle of an open set in a vector space, then jy is the canonical isomorphism
as before.
Definition 12.12. A (linear Ehresmann) connection on a vector bundle
π : Ε —> Μ is a smooth distribution Η on the total space Ε such that
(i) Η is complementary to the vertical bundle:
TE = U® VE.
(ii) Η is homogeneous: Τνμτ (Hy) = Hry for all у G E, r G R, where
μτ : Ε —> Ε is the multiplication map given by μτ : у \-ь ту.
The subbundle H is called the horizontal distribution (or horizontal
subbundle).
The statement ТЕ = Η Θ VE means that for every у G Ε we have
the internal direct sum decomposition TyE = Hy Θ VyE. Any υ G ТЕ
has a corresponding decomposition υ = pvv + p^v. Here, pv : υ и· ρνυ
and ph : υ н-> рь^ are the obvious projections onto VyE and 7^ referred
to respectively as the horizontal and vertical projections. Note well that
without a choice of horizontal distribution Η there is no vertical projection
pv = 1 - Ph· For у G E, an individual element w G TyE is horizontal
if w G Hy and vertical if w G VyE. A vector field X G 3C(J5) is said to
be a horizontal vector field (resp. vertical vector field) if X(y) G Hy
(resp. X(y) G VyE) for all у G 22. On the right hand side of Figure 12.2 we
see a schematic representation of the field of horizontal spaces. Notice that
these spaces are tangent to the zero section (which we identify with M).
This must be the case and is a consequence of the homogeneity condition
(ii) from the definition. On the left hand side we see a particular tangent
space to Ε together with a vector and its vertical projection.
Exercise 12.13. Show that Η = π*ΓΜ.
Theorem 12.14. Every vector bundle admits a connection.
12.4. Ehresmann Connections
513
Figure 12.2. Horizontal distribution
Proof. First notice that we may easily define a connection on a trivial
bundle prx : Μ χ V -» Μ. Given a fixed υ G V, let iv : Μ -» Μ χ V be
defined by 2υ(_ρ) := (ρ, ν). Next define Ή(ρ?ί;) := Tiv(TpM) for each p. We
then have Трт1(Н(р^)) = TpM. Since for any scalar α we have μα°4 = iav,
we also have Τμα ο Tiv = Tiav so that
Τμα(Η{ρ,υ)) = Τμα (Tiv(TpM)) = Tiav(TpM) = U{p,av) = UaM.
For a general vector bundle π : Ε —> Μ, let {Ua} be a locally finite
cover of Μ such that the bundle is trivial over each Ua. Then we may
choose a connection Ka on each π_1(ί7α). Let {pa} be a partition of unity
subordinate to {/7a}. For each у G E, define Ly : Τπ^Μ —> T^E by
L2/(^) := Σ РаЫу))и>а,
{a:n(y)eUa}
where for each a, the vector wa is the unique vector in Ha such that Tn-wa =
v. It is easy to check that Ly is linear and satisfies Tyn о Ly = ΊάτρΜ· The
distribution we seek is then defined by Ly (Γ7Γ(2/)Μ) for each y. We leave
it to the reader to check that this distribution is smooth and satisfies the
required conditions. D
Theorem 12.15. If К is a connection on π : Ε —> Μ and fj Ν -ϊ Μ α
smooth map, then the distribution f*T-L = (Tf)~l(%), where f := ρτ2\**Ε,
defines a connection on the pull-back bundle f*E —> N. This is referred to
as the pull-back connection:
f*H С Tf*E^^TE
\ Tf \
TN ^TM
514
12. Connections and Covariant Derivatives
Proof. First note that / is the restriction of the projection TV χ Ε —> Ε and
so f*H := {TJ)-l(U) can be defined: For (q,y) G f*E, we let (/*«)fo,y) =
{Тш1)~1иу. By Exercise 12.11 we can identify T(f*E) with {(v,w) €
TNxTE : Τ/·ι> = Τπ·^}. Under this isomorphism, (Vf*E), ^ corresponds
to {OJ x VyE while (f*H){qty) corresponds to {(v, w) e TNx Η :Tf-u =
Τπ · ν}. Thus we have the decomposition
(г;,рьгу) + (0,pvw),
which is easily seen to be unique. Hence the distribution f*% is
complementary to Vf*E. Under the same identification, multiplication ma on f*E is
гпа{ч->у) — (Ят^аУ) and Tma(v,w) = (ν,Τμανυ), which makes it clear that
the second defining condition for a connection holds for f*H since it holds
for H. □
Given a vector υ G TpM and a choice of у G Ep, there is a unique
vector vy e Ήυ С ΤυΕ such that Τυπ · vy = v. This vector is called the
horizontal lift of υ to TyE. The idea works for fields too. Given a vector
field X G X(M), there is a unique vector field X G X(E) such that X(y) is
horizontal for all у G Μ and Tyn-X(y) = X(n(y)). Thus X G Γ (Ή) С X(i?).
This horizontal vector field on the total space Ε is called the horizontal
lift of X. Clearly X is π-related to X.
Proposition 12.16. Let π : Ε —> Μ be a vector bundle with connection H.
IfX,Ye X(M) and f G C°°(M), then
(i) axXbY = aX + bY for all a,beR;
(ii) /Χ = (/οπ)Χ;
(iii) [X,y] = ph[X,y].
Proof, (i) and (ii) are obvious, (iii) follows from the easy to check equalities
π*[Χ, Υ] = [Χ, Υ] = π* [Χ, У] and the uniqueness of horizontal lifts. D
Definition 12.17. Let σ : TV —> Ε be a section of Ε along a map / : N -4
M. We say that σ is a parallel section if Γσ-г; is horizontal for all υ G TN.
If 5 is a section of Ε and с : / —> Ε is a curve, then we say that 5 is parallel
along с provided 5 о с is parallel.
Exercise 12.18. Recall that a section of f*E must have the form s : q И
(g, σ5(ς)), where σ is a section along /. Show that if 5 is parallel with respect
to the pull-back connection on /*22, then as is parallel.
Exercise 12.19. Let [0,6] be an interval and let t denote the standard
coordinate on [0, 6]. Suppose that π : Ε —> [0, 6] is a vector bundle over [0,6]
with connection. Let д denote the horizontal lift of d/dt.
12Λ. Ehresmann Connections
515
(a) Show that if с : [0, α] —> Ε is an integral curve of <9, then c(a) G Ea.
[Hint: Show that π о с is an integral curve of d/dt.]
(b) Let 0 < to < b. Show that there is a fixed e > 0 depending only
on to such that all maximal integral curves of д originating in the
fixed fiber Eto are defined at least on [io,e). [Hint: Endow Ε with
a bundle metric and consider all integral curves originating in the
unit sphere in Eto and then use property (ii) of Definition 12.12.]
(c) Using (a) and (b), show that integral curves of д all have domain
equal to [0,6].
Finding a horizontal lift of a vector field in X(M) is trivial and
automatic. On the other hand, finding a parallel section along a given map with
prescribed value a(q) for some q is generally nontrivial and may not exist.
However, we have the following
Theorem 12.20. Let π : Ε —> Μ be a (smooth) vector bundle with a
connection %. Suppose that с : [α, 6] —> Μ is a smooth curve. For each
и G Ec^a), there is a unique parallel section ac^u along с such that aCyU(a) = u.
If Pc · Ec(a) ~^ Ес(Ъ) denotes the map which takes и G Ec(a) to ac,u{b), then
Pc is linear.
Proof. Without loss of generality we may take α = 0. Let д = д/dt denote
the standard coordinate vector field on the interval [0,6] and let д denote
its horizontal lift in the pull-back bundle c*E with respect to the pull-back
connection с*%. Let cu denote the maximal integral curve of д in c*E with
cu(0) = (0,u) G c*E. We have
τ: (Pri ° cu) = Τρτλ о си = Tprx о д о си = д о ргх о си.
at
Thus prx ocu is an integral curve of д = д/dt and so prx ocu(t) = t. Prom this
we see that cu(t) = (i, pr2ocw). By Exercise 12.19 we know that cu is defined
on [0, 6] and that cu(b) G (c*E)b. Let ac,u '·= Pr2 ocu on [0,6]. Then ac?w is a
section of Ε —> Μ along с which is parallel since cu is horizontal (see Exercise
12.18). We define Pcu := aC)U(b) for и G Ec^)· The uniqueness follows from
the uniqueness of integral curves and we leave this to the reader. For any
rGR, the field raCyU is parallel. Indeed, (rac,u) = T/^r ° &c,u is horizontal
since ΤμΤ preserves %. But then Pc(ru) = rPc(u) so Pc is homogeneous.
We aim to show that Pc = Jq1 о ТРс о j0, which is a composition of linear
maps. Let vo G ToEc(0) so that vo = 7(0), where 7 is defined by 7(i) = tv
for an appropriate υ G Ec^y Then j^"1^ = v. We have
T0Pcvo = ^(Pco7)(0).
516
12. Connections and Covariant Derivatives
But also Pc ο 7(ί) = Pc(tv) = tPc(v) so that
ToPcvo = jo (Pc (v)) = Jo о Pc о j^vo.
Thus jo о Pc о Jq1 = TqPc or Pc = Jq1 о TqPc о jo, and we conclude that Pc
is linear.
Since Pc has inverse Pc*- where c*~(i) := c(b — i), we see that it is a
linear isomorphism. D
Definition 12.21. Let с : [α, b] —> Μ be as in the theorem above. The map
Pc is called parallel translation along с from c(a) to c(b). Let с be any-
smooth curve in M. For t\ and ^ in the domain of c, let P{c)\2 : -Ec(ti) ™>
Ec(t2) be defined as P(c)£ := ΡφιΜ if t2 > h and P(c)\\ := P^2 ^ if
ii > ί2·
We also say that the curve ac?w of Theorem 12.20 is a parallel lift
or horizontal lift of the curve c. The map Pc is also sometimes called
parallel transport. Suppose that с : [α, b] —> Μ is a (continuous) piecewise
smooth curve (Definition 2.121). Thus we may find a monotonic sequence
ίο?ίι? · · · ,ij = t such that C{ := c\u t-\ (or cL t -ι) is smooth.1 In this
case we define
P(c)\0 := Р{с)\^ о ■ ■ ■ о Р(с)Ц.
Exercise 12.22. The map P{c)\0 : Ec(t0) -> Ec(t) ls a linear isomorphism
for all t with inverse P(c)
to
t ·
Parallel transport behaves nicely with respect to reparametrization. This
is the content of the next theorem, which we ask the reader to prove in
Problem 1.
Parallel transport
Theorem 12.23. Let с : [α,6] -ϊ Μ be a smooth curve. Ifr : [a\bf] -> [a,b]
is a smooth map with dr/dt > 0, then for 7 := с о r we have Pc = ΡΊ.
xIt may be that t < to .
12.4. Ehresmann Connections
517
It is often convenient to use special vector bundle charts constructed
using parallel translation. Let ({7, x) be a chart on Μ centered at ρ G Μ
and such that x({7) is a ball Br(0) of radius R. Consider the family of
curves cu : [0, R] -> Μ given for each u G Sn_1(#) = dBR(0) by
cu(i) :=х_1(*и).
Define a frame field (ei,..., e&) for Ε over U as follows: Let (ei(0),..., e/c(0))
be an ordered basis of Ep. If q G /7, then let еДд) be defined as
ег(д):=Р(си)[)(ег(0)),
where (u,i) is the unique element of Sn~1(R) χ [0,1] such that cu(t) = q.
We say that (ei,..., e/-) is radially parallel with respect to the spherical
chart (/7, x). Notice that by composing with a dilation of W1 if necessary, we
may choose R to be any positive number. Also, if we use a radially parallel
frame field to define a vector bundle chart φ = (π, Φ) in the usual way, then
Φ will be constant along each curve cu.
We still have more to learn about the geometric meaning of a connection.
We will eventually be led to the notion of curvature. At this point let us just
consider what it means when a connection is integrable as a distribution.
Definition 12.24. A connection on a vector bundle is called flat if it is
integrable as a distribution.
Let us agree to call a connection on a vector bundle Ε a trivial
connection if given any и G Ε there is a parallel section s such that s(n(u)) = u.
Finding such sections is not always possible (even locally). In fact, we have
the following characterization.
Theorem 12.25. Let π : Ε —ϊ Μ be α vector bundle with connection %.
The following assertions are equivalent:
(i) For any simply connected open set U С Μ, the restriction of К
to n~l(U) is a trivial connection on the restricted vector bundle
tt-1(U)-^U.
(ii) К is flat.
Proof. If (i) holds, then given any и G Ε there is a parallel section s
defined on U. Then it is easy to see that s(U) is an integral manifold of the
distribution K. Since we can find such an integral manifold through any u,
we see that % is integrable (i.e. flat).
If % is flat, then certainly Ή\ν is flat for any open U. Suppose that U
is simply connected. By the Probenius theorem there is a maximal integral
manifold Lu of Ή\ν through any и G π-1 (U). By Theorem 12.20, we see
that any smooth path in U can be lifted uniquely to Lu. This is enough to
518
12. Connections and Covariant Derivatives
imply that n\L : Lu —> U is a covering space (see [Span], 2.4.10). Since
U is simply connected, the lifting theorem for covering spaces (Theorem
1.95) implies that n\L has an inverse. The desired parallel section is then
We now come to the task of relating covariant derivatives with Ehres-
mann connections on vector bundles. Denote the vector bundle isomorphism
from VE to Ε along π by p:
ρ : VE -> E,
ρ : wy h-> w.
For each y, the map ρ just gives the canonical identification of TyEp with
Ep, and on each fiber, it is the inverse of j. If we have a connection on
π : Ε —> Μ, then we have an associated connector (or connection map),
which is the map к : ТЕ —> Ε defined by
κ (υ) :=p(pv^) =Зуг(руУ)
for υ G ΤυΕ. The connector is a vector bundle homomorphism along the
map π : Ε -> Μ:
TE^^E
An interesting fact is that given the appropriate definition of vector space
structure on the fibers, ТЕ is also a vector bundle over TM via the map Τπ
(Problem 11 from Chapter 6). Recall that the addition and scalar
multiplication on a fiber Τπ~ι(χ) of this bundle are defined by
uSv:=Ta- (u,υ) for u,v eTE with Τπ-η = Τπ·υ = χ,
cQv := Τμ0 · υ for υ G ТЕ and с G F,
where a(j/i, y2) := y\ + У2 for (2/1,2/2) G £ Θ Ε and дсу := су for у e Ε and
cGF.
Lemma 12.26. Suppose that f : RK —> Rk is a smooth map such that
f(av) = af(v) for all υ G Шк and a G M. Then f is linear. Similarly, if
f : CK —> Ck is a smooth map such that f(av) = af(v) for all ν G C^ and
a G C, then f is linear.
Proof. Let / : RK -> Rk as in the statement. Then Df(0)v = ^\t=Qf(tv)
= i\t=0tf(v) = f(v). Thus / = Df(0) and so / is linear. If / : CK -+ Ck
is a smooth map such that f(av) = af(v) for all υ G CK and α G С then
the first part shows that / : CK —> Ck is linear over R. But by hypothesis
f{iv) = if{v), so / is actually complex linear. D
12.4. Ehresmann Connections
519
Corollary 12.27. Let πι : Ει —> Μι αηάττ2 : ^2 —>■ Λ^2 be¥-vector bundles.
Let f : Ει -^ Ε2 be α fiber bundle morphism over f : Μι —ϊ Μ2. If f is
homogeneous on each fiber, so that f(av) = af(v) for all ν G E\ and aGF,
then f is linear on fibers, and so it is a vector bundle morphism.
Proof. Use vector bundle charts and Lemma 12.26. D
Lemma 12.28. Let μτ : Ε —> Ε be multiplication by r. Then for any ρ G Μ
and y,w G Ep we have
Τμτ (jyw) = jry (rw) = rjryw.
Proof. We have
(ry + trw)
i=0
= Jry (rw) = rjryw. D
The connector к gives a vector bundle homomorphism along птм :
Τ Μ —ϊ Μ. More precisely, we have
Theorem 12.29. If к is the connector for a connection on a vector bundle
π : Ε —> Μ, then к gives a vector bundle homomorphism from, the bundle
Τπ: ТЕ -> Τ Μ to the bundle π: Ε -> Μ along the map пТм- Τ Μ -> Μ,
TE~J1-^E
Τπ\
τ τ
Τ Μ ^Μ
Proof. It is easy to check that the above diagram commutes. Thus we
must show that к is linear on fibers. Let Yy G (Τπ)~ (Χρ) where n(y) =
p. We may write Yy = Hy + Vy, where Hy and Vy are horizontal and
vertical respectively Since Xp = Τπ (Yy) = Τπ (Hy), we see that Hy is the
horizontal lift Xy of Xp to TyE. Also, Vy = jyw for a unique w G Ep. So
the decomposition may be written as Yy = Xy +jyw. By definition we have
n(Yy) = w.
Using the homogeneity property of the horizontal distribution together with
Lemma 12.28 above we have
Τμτ (Yy) = Τμτ (Xy) + T^{jyw) = Xry + jryrw.
Thus κ (Τμτ (Yy)) = rw. Therefore, we have κ (Τμτ (Yy)) = гк (Yy). If we
denote the scaling operation for the vector bundle structure on ТЕ —> Τ Μ
using Θ as before, then this last statement is just к (r Q Yy) = гк (Yy). The
result now follows from the Corollary 12.27. D
Τμτ (jyw) =
dt
VLr{y + tw) =
t=Q
dt
520
12. Connections and Covariant Derivatives
Once we have a connection on our bundle then the addition in the vector
bundle ТЕ —ϊ Τ Μ can be described in a convenient form. Any element of
ТЕ that lies over the same element Xp G Τ Μ can be written in the form
Xy+jyW for some w G Ep, where Xy is the horizontal lift of Xp to the point
y. Let Xyi + jyiw\ and ХУ2 + jy2W2 be two such expressions. Then the sum
of these two elements using the addition in the vector bundle ТЕ —ϊ ΤΜ
is given by ХУ1+У2 + 3yi+y2 (wi + ^2)? where ХУ1+У2 is the horizontal lift of
Xp to the point у ι +У2-
Exercise 12.30. Prove the last statement.
Exercise 12.31. Deduce from the previous theorem that (kte, к) '- ТЕ -Л
Ε Θ Ε is a vector bundle isomorphism along the tangent bundle projection
πΤΜ : Τ Μ ->> Μ.
Using this notion of connection with associated connector к we can get
a covariant derivative. If К is a connection on a vector bundle Ε —ϊ Μ with
connector к, then for a section σ G Tf(E) along a smooth map / : Ν -> Μ
we make the definition
ν{σ := κ (Τρσ · υ) for υ G TpN.
If V is a vector field on JV, then (ν£σ)(ρ) := Vy, ^.
Theorem 12.32. Let π : Ε —> Μ be α vector bundle. Suppose that the
vector bundle is endowed with a connection К and associated connector к.
Then for each smooth map f : Ν —ϊ Μ, the map V·^ defined above is a
covariant derivative along f (Definition 12.7/ If f = 1<1м, then we obtain
a Koszul connection.
Conversely, if V is a Koszul connection on π : Ε —ϊ Μ, then we may
define a connection by
Hy := {Ts · и - JyVus : s G Г(Я), s(n(y)) = y, ue Tn(y)M}.
The resulting connection, in turn, gives back the Koszul connection according
to Vv5 := к (Tps · υ) for ν G TpM.
Proof. We write V for V·^ when no confusion is likely. We start by simply
noting that (i) and (iv) of Definition 12.7 are easily seen to be true and leave
these as an exercise. Notice that if g : Ρ —> N and / : TV —>· Μ are smooth
and и G TP, then for each σ G Tf(E), we have Vw (s о g) = VTg-us- Indeed,
Vu (σ о д) = (κ ο Τ(σ од))и = κ (Τσ (Тд · и)) = Vt9-u^·
This gives (v) of Definition 12.7. Next we claim that if σι, θ2 G Tf(E) and
и G TPN, then
Vw (σι + σ2) := Vuai + νυσ2,
which implies (ii) of Definition 12.7.
12.4. Ehresmann Connections
521
Proof of the claim: Recall the definition of addition in the vector bundle
Τπ : ТЕ ->> Τ Μ in terms of the tangent lift of a : (u, v) H> u+v. Ifu = 7(0)
for a smooth curve 7 in N with 7(0) = p, then
d
Τσι · и Ш Τσ2 · u := Τα(Τσι · и, Τσ2 · и) =
rfi
(σι ο 7 + σ2 ο 7)
d_
~dt
(σι + σ2) ο 7 = Γ(σι + σ2) · гг.
Now we use the above together with the fact that к is a bundle homomor-
phism along πτΜ· We have
Vw (σι + σ2) = κ (Γ(σι + σ2)η) = к(Тащ Ш Τσ2^) = V^i + Vua2.
Claim: If h e C°°(N,¥) and σ G Г/(Я), then Vuha = u(h)a(p) +
h(p)Vua, which is (iii) of Definition 12.7.
Proof: Let и G TpN and let σ : N —> Ε be a section along a smooth
map f : N -> M. We begin by calculating Τμ : TRxTE -+ ТЕ, where
μ : R χ £7 —> Ε is scalar multiplication in the vector bundle Ε —> M. So
let (a,y) G Κ χ Ё and (&^Ια'νν) G TflM x Γνβ· Το find Τμ ' (baila'vv)
we calculate Τμ · (0α, г;у) and Τ μ · (Ь ^| , 0y) separately. Let c be a smooth
curve with c(0) = у and c(0) = гу Then
Γμ.(Οα,ι>ν) =
di
μ(α,φ)) =
rfi
Ma(c(0)
= Τμα · г^ =: α Θ г;у,
where Θ refers to multiplication in the vector bundle structure of ТЕ —ϊ
TM. Next, let с be the curve in the manifold R given by c(t) := a + tb so
that c(0) = bft\a. Then
d
T"(bI, ·°») =
i.
МФ),у)= ^
((α + W)y)
(ay + %) = д,у(6у).
We now have
Τμ'\~αΎ\ 'vy)=aQvy+:)ay(hy}-
Next we let с be a curve in N with c(0) = ρ and c(0) =«£ Tp7V. Then
Tph-u= —
at
h(c(t)) = (hoc)'(t)
dt
ft(c(0))
= u[b]
h(p)
522
12. Connections and Covariant Derivatives
Now let h χ σ : TV —>· R χ Ε denote the map defined by (h χ σ) (χ) =
(/ι(χ),σ(χ)) and let σ := ha. Then using the linearity of к and what we
developed above we have
Vuha = κ(Τσ · и) = κ [Τμ ο Τ (h χ σ) (и)]
d
= κΤμ(η[Κ\ — ,Γρσ·τζ)
ν ώίΙ/ι(ρ)
= /с {h(p) Θ (Τσ · и) + Λι(ρ)σ(ρ)ΜΛ]σ(ρ))}
= Η(ρ)κ(Τσ · τζ) + τζ[/ι]σ(ρ) = h(p)Vua + u[h]a(p).
The proof that
Wy := {J2/VW5 - T5 · и | 5 G Г(Я), u G Τπ(2/)Μ}
defines a connection that gives back the original covariant derivative is left
to the reader as Problem 9. D
Example 12.33. In Chapter 4 we saw that each hypersurface in Rn+1 has
a natural covariant derivative (the Levi-Civita covariant derivative). This
means that there is a corresponding horizontal distribution. Let us consider
the canonical connection on Sn С JRn+1. We may identify TSn with the
subset of Sn χ Rn+1 given by {(p,u) e Sn χ Mn+1 : {p,u) = 0}. Under
this identification, the velocity of a curve с has the form с = (c,cf). Now
let us find a representation of T(TSn) as a subset of TSn χ Μ2η+2. A
section of TSn along a curve с has the form σ : t ь-> (c(i),x(i)) and so we
take σ = (с, χ, d,x') G TSn χ Μ2η+2. Note that since (c,x> = 0, we have
(c, x;) = — (c;,x). Also, since (c, c) = 1, we have (c,cf) = 0. It follows that
we may make the identification
T(TSn) = {((p, u), (t;,«;)) € TS" χ R2"+2 : (p, V) = 0, (p,«;} + (n, V} = 0}.
The vertical space V(p?w) in T(pu)(TSn) is
V(p,„) = {((P,n), (0,«;)) € T(PiU)(T5")}.
Recall that in Chapter 4 we obtained the Levi-Civita covariant derivative
by projection of the ambient derivative back into the tangent space. In
the present context, this means that ((p, u), (v,w)) is horizontal if w is a
multiple of p. But since we also have (p, w) + (г/, г;) = 0, the criterion for
being horizontal becomes w = — (u, v) p. A little thought reveals that we
should take К to be defined by
Нш := {((p,u), (v, - (u,t;>p)) G T(P;W)T5n}.
Exercise 12.34. Let p,q G S2 С Μ3 be such that (p,g) = 0. The great
circle containing ρ and q can be parametrized as
c(t) := (cos ί) ρ + (sin ί) g
12.4. Ehresmann Connections
523
Figure 12.3. Parallel transport around path shows holonomy
for 0 < t < 2π. Using the notation of the previous example, show that
if σ = (с, χ) is a parallel section along c, then both (x,x) and (c',x) are
constant along с
Figure 12.3 shows parallel translation of a vector at the north pole
around a loop comprised of segments of great circles. Notice that
parallel translation around this loop results in a map which is not the identity
map. In general, if π : Ε —> Μ is a vector bundle with connection and
с : [α, b] —> Μ is a piecewise smooth closed curve with ρ = с(а) = c(b), then
we obtain an isomorphism Pc G GL(Ep). This map is called the holonomy
of the curve.
Definition 12.35. The subset Gp С GL(£P) consisting of the maps Pc as
с ranges over all piecewise smooth closed curves с with ρ = с(а) = c(b) is
called the holonomy group at ρ for the connection.
It is easy to see that any piecewise smooth curve с : [α, b] —> Μ induces
a group isomorphism between Gc(a) and Gc^ given by
g^PcogoP-\
Thus for a connected manifold we may speak of the holonomy group of the
manifold, and this is well-defined up to group isomorphism.
One may use parallel translation to recover the covariant derivative and
this is a very natural viewpoint:
Theorem 12.36. Let π : Ε —> Μ be α vector bundle with connection %.
Let f : N —> Μ be α smooth map. Ifue TPN and σ is a section of Ε along
524
12. Connections and Covariant Derivatives
f, then for any smooth curve с : (—ε, ε) —> N with c(0) = и we have
ν/σ = lim (P(f о сУоУ1 a(c(t)) - a(c(0))
u t-+o t '
where P(f о c)q is the parallel transport along foe from (/ о с) (0) to
(foc)(t).
Proof. Use parallel transport to obtain parallel frame fields ei,..., e/~ along
foe with ei(0),..., β/~(0) a basis of E^foc^0y Then we may write
σ о с = 2_] σ%£ϊ
for unique smooth functions σι defined on (—ε,ε). Then
(P(f о с)*)"1 σ ο c(t) = (Р(/ о с)*)"1 Σ At)ei(t)
Let Do denote J^L. Then using the chain rule for connections and the fact
that S7gtei(t) = 0 we have
V„a = νέ(0)σ = Vtc-Do0"
= VA,(aoc) = VDo53ai(i)ei(i)
= Σ
= lim
ί->0
~dt
eM
It
[(P(f о с)*)"1 σο c(t)
k-l
(P(foCyoyla(c(t))-a(c(0))
t
D
The most important special cases are the case of / = id^, where we
recover a basic Koszul connection, and also the case where / itself is a curve
с : I -* M. However, recall that we often omit the superscripts indicating
maps on the various covariant derivative operators.
Corollary 12.37. Let π : Ε —> Μ be a vector bundle with connection H. If
ν G TPM and s is a section of E, then for any smooth curve с : (—ε, ε) —> Μ
with c(0) = ν we have
(PJcYq)-1 s(c(t)) - s(c(0))
vvs = lim ,
i->0 t
where Р(с)г0 is the parallel transport along с from c(0) to c(t).
Now recall that if с : I —> Μ is a smooth curve and σ : 7 —> Ε is a
section along c, we define V_a_a to be a section along с given by
(Va_a) (t) := Vei σ.
\ dt / dt\t
12.5. Curvature
525
We have
Corollary 12.38. Let π : Ε —> Μ be α vector bundle with connection K.
Let с : I -* Μ be α smooth curve and suppose that σ : I —> Ε is a section
along с Then we have
V di ) e->0 б
Exercise 12.39. Derive the above two corollaries from Theorem 12.36.
Corollary 12.40. Let π : Ε —> Μ be a vector bundle with connection %.
Let f : N —> Μ be a smooth map. If σ is a section of Ε along f, then σ
is parallel if and only if ν{σ = 0 for all и G TN. In particular, a section
s G T(E) is parallel along a curve с if and only if V^/^s о c = 0.
If one approaches parallelism solely via covariant derivatives, then one
could take ν£σ ξ 0 as the definition of parallel. This is a common equivalent
approach.
Warning: There is a subtle point to be made here. If s is a section of
Ε and с : (—ε, ε) —> Μ is a curve, then Vc^s is zero whenever c(t) = 0, but
for a section σ : I -* Ε along с it is possible that (V^/^σ) (i) is nonzero
even when c(t) = 0. For example, if cv : I —> Μ is a constant curve taking
the fixed value ρ G M, and σ : 7 —> TpM is a smooth map, then σ can be
considered a section along cv and then V^/^σ = σ' (the ordinary derivative
of a curve in the vector space TpM).
Exercise 12.41. Show that σ G Γ(Μ, Ε) is a parallel section if and only if
σ о с is parallel along с for every curve с : I —> M.
Exercise 12.42. Show that if t ь-> a(t) is a curve in Ep, then we can consider
σ as a section along the constant map cp\t\-^p and then ν^σ(ί) = a'(t) G
Ep.
Exercise 12.43. Let V be a connection on Ε -^ Μ and let a : [a, 6] —> Μ
and β : [α, 6] -* £α«ο). If Χ(ί) := Ρ(α)ί0(/3(ί)), then
(VftX)(i) = P(a)i0(/3/(i))-
Note: /3;(£) G £а(г0) ^ог а^ *· [Hint: Use a parallel frame field along the
curve.]
12.5. Curvature
An important fact about covariant derivatives is that they do not need to
commute. If σ : Μ —> Ε is a section and X G X(M), then Wχσ is a section
also, and so we may take its covariant derivative VyV^a with respect to
some Υ G X(M). In general, VyVxa φ VxVya. A measure of this
526
12. Connections and Covariant Derivatives
lack of commutativity is the curvature operator that is defined for a pair
Χ, Υ € X(M) to be the map F(X, Y) : T(E) -+ T(E) given by
F(X, Υ)σ := VxVya - WYVxa - V^yja,
i.e. F(X,r):=[Vx,Vy]-V[x,r].
Theorem 12.44. For fixed σ, the map (Χ,Υ) >-»· F(X,Y)a is C°°(M)
bilinear and antisymmetric. Also, F(X,Y) : T(E) ->· T(E) is a C°°(M)
module homomorphism; that is, it is linear over the smooth functions,
F(X,Y)(fa) = fF(X,Y)(a).
Proof. We leave the proof of the first part as an easy exercise. For the
second part, we calculate:
F(X, Y)(fa) = VxVy/σ - Vy Vx/σ - V[xx]fa
= Vx(/Vya + (Yf)a) - Vy(/Vxa + (Xf)a)
-/V[x,y]a-([X,y]/)a
= /Vx Vya + (Xf) Vya + (Yf)Vxa + X(Yf)
- /VyVxa - (Yf)Vxa - (Xf)VYa - Y(Xf)
-fV[xy]a-([X,Y]f)a
= /[Vx, Vy] - /V[x,y]a = /F(X, Υ)σ. Π
Thus we also have F as a map F : X(M) χ X(M) -» Γ(Μ, End(£)). But
F is C°°(A/) bilinear in the first two slots also.
Exercise 12.45. Show that F is C°°(M) bilinear in the first two slots. Let
vp,wp e TPM and Up e Ep be given. Let Χ,Υ e X(M) and σ <Ε T(E)
such that X(p) = vp,Y(p) = wp and σ(ρ) = σρ. Argue that F(Xp,Yp)ap is
well-defined and that the map {vp,Wp,ap) \-> F(vp,Wp)ap is multilinear.
In light of this exercise we see that for each vp,wp G TpM, F(vp,wp)
is a linear map Ep —> Ep. In other words, F(Xp,Yp) G End(Ep) (a.k.a.
L(Ep,Ep)). But F(Xp,Yp) is antisymmetric, and so we obtain, for each p,
an element of End(£p) ® /\2T*M. Finally, we see that F may be considered
as a section of End(E) ® Λ2Γ*Μ. The space of such sections is denoted
Q2(M,End(E)).
[ Curvature is an End(i£)-valued 2-form. |
We will have many occasions to consider differentiation along maps, and
so we should also look at the curvature operator for sections along maps.
12.5. Curvature
527
f(s,t)
Я*.0)
Figure 12.4. Parallel translation around loop
Let / : TV —> Μ be a smooth map and σ a section of Ε —> Μ along /. For
U,V e X(N), we define a map F'(E/, V) : Tf(E) -» Tf(E) by
F/(C/, 1/)σ := VuVva - V^Vf/σ - V^ja
for all σ e Tf(E).
Note that if s e T(E), then sof e Tf(E), and ifUe X(N), then Г/оС/
G Tf(TM) =: Xf(M) is a vector field along /. As a matter of notation, we
let F(Tf o[/,T/o V> denote the map ρ ^ F(Tf · Up, Tf · Vv)s (p), which
makes sense because of Theorem 12.44. Thus F(Tf o[/,T/o V)s e Tf(E).
Then we have the following useful fact:
Proposition 12.46. Let X e Tf(E) and C/, V e X(N). Then
Ff{U, V)X = F(Tf о С/, Tf о V)X.
Proof. Exercise!
D
The next theorem is an important step in understanding the geometric
meaning of curvature in terms of parallel transport. Refer to Figure 12.4.
Theorem 12.47. Let π : Ε —> Μ be a vector bundle with connection.
Let u,v G TpM and у G Ev. Let U be a neighborhood of the origin in
Ш2 and denote standard coordinate frame fields on Ш2 by d\, 82 and their
values at the origin by <9i(0) and ^(O). Suppose that we have a smooth map
f : U-> Μ withf(0) =p, Tpf-di(Q) =u andTpf-d2{0) =v. For small s,t,
let ysj denote the element of Ep obtained by parallel translation of у around
the piecewise smooth loop obtained by successively tracing the following four
curves:
(1) cf :σ^/(σ,0)?0<σ<5;
528
12. Connections and Covariant Derivatives
(2) cj :ги/(в,г), 0 < r < i;
(3) C3' :(ти /(5 — σ, ί), Ο < σ < s;
(4) cf :r^/(0,i-r), 0<r<i.
ГЛеп,
F(u,v)y = — lim — .
Proof. Let У be a section along / defined as follows: If we choose e > 0
sufficiently small, then for each (s, i) G [0, e] χ [0, e] the four curves described
above will be defined. For each such (s, i), let Υ(s, i) be the result of parallel
translation of у along the first two curves c\' and c^ · We leave it to the
reader to argue that Υ is smooth. Notice that у = Υ(0, 0). By Proposition
12.46 we have
F(u,v)y = Ff(dl(0),d2(0))y = VA(0)Vfty - V^V^F
since [<9i, $2] = 0. Since Υ is parallel along the curves c^' , we have Vq2Y — 0
and so
F(u,v)y = -Vd2(0)VdlY.
If we let Pt denote parallel translation along г ь-> /(0,r) from ρ to /(0,£),
then
„ v y lim ΡΓ1 (Vfty) (0,0 - (Vfty) (0,0) 1im ^T1 (Vft(o,t)y)
Vft(0)Vfty = lim -t lim } .
On the other hand, if PSit denotes parallel translation along σ н-> /(σ, t)
from /(0,£) to /(s,i), then
р-/у(д,<)-у(о,0
Vft(o,t)y = bm ·
Putting things together we have
Pt-[p-tlY(s,t) - Pt~lY(0,t) yst-y
F{u,v)y = - lim - д'* V } - ^— = - lim ^—У-. D
Next we will try to understand curvature in terms of the horizontal
distribution. Let us define an object that in some sense measures how far
the horizontal distribution is from being integrable. First note that the
horizontal lifts of local frame fields on the base manifold Μ give vector fields
on the total space that span the distribution. If the bracket of these lifted
fields were always horizontal, then the distribution would be integrable—
the connection would be flat. This suggests the following: For every pair of
horizontal fields Z\, Ζ2 on E, let
C(ZuZ2):=pv([ZuZ2}).
12.5. Curvature
529
Lemma 12.48. Let Γ(Ή) denote the C°°(E)-module of horizontal fields and
let T(V) denote the C°°(E)-module of vertical fields. The map С : Γ(Ή) χ
Γ(Η) —> T(V) given by (Z\, Z2) \-ъ C{Z\,Z<i) is a module homomorphism.
Proof. Let / € C°°{E). Then we have
C(fZu Z2) = pv([/Zb Z2]) = pv(/[Zb Z2] - (Z2/) Z2)
= /pv([Z1,Z2])-pv((Z2/)Z2) = /pv([Z1,Z2])
= /C(ZbZ2).
The analogous result for the second factor follows because С is clearly skew-
symmetric. Additivity is obvious. D
Corollary 12.49. For у G E, the value C(Z\,Z2)(y) depends only on the
values Z\(y) and ^(y). Also, С is well-defined on locally defined horizontal
fields.
Proof. The corollary says that С is "tensorial". The proof is just like the
proof of Theorem 7.32 and can also be derived from Proposition 6.55. D
Because of this corollary, we may think of С as a map Η χ Η —> V.
Theorem 12.50. Let π : Ε —> Μ be a vector bundle with connection. Then
for u, υ G TPM and у G Ep we have
F(u,v)y = -j-1(C(Uy,vy)),
where uy,vy denote the horizontal lifts of и and ν to the point y. Here
jy : TyEp —> Ep is just the canonical map as usual.
Proof. Choose C7, V G X(M) with U(p) = и and V(p) = v. We may assume
that [C/, V] = 0 near p. Let ψ8 and φι be local flows of U and V respectively.
Also, let ψ8 and φ% be the flows of the horizontal lifts U and V. Observe
that since ψ3 ο π = π ο φ3, we have that <ps(y) is the parallel translation of
у along the curve σ \-> φσ ο 7г(у), 0 < σ < s. Similarly, ^(y) is the parallel
translation of у along the curve τ \-^ фт о 7г(у), 0 < τ < t.
Now consider the curve с defined for small t by
Ф) := ф_л о φ_νϊ оф^до ψ^.
Then, by Theorem 12.47, we have
π, ч v У** ~ У v VyTtrf ~ У у c(t) - c(0)
F(u.v)y = - hm —■ = - hm —— = - hm -^ —.
s,i->0 St £->0+ ί £->0+ t
But by Theorem 2.112,
-flim ^ = -^с(О) = -rf[U,V](v).
530
12. Connections and Covariant Derivatives
Finally, notice that [C/, V](p) = 0 so [U ,V](y) is vertical and is equal to
MW,V])(y) = C(U,V)(y) = С(иУ,уУ). D
Unwinding the definitions, the result of the previous theorem can be
written as
(F(U,V)s)(p) = -K([U,V]a(p)),
where s G Г(Е) and к is the connector associated to the connection.
Theorem 12.51. Let π : Ε —> Μ be α vector bundle with connection. If
the curvature F vanishes, then the connection is flat. In particular, given
any у G E, there is a locally defined parallel section s with s(p) = у where
π (у) = р.
Proof. Let Z\,Z*i be locally defined horizontal vector fields. Then for any
у in the common domain of Zi,Z2, let u,v be the images of Z\(y),Z2{y)
under Τπ. Thus Z\(y) and ^(y) are the horizontal lifts of и and v. Then
we have
-3y1(Pv([ZuZ2]y))=F(u,v)y = 0.
But й1 is an isomorphism, and so pv([Zi,Z2]y) = 0, which means that
\Z\,Z*^y is horizontal. Since у was arbitrary, [Ζι, Ζ2] = 0. We conclude that
Η is integrable by the Probenius theorem. Now apply Theorem 12.25. D
12.6. Connections on Tangent Bundles
A connection on the tangent bundle Τ Μ of a smooth manifold Μ is called
a linear connection on Μ. It is traditional to use special notation for the
components of the connection form when using coordinate frames {^r}· In
this case, formula (12.3) becomes
A;
The Γ^· are a special case of the components of the connection forms.
If X = Xi£j and Υ = y^gfj, then
/QYk \ β
(12.7) VXY = I -fi-jXJ + TkijXlY3 I £-£ (summation convention),
which is formula (12.4) in the current context. The functions Γ^ are called
Christoffel symbols.
It is a consequence of Proposition 7.37 that for each X G X(M) there
is a unique tensor derivation Vx on TJ(M) such that Vx commutes with
contraction and coincides with the given covariant derivative on X(M) (also
denoted V*) and with Cx on C°°{M).
12.6. Connections on Tangent Bundles
531
To describe the covariant derivative on tensors more explicitly, consider
ω € 7j°(M). Since we have the contraction У®ол-> С (Υ <g> ω) = ω(Υ), we
should have
νχω(Υ) = VXC(Y ® ω)
= C(Vx(Y®u))
= C{VXY <g> ω + Υ ® νχω)
= ω(νχΓ) + (νχω)(Γ).
So we define (Vxoj)(Y) := Vx(w(y)) - u(VxY). This implies that for a
local frame field E\,..., En with dual frame field Θ1,..., θη we have
к
where ω%» are the connection forms satisfying VxEj = J2^^j(X)Ei. More
generally, if Τ G TJ, then
(12.8)
(VxT)K,...)u,r,y1)...,n)
r
= Χ(Ύ(ωι,... ,ur, Ylt..., Ys)) - Σ Τ(ωι, .Vxujf ...,ur,Yu...)
3=1
S
г=1
Definition 12.52. The covariant differential of a tensor field Τ G Tf is
denoted by VT and is defined to be the element of 7^ given by
VTtV,..., ω1, X, Yu ..., Ys) := VxT(u\ ..., ω\ Yu ..., Ys).
For any fixed frame field £Ί,..., En, we denote the components of VT
by ViTh-ir.
J г Jl-Js
Remark 12.53. We have placed the new variable at the beginning as
suggested by our notation V^T^ -s for the components of VT but in opposition
to the equally common notation T*·1 · .{. This has the advantage of meshing
well with exterior differentiation, making the statement of Theorem 12.56
as simple as possible.
The reader is asked in Problem 2 to show that T(X, Y) := VxY-VyX-
[X, Y] defines a tensor and so T(XP, Yp) is well-defined for Xp, Yp G TpM.
This tensor is called the torsion tensor. If Г vanishes identically, then we
say that the connection is torsion free. We have already noted that the
532
12. Connections and Covariant Derivatives
Levi-Civita connection defined in Chapter 4 for a hypersurface is torsion
free.
We shall have more to say about torsion further on, but for now, notice
that if the connection is torsion free, then
0 = Vdtd3 - Vdjd% = Σ (Γ6 - Г"г) дк,
к
and so the Christoffel symbols are symmetric with respect to the lower two
indices;
yk _ -р/с
However, even for torsion free connections, this is generally not true for
undefined by V#Mej = Σиипек unless the elements t\,..., en of the frame field
have pairwise vanishing Lie brackets. But this amounts to saying that they
are locally coordinate frame fields for some chart.
12.7. Comparing the Differential Operators
On a smooth manifold we have the Lie derivative С χ : TJ(M) —>· TJ(M)
and the exterior derivative d : Qk(M) —» ΩΛ+1(Μ), and in case we have a
torsion free covariant derivative V, that makes three differential operators
which we would like to compare. To this end, we restrict attention to purely
covariant tensor fields Tf(M).
The extended covariant derivative on tensor fields Vx : T^(M) ->
T^(M) respects the subspace consisting of alternating tensors, and so for
each к we have a map
VX : Lkah(M) -> Lkah(M)
and these combine to give a degree preserving map
Vx : Lalt(M) -+ Lait(M).
In other notation,
Vx : Ω(Μ) -> Ω(Μ).
It is also easily seen that not only do we have Vx(a ® β) = V^a ® β + α <g>
Vx/3, but also
Vx(a Λ β) = S/χα Л/5 + аЛ 4χβ.
Recall (12.8) and the similar formula (7.10) for the Lie derivative. If V
is torsion free so that £χΥ{ = [X,Yj\ = V'χΥι — V^X, then we obtain the
following modification of formula (7.10) which incorporates V.
12.7. Comparing the Differential Operators
533
Proposition 12.54. For α torsion free connection, we have the following
equality for any S G T^(M):
+ Y/S(Y1,...,Yi-1,VYiX,Yi+1,...,Ya).
(CxS){Y1,...,Ys) = {VxS)(Y1,...,Ys)
(12.9)
i-
Proof. See Problem 10. D
Corollary 12.55. CxS — S/xS is a tensor.
For ω G Ω/ε(Μ), we have that Vcj is a covariant tensor field but now not
necessarily alternating.2 We search for a way to fix this. By antisymmetriz-
ing, we get a map Qk(M) —>· Qk+l(M) which turns out to be none other
than our old friend the exterior derivative as will be shown below.
Theorem 12.56. If' V is a torsion free covariant derivative on M, then
d=(k + \)k\toV
or in other words, if ω Ε Qk(M), then
к ^
Proof. First we show that
к
(AhoVu)(X0,Xl,...,Xk) = 1^-iY^(-lY(VxiLj)(Xo,...X,...,Xk).
г=0
Consider the subgroup Η consisting of the permutations of {0,1,..., fc}
that fix 0. The cosets of Η are Η = #o, #ъ · · · > Η к where Hj is the set of
permutations that send 0 to j. If we decompose the group of permutations
into these cosets, then
(A\to^)(X0,Xi,...,Xk) = jj^Yy^sgn(a)(VX(rou)(X<Tl,...,Xak)
к
= 7ΓΤΰ\Έ Σ ^η(σ)(4Χσοω)(Χσι,...,Χσ,)
2However, Vu; can be viewed as being in Qk(M) <8>c°° Ω1(Μ).
534
12. Connections and Covariant Derivatives
To finish we calculate as follows:
к ^
г=0
+ 22 (-l)r+Suj([Xr,Xs],Xo,---,Xr,-..,Xs,...,Xk)
l<r<s<k
к
= ^(-1)гХг(о;(Хо,...,^г,...,^))
г=0
+ Σ (-l)r+su(VXrXs - VXsXr , Xo, · · ·, £, · ■ ·, Xs,..., Xk),
l<r<s<k
which is equal to
к ^
г=0
l<T<S<k
- ς (-i)M*o,...,vXsxr,...,x;,...,xfc)
l<r<s<k
к
= Σί-^χΜΧο,. · ·, Χ» · · ·, Xk) (by using 12.8). D
г=0
12.8. Higher Covariant Derivatives
Now let us suppose that we have a connection V^ on every vector bundle
Ei —> Μ in some family {^}ге/ · We then also have the connections V^1*
induced on the duals E* —> M. By demanding a product formula be
satisfied as usual we can form a related family of connections on all bundles
formed from tensor products of the bundles in the family {Ei,E*}iej. In
this situation, it might be convenient to denote any and all of these
connections by a single symbol as long as the context makes confusion unlikely.
In particular, we have the following common situation: By the definition of
a connection we have that X н-> V^a is C°°(M) linear and so Vеσ is a
section of the bundle T*M ® E. We can use the Levi-Civita connection or
any torsion free connection V on Μ together with V^ to define a connection
on Ε ® T*M. To get a clear picture of this connection, we first notice that
a section ξ of the bundle E®T*M can be written locally in terms of a local
frame field {Θ1} on T*M and a local frame field {ei} on E. Namely, we may
write ξ = Σφ;®^. Then the connection νβ®τ*Μ on Ε®Γ*Μ is defined
so that a product rule holds. Let η„ and аЛ be the connection forms for V
12.8. Higher Covariant Derivatives
535
and V^ respectively, so that locally, using the summation convention, we
have
Vx»T*M£ = Vf И*) ® 0" + C^i ® Vx^
= (*#* + CVf e<) ® ^ + ije< ® Vx^
= (X£er + его;Г(Х)С) ® 0" - τ£(Χ)ξ£βΓ ® 0"
= (X£ + о;Г(Х)С - 7?(*)ί£) er ® Θ».
Now let ξ = VEa for a given σ e T(E). The map X ^ Vf^T*M(V^a) is
C°°(M) linear, and VE®T*M(VEa) is an element of Γ (Ε ® Г*М ® Г*М),
which can again be given the obvious connection. The process continues,
and denoting all the connections by the same symbol we may consider the
Jfc-th covariant derivative Vka <E T(E ® T*M®k) for each σ eT(E).
It is sometimes convenient to change viewpoints just slightly and define
the covariant derivative operators Vxbx2}...}xfc : T(E) —> T(E). The
definition is given inductively as
Vxbx2a :=Vxx Vx2a - ννΧιχ2σ>
Vxlv..,xfca :=Vxx (Vx2,x3,...,xfca)
- ννΧιχ2,χ3,..·Λσ νχ2ιχ3ι...,νΧιχΛσ.
Then we have the following convenient formula, which is true by definition:
Warning: V^V^r is a section of £", but it is not the same section as
Vdidjτ since in general
VftV^r φ VdiVdja - Vvd.dja.
Now we have that
Vxbx2a - Vx2,x1a = Vxx\7χ2σ - ννχιχ2σ ~ (V^V^a - VyX2x^J
= VXl\7χ2σ - Vx2VXla - V(Vxix2-vX2Xi)^
= F(Xi, Χ2)σ - ντ(χ1?χ2)σ.
So, if V (the connection on the base M) is torsion free, then we recover
the curvature
Vxbx2a - Vx2,Xla = F(Xi, Χ2)σ.
One thing that is quite important to realize, is that F depends only on the
connection on E, while the operators Vxbx2 involve a torsion free
connection on the tangent bundle TM.
536
12. Connections and Covariant Derivatives
12.9. Exterior Covariant Derivative
The exterior covariant derivative essentially antisymmetrizes the higher co-
variant derivatives just defined in such a way that the dependence on the
auxiliary torsion free linear connection on the base cancels out. Of course
this means that there must be a definition that does not involve this
connection on the base at all. We give the definitions below, but first we point out
a little more algebraic structure. We can give the space of £-valued forms
Ω(Μ,Ε) the structure of a (il(M),il(M))-ubi-modu\en. This means that
we define a product Λ : Ω(Μ) χ Ω(Μ, Ε) ->> Ω(Μ, Ε) and another product
Λ : Ω(Μ, Ε) χ Ω(Μ) -> Ω(Μ, Ε) which are compatible in the sense that
(α Λ ω) Λ β = α Λ (ω Λ β)
for ω Ε Ω(Μ, Ε) and α Ε Ω(Μ). These products are defined by extending
linearly the rules
α Λ (σ ® ω) := σ ® α Λ ω for σ Ε Γ(2?) and α, ω <Ε Ω(Μ),
(σ ® α;) Λ α := σ ® α; Λ α for σ Ε Γ(2?) and α, ω Ε Ω(Μ),
αΛσ = σΛα = σ®αίθΓαΕ Ω(Μ) and σ Ε Γ(2£).
More precisely, we extend by using the universal properties of multilinear
products. It follows that
α Λ ω = (-1)к1ш Λ α for α Ε Ω*(Μ) and ω Ε Ω*(Μ, Ε).
In the situation where σ Ε Γ(Ε) := Ω°(Μ, J5) and α; Ε Ω(Μ), we have all
three of the following conventional equalities:
σω = σΛω = σ<8>ω (special case).
The proof of the following theorem is analogous to the proof of the existence
of the exterior derivative.
Theorem 12.57. Given α connection V on α vector bundle π : Ε —> Μ,
there exists α unique operator dv : Ω(Μ,Ε) —> Ω(Μ,Ε) such that
(i) (F(tok(M,E)) ci1w(M,£);
(ii) For α Ε Ω*(Μ) and ω Ε Ω*(Μ, Ε) we have
dv(a Λ ω) = da Λ ω + (-1)ка Л dvcj,
άν(ω Λ α) = dvcj Λ α + (-1)*ω Λ da;
(Hi) dva = Va/oraEr(£;).
/η particular, if α Ε Ω* (Μ) and σ Ε Ω°(Μ, Ε) = Γ (Ε), we have
d7(a ® a) = dva Λ α + σ ® da.
12.9. Exterior Covariant Derivative
537
It can be shown that if we use Qk(M;E) ^ L^t(X(M),T(E)), then we
have the following formula:
dvu(X0,...,Xk)= J] (-lYVxMxo,---X,---,Xk))
0<i<k
+ Σ (-1)*'ш([ХиХа],Х0,...,Х{,...,х),...,Хк).
0<i<j<k
Definition 12.58. The operator dv whose existence is given by the previous
theorem is called the exterior covariant derivative.
Exercise 12.59. We shall sometimes use the notation dE rather than dv.
This is especially useful when two possibly unrelated connections on possibly
different bundles are involved in the discussion. For example, we may deal
with a connection V™ on Τ Μ and at the same time deal with a connection
V^ on Ε —> Μ. Then it is conceivable that we may have to work with both
dE and d™. Failure to notice these notational possibilities can result in
serious confusion.
Note that we have the following special case for μ G Ω1(Μ, Ε):
άνμ{Χ, Υ) = Vx (μ(Υ)) - Vy (μ(Χ)) ~ μ([Χ, Υ])
= <Ρ(μ(Υ))(Χ)-<Ρ(μ(Χ))(Υ)-μ([Χ,Υ]).
Example 12.60. Let us consider the case where Ε = Τ Μ with connection
V™. Bundle maps Τ Μ ->> Τ Μ may be regarded as elements of Ω1 (Μ; ΤΜ)
(think about this). In particular, the identity map id^M^) = v can be
viewed as a TM-valued 1-form. If we take the exterior covariant differential
of θ := idTM, we obtain an element d™0 of Ω2(Μ; Τ Μ). For Χ, Υ e X(M),
we compute
ά™θ(Χ, Υ) = d™ (Θ(Υ)) (Χ) - d™ (Θ(Χ)) (Υ) - Θ([Χ, Υ})
= d™ (Υ) (Χ) - d™ (Χ) (Υ) - [Χ, Υ]
= VXY -VYX -[Χ,Υ]=Τ(Χ,Υ).
So we see that d™9 gives the torsion of the connection V™.
is a torsion free covariant derivative on Μ and Vе is a connection
on the vector bundle Ε —>· Μ, then as before we get a covariant derivative
V on all the bundles Ε <g> /\kT*Μ and as for the ordinary exterior derivative
we have the formula
dv = dE = (fc + l)AltoV,
538
12. Connections and Covariant Derivatives
or in other words, if ω G Ω^(Μ, 22), then
к
г=0
Let us look at the local expressions with respect to a moving frame. Let
{ei} be a frame field for Ε on U С Μ. Then locally, for ω e ftk(M,E), we
may write
where ω1 Ε Ω/ε(Μ). Using the summation convention, we have
dvu = dv {ei ® о;*) = ег ® do;* + dve; Λ α/
= ej ® do;·7 + ujej А α/
= ej ® do;·7 + ej ®ω\ Αω1
= ej ® ido;·7 + ω? Л α/J .
Thus the "(fc + l)-form coefficients" of άνω with respect to the frame {ej}
are given by duji + ω\ Α ω1.
We can extend our wedge operation to a bilinear operation between
Ω(Μ, End(JS)) and Ω(Μ, Ε) in such a way that
(A ® α) Λ (σ ® /3) = Α(σ) ® α Λ /3.
To understand what is going on a little better, let us consider how the action
of Ω(Μ, End(JS)) on Ω(Μ, Ε) comes about from a slightly different point of
view. We can identify Ω(Μ, End(JS)) with Ω(Μ, E<g>E*), and then the action
of Ω(Μ,Ε ® 2?*) on Ω(Μ, 2?) is given by tensoring and then contracting:
For α,/З G Ω(Μ), 5,σ G Γ(£), and 5* <G Г(£), we have
(5 ® 5* ® α) Α (σ ® /3) := C(s ® 5* ® σ ® (α Λ /3))
= 5*(σ)5® {α Α β).
Prom Vе we get related connections on 2?*, Ε ® £*and Ε <g> E* <g> E. The
connection on Ε (g> E* is also a connection on End(JS). Of course, we have
vEnd(K)«£; (L Θ σ) = (vEnd(£;)L) ^ σ + L ^ νΕσ)
and after contraction,
Vf (ВД) = Vf C(L ® σ) = C(V?ld(B)®BL ® σ + L ® Vf σ)
= (V*nd(£)L)(a) + L(Vf σ) = (Vjnd(£)L)(a) + L(Vf σ).
But X ►-» L(Vf σ) is just L Λ V*V. So we have
VE(L(a)) = (VEnd(^L)(a) + L Λ V£a.
12.9. Exterior Covariant Derivative
539
The connections on End(E) and End(E)<g>E give the corresponding exterior
covariant derivative operators
dEnd(E) . i)*(M,End(£)) -> nk+1(M,End(E))
and
dEnd(E)®E . ii*(M,End(£) ® Д) -> ΩΛ+1(Μ,Εηά(£;) ® Я).
It must be expected that many readers will find this formalism a bit
daunting but it is not really as bad as it seems. Local calculations turn
out to be quite natural as we shall see below. We encourage the reader to
seek as much exposure as possible. To this end, we recommend [BaMu],
[Poor], and [Dar]. Also, part of what might be intimidating is really just
the bulkiness of the notations. The following notational convention makes
things more palatable:
Notation 12.61. Let us now agree that, whenever convenient, we may
write simply V for vE*,VEnd(E\ VEnd(E)®E, etc. and dv for what we have
called dE\ dEndW etc.
Proposition 12.62. For Φ e Ω*(Μ, End(E)) and ω e tok(M,E), we have
άν(Φ Α ω) = άνΦ Α ω + (-1)*Φ Λ άνω.
Proof. We have
dE((A ® α) Α (σ ® β)) := άΕ(Α(σ) ® (α Α β))
= VE(A(a)) Α (α Α β) + Α(σ) ® d (α Λ /3)
= (-l)k (α Λ νΕ(Α(σ)) Λ β) + Α(σ) ® d (α Λ /3)
= (-l)k (α Λ {VEnd(E)Α Λ σ + Α Λ Vеσ) Λ /з)
+ Α(σ) ® da Λ /3 + (-Ι^^σ) ® α Λ d/3
= (VEnd(E)A) Λ α Λ σ Λ /3 + (-l)kA Λ α Λ (νΕσ) Λ β
+ Α(σ) ®άα/\β + (-Ι^^σ) ®α/\άβ
= (VEnd(EU Λ α + A ® da) Λ (σ ® β)
+ (-l)k (Α ® α) Λ (νΕσ Λ /3 + σ ® άβ)
= dEnd{E)(A ® α) Λ (σ ® /3) + (-l)k (A® a) A dE (σ ® β).
By linearity we conclude that for Φ e ftk(M, End(E)) and ω e Ω(Μ, Ε) we
have
<^(ф л ω) = άΕηά{Ε)Φ Αω + (-1)*Φ Λ Α.
So in light of our notational conventions we are done. D
540 12. Connections and Covariant Derivatives
Remark 12.63. In the literature, it seems that the different natures of
(dv) and (V) are not always appreciated. For example, the higher
derivatives given by (dv) are not appropriate for defining k-th order Sobolev
spaces since (dv) is zero for any flat connection.
12.10. Curvature Again
The space Ω(Μ, End(JS)) is an algebra over C°°(M) where the multiplication
is according to (L\ ® ωχ) Λ (L^ ® ω^) = (L\ о L2) ® ωχ Λ ω^. Now Ω(Μ, Ε)
is a module over this algebra because we can multiply (using the symbol Λ)
as follows:
(L ® α) Λ (σ ® β) = La <g> (α Λ β).
As usual, this definition on simple elements is sufficient. If Χ,Υ Ε Χ (Μ)
and Φ Ε Ω2(Μ, End(E)), then using Ω2(Μ; Ε) 9* L2lt(X(M), Г(Я)) we have
(ΦΛσ)(Ι,7) = Φ(Ι,7)σ
for any σ Ε Ω(Μ,Ε).
Proposition 12.64. The map dv о dv : Ω*(Μ, J5) -» Ω*+2(Μ, Я) is given
by the action of F, the curvature 2-form ofVE7
<F ο άνμ = F Λ μ for μ Ε Ω* (Μ, Ε).
Proof. Let us check the case where к = 0 first. Prom formula (12.10) above
we have for σ Ε Ω°(Μ, Ε),
(dv ο άνσ) (Χ,Υ) = VX (dva(Y)) - Vy (dva(X)) - σ([Χ,Υ\)
= VxVya -VxVya - σ([Χ,Υ})
= F(X,Y)a = (FAa)(X,Y).
More generally, we just check d? о d? on elements of the form μ = σ ® θ:
d? οάνμ = άνοάν{σ®θ)
= dv {dvaΛθ + σ®άθ)
= (dvdva) Αθ-άνσΑάθ + άνσΑάθ + 0
= (F Λ σ) Λ θ = F Λ (σ Λ θ) = F Λ (σ ® 6·)
= F Λ μ. D
Let us take a look at how curvature appears in a local frame field. As
before restrict to an open set U on which ец = (e\,..., er) is a given local
frame field and then write a typical element s € £lk(M, E) as s = εη, where
12.11. The Bianchi Identity
541
η = (771,..., η7*)1 is a column vector of smooth /c-forms. With шц = {ω1·) the
connection forms we have
dvdV5 = dvdv (euVu) = dv {^υάηυ + dveu Α ηυ)
= dv (eu (άηυ + ων Α ην))
= dveu Α (άηυ + ^υ Λ т/с;) + ее/ Λ άν (άηυ + ων Λ τ^/)
= ec/CJc/ Λ (άηυ + ωυ Λ 77^/) + β[/ Λ dv (dijc/ + ^f/ Λ τ#/)
= ευωυ Λ (rfr7c/ + ^с/ л Vu) + ее/ Л deje/ Αηυ - еи Αωυ Α άηυ
= βυάωυ Αηυ + е-и^и Αωυ Αηυ
= еи (άωυ +ων Α ωυ) Α ηυ-
The matrix άωυ+ωυΑωυ represents a section of End(2?)t/® Д T*U. In fact,
we will now check that these local sections paste together to give a global
section of End(£) <g> /\2T*M, or in other words, an element of Ω2(Μ, End(£)),
which is clearly the curvature form: F : (X,Y) *-> F(X,Y) e T(End(E)).
Let Fu = άωυ +ωυ Αωυ and let Fy = άωγ + ω γ Α ωγ be the corresponding
form for a different moving frame eu := ey<7, where g : U Π V —> GL(Fr),
and r is the rank of E. What we need to verify is the transformation law
Fv = g-lFug,
which we met earlier in equation (6.2). Recall that ωγ = g~1ωug + g~ldg.
Using d (g~l) = —g~ldgg~l, we have
Fy = άωγ + ωγ Α ωγ
= ά(g~lωug + g~lάg)
+ (g~lωug + g~ldg) A (g~lωug + g~ldg)
= d (g'1) A ωug + g~Xdwvg - g~luJu Adg + d(g~l) A ag
+ g~lωu A ωug + g~ldgg~l A ωug
+ g~lωu Aag + g~ldg A g~ldg
= g~lάωug + s~ W А ωug = g~lFvg,
where we have used that g~lag A g~lag = a(g~1g) = 0.
12.11. The Bianchi Identity
In this section we give several versions of the so-called Bianchi identity for
a connection V on a vector bundle Ε —> Μ. Perhaps the simplest version
to understand is the following: If /7, V, W G X(M), then
[Vf/, [Vv, Vw]] + [Vv, [Vw, Vt/]] + [Vw, [Vt/, Vv]] = 0 (Bianchi identity),
542
12. Connections and Covariant Derivatives
where [Vc/, Vy] := Vc/ о Vy - Vy о V[/. This identity follows trivially
once we observe that the set of linear operators on any vector space is a Lie
algebra under the commutator bracket operation [A,B] := AoB — BoA. So,
in this form, the Bianchi identity is just an instance of the Jacobi identity.
Let (E/,x) be a chart on M, and let Εμν : T(E)\U -» T(E)\U be the local
curvature operator defined by
^=[VeM>Vel,]=F(a/1,av)>
where δμ = δ/δχμ, etc. Then we have the following version of the Bianchi
identity:
[V^, F„x] + [Vdu, FXfi] + [V^, F^} = 0 (Bianchi identity).
Another revealing form of the Bianchi identity depends on our discussions
in the last section and in particular on Proposition 12.62. We have, for any
unvalued form 77,
{dV)3 η = dV ({dV)2vj = dV(F A^ = dVFf\V + FA άνη.
But equally,
(dvYV=(dv)2(dvV)=FAdvv
so it must be that d^F Λ η = 0 for any η and so we obtain
dvF = 0 (Bianchi identity).
Exercise 12.65. Use a calculation in local coordinates and with respect
to a local frame ei,..., e& to show that the above versions of the Bianchi
identity are equivalent.
12.12. G-Connections
If the vector bundle π : Ε —> Μ has a G-bundle structure, then there
should be certain connections that respect this structure. These are the
G-connections. It is time to confess that we have come face to face with a
weakness of our approach to connections. It is much easier and more natural
to define the notion of G-connection if one first defines connections in terms
of horizontal distributions on principal bundles. We treat this in the online
supplement [Lee, Jeff]. However, we can still say what a G-connection
should be from the current point of view without too much trouble.
Recall that if π : Ε —> Μ is a rank к vector bundle with typical fiber V
that has a G-bundle structure where G acts on V by the standard action as
a subgroup of GL(V), then we have the bundle of G-frames
FG(E):={JpeMFG(Ep),
where FG(EP) := {u <E GL(V,£P) : и = φ~ι{ρ,-) for some (/7,0) <E Ag}
and Ag is the maximal G-atlas defining the structure. The elements of
12.12. G'-Connections
543
Fg{Ep) are called G-frames. Having fixed a basis (ei,... , e/~) for V, each
element of Fq(Ep) can be identified as a basis (г/i,..., щ) for Ep according to
и : χ ь-> Σ x%ui' If F(Ep) ls the set of all frames at p, then Fg{Ep) С F(EP)
and Fg{E) is a subbundle of F(E). Now let с : [α, b] —> Μ be a smooth
curve. If (i6i,..., щ) is a frame, then (Pci6i,..., РсЗД) is also a frame. Thus
we get a map Pc : F(Ec(a)) —> F (J5c(a)). The following is then a workable
definition of G-connection:
Definition 12.66. Let π : Ε —> Μ be a vector bundle with a G-bundle
structure as above. A connection on the bundle is called a G-connection
if parallel transport Pc takes G-frames to G-frames for all piecewise smooth
curves c.
A G-frame field is a frame field which is a G-frame at each point of its
domain.
Exercise 12.67. Show that if V is the covariant derivative associated to a
G-connection on J5, then for each G-frame field the associated connection
forms take values in the Lie algebra of G thought of as a matrix subgroup
of GL(n,F).
Let π : Ε —> Μ be a vector bundle with metric h = (·, ·) and standard
fiber Rk. The metric h gives a reduction of the structure group to 0(n). In
this case, an 0(n)-frame is an orthonormal frame. A connection on E is an
0(n)-connection if and only if the associated covariant derivative V satisfies
(12.11) υ (si,s2) = (Vv5b 52> + (si, Vvs2)
for all 5i, 52 e T{E) and all υ e TM.
Exercise 12.68. Prove this last assertion.
One may also consider metrics which are nondegenerate but not
necessarily positive definite. In this case, the structure group reduces to one of
the semiorthogonal groups 0(k,n — k).
Definition 12.69. A covariant derivative satisfying (12.11) above for some
metric h on a vector bundle Ε —> Μ is called a metric covariant
derivative (or metric connection).
A simple partition of unity argument shows that if h is a given metric,
then there exists a (nonunique) metric connection for h (Problem 11). For a
metric connection, parallel transport is an isometry (Problem 12).
Furthermore if h is the metric and V is metric with respect to /ι, then it is easy to
check that V/ι = 0.
544
12. Connections and Covariant Derivatives
Proposition 12.70. Suppose that h = (·,·) is a metric on a vector bundle
Ε -ϊ Μ. If V is a metric covariant derivative, then the corresponding
curvature satisfies
(F(X, 1>ι, σ2) + (σι, F(X, Υ)σ2) = 0
for all Χ,Υ e X(M) and all σλ,σ2 e T(E).
Proof. Let X and Υ be fixed vector fields. Without loss of generality we
may assume that [X, Y] = 0 (recall Exercise 7.34). It is also enough to show
that (F(X, Υ)σ, σ) = 0 for all σ. We have
(F(X,Y)a,a) = (VxVya,a) - <σ, VxVya)
= Χ{νγσ,σ)-{νγσ,νχσ)
-Υ{νΧσ,σ)-{νχσ,νγσ)
= ^(ΧΥ(σ,σ)-ΥΧ(σ,σ)) = 0
since [Χ, Υ] = 0. D
We shall study metric connections on tangent bundles in the next chap-
Problems
(1) Prove Theorem 12.23.
(2) Let Μ have a linear connection V and let T(X, Y) := VXY - VYX.
Show that Τ is C°°(M)-bilinear (tensorial). The resulting tensor is
called the torsion tensor for the connection.
(3) Show that a holonomy group is indeed a group. Show that the holonomy
at any point of a sphere is isomorphic to the special orthogonal group
SO(2).
Second order differential equations and sprays
(4) A second order differential equation on a smooth manifold Μ is a vector
field on ΓΜ, that is, a section X of the bundle TTM (second tangent
bundle) such that every integral curve a of X is the velocity curve of
its projection on M. In other words, a = 7 where 7 := птм ° ос. А
solution curve 7 : I —> Μ for a second order differential equation X is,
by definition, a curve with ά(ί) = X(a(t)) for all τ Ε I.
Problems
545
In the case Μ = Mn, show that this concept corresponds to the usual
system of equations of the form
v' = f(y,v),
which is the reduction to a first order system of the second order
equation y" = f(y,yf). What is the vector field on TRn = RnxRn which
corresponds to this system?
Notation: For a second order differential equation X, the maximal
integral curve through υ G Τ Μ will be denoted by av and its projection
will be denoted by ην := птм ° а-
(5) A spray on Μ is a second order differential equation, that is, a section
X of TTM as in the previous problem, such that for ν e Τ Μ and sgR,
a number t G Ш belongs to the domain of η8ν if and only if st belongs to
the domain of 7V, and in this case
7sv(t) = 7v(s*)·
Show that there are infinitely many sprays on any smooth manifold M.
[Hint: (i) Show that a vector field X e X(TM) is a spray if and only
if Τπ ο Χ = ιάτΜ, where π = птм'-
TTM
TM ——>TM
ιάτΜ
(ii) Show that X G X(TM) is a spray if and only if for any sGR and
any υ e TM,
Xsv = Τμ8(3Χυ),
where μ8 : υ ι-» sv is the multiplication map.
(iii) Show the existence of a spray on an open ball in a Euclidean
space.
(iv) Show that if X\ and X2 both satisfy one of the two
characterizations of a spray above, then so does any convex combination of X\
and X2]
(6) Show that if one has a linear connection on M, then there is a spray
whose solutions are the geodesies of the connection.
(7) Show that given a spray on a manifold there is a (not unique) linear
connection V on the manifold such that 7 : I —> Μ is a solution curve
of the spray if and only if V#t7 = 0 for all t e I. Note: 7 is called
a geodesic for the linear connection or the spray. Does the stipulation
that the connection be torsion free force uniqueness?
546
12. Connections and Covariant Derivatives
(8) Let X be a spray on M. Equivalently, we may start with a connection
on Μ (i.e. a connection on TM) which induces a spray. Show that the
set Οχ := {υ Ε TM : 7^(1) is defined} is an open neighborhood of the
zero section of TM.
(9) Finish the proof of Theorem 12.32.
(10) Prove formula (12.9).
(11) Let h be a metric on a vector bundle Ε -* M. Show that there exists a
metric connection for h. [Hint: Use orthonormal frames defined on each
open set of a locally finite cover.]
(12) Show that parallel transport along curves with respect to a metric
connection in a vector bundle with metric is an isometry of scalar product
spaces.
Chapter 13
Riemannian and
Semi-Riemannian
Geometry
"The most beautiful thing we can experience is the mysterious.
It is the source of all true art and science."
- Albert Einstein
In this chapter we take up the subject of semi-Riemannian geometry,
which includes Riemannian geometry and Lorentz geometry as important
special cases. The exposition is inspired by [ONI], which we follow quite
closely in some places (also see [LI]). Recall that by definition, a semi-
Riemannian manifold (M, g) has a well-defined index denoted ind(M) or
ind(p). In the case of an indefinite metric (ind(M) > 0), we will need a
classification:
Definition 13.1. Let (V, (·, ·)) be a scalar product space. A nonzero vector
ν e V is called
(1) spacelike if (г;, г;) > 0;
(2) lightlike or null if (υ, υ) = 0;
(3) timelike if (ν, ν) < 0;
(4) nonnull if ν is either timelike or spacelike.
The terms spacelike, null (lightlike), and timelike indicate the causal
character of a vector. The word causal comes from relativity theory and is most
apropos in the context of Lorentz manifolds defined below. If (M,g) is a
547
548
13. Riemannian and Semi-Riemannian Geometry
LIGHTLIKE TIMELIKE
SPACELIKE
Figure 13.1. Lightcone
semi-Riemannian manifold, then each tangent space is a scalar product space
and the above definition applies. Recall that we define \\v\\ = \(v,v)\ ,
which we call the length of v.
Note: We have so far left the causal character of the zero vector
undefined. It may seem reasonable that it should be considered null. A second
possibility is that the zero vector should have all three causal characters.
Actually, we shall see that if the index of the scalar product is one, then it
is convenient to consider the zero vector as being spacelike.
Definition 13.2. The set of all null vectors in a scalar product space is
called the nullcone or lightcone. If (M, g) is a semi-Riemannian manifold,
then the nullcone in TpM is called the nullcone at p.
Definition 13.3. Let I С R be some interval. A curve с : / —> (M,g) is
called spacelike, null, timelike, or nonnull, according as c(t) G TC^M is
spacelike, null, timelike, or nonnull, respectively, for all t G J.
While every smooth manifold supports Riemannian metrics by
Proposition 6.45, the existence of an indefinite metric on a given smooth manifold
has an obstruction:
Theorem 13.4. A compact smooth manifold admits a continuous C°
indefinite metric of index к if and only if its tangent bundle has a C° rank к
subbundle.
This result is Theorem 40.11 of [St].
Definition 13.5. Let (M,g) be semi-Riemannian. If с : [α, b] —> Μ is a
piecewise smooth curve, then
rb
Lc{a)Ab){c)= / |(c(t),c(t)>|1/2rft
J a
is called the arc length or simply length of the curve.
13. Riemannian and Semi-Riemannian Geometry
549
The word length could cause confusion since the length of a null curve is
zero. Thus for indefinite metrics, arc length can have some properties that
are decidedly not like our ordinary notion of length. In particular, a curve
may connect two different points and the arc length might still be zero! The
word length is therefore sometimes reserved for timelike or spacelike curves.
Definition 13.6. A positive reparametrization of a smooth curve с :
[α, b] —> Μ is a curve defined by composition с о h : [α;, У] —> Μ, where
h : [α;, У] —> [α, b] is a smooth monotonically increasing bijection. Similarly,
a negative reparametrization is given by composition with a smooth
monotonically decreasing bijection h : [a', У] —> [a, b]. By a
reparametrization we shall mean either a positive or negative reparametrization.
The above definition can be extended to piecewise smooth curves.
Suppose с : [α, b] —> Μ is a continuous curve such that, for some partition
α = to < t\ < · · · < tk = b, we have that с is smooth on each [ii-i,ii]·
A positive reparametrization of с is a curve с о h : [α;, У] —> Μ, where
h : [а',У] —> [a,b] is a monotonically increasing continuous bijection that is
smooth on each interval h~l ([U-i,ti]). Negative reparametrization is
defined similarly.
Remark 13.7 (Important fact). The integrals above are well-defined since
c(t) is defined and continuous except for a finite number of points in [a, b].
Also, it is important to notice that by standard change of variable arguments,
a reparametrization 7 = с о h does not change the arc length of the curve:
/ |(έ(ί),έ(ί)>|1/2Λ= / \(^u)Mu))\1/2du.
Thus the arc length of a piecewise smooth curve is a geometric property of
the curve; i.e. a semi-Riemannian invariant.
Definition 13.8. Let (M,g) be semi-Riemannian. Let I be an interval
(possibly infinite). If с : / —> Μ is a smooth curve with ||c|| = 1, then we
say that с is a unit speed curve.
If с : I —> Μ is a curve such that ||c|| is never zero, then choosing a
reference to G /, we may define an arc length function ί : I —> Г С Ш by
(13.1) £(t):= f \(c{t),c(t))\1'2 dt.
For a finite interval of definition [a, b], the reference to is most often taken
to be the left endpoint a. Since di/dt = |(c(t), c(t))^'2 > 0, we may invert
to find l~l and then reparametrize:
7(5) :=с(Г\в)).
550
13. Riemannian and Semi-Riemannian Geometry
It is easy to see from the chain rule that the resulting curve is a unit speed
curve. Conversely, if 7 is a unit speed curve, then the arc length from 7(51)
to 7(52) is 52-51. Often one abuses notation by writing s = £(t) and then
ds/dt instead of di/dt. The use of the letter s for the parameter of a unit
speed curve is traditional, and we say that the curve is parametrized by-
arc length. In the case of timelike curves, people sometimes use the letter
r instead of s and refer to it as a proper time parameter.
13.1. Levi-Civita Connection
In this chapter we will use the term "connection" to be synonymous with
covariant derivative. Let (M,g) be a semi-Riemannian manifold and V a
metric connection for Μ (see Definition 12.11). By definition, we have
X(Y,Z) = (VxY,Z) + (Y,VxZ)
for all X, У, Ζ G X(M). It is easy to show that the same formula holds for
locally defined fields. Recall that the operator Γ : X(M) χ X(M) -» X(M)
defined by T(X, Y) = VXY - VYX - [X, Y] is a tensor called the torsion
tensor of V. Prom the previous chapter we know that (X, Y) ь-> T(X,Y)
defines a C°°(M)-bilinear map X(M) χ Χ(Μ) -» X(M). The isomorphism
(7.6) implies that Γ gives a section of T%(TM;TM). That is, Γ can be
thought of as defining a TpM-valued 2-tensor field at each p, so if Xp, Yp G
TpM, then Τ (Χρ,Υρ) is a well-defined element of TpM. Recall from our
study of tensor fields that Τ (XP,YP) is defined to be T(X,Y)(p) for any
fields Χ, Υ such that X{p) = Xp and Y{p) = Yp.
Requiring that a connection be both metric and torsion free, pins down
the metric completely.
Theorem 13.9. For α given semi-Riemannian manifold (M,g), there is a
unique metric connection V such that its torsion is zero, Τ = 0. This unique
connection is called the Levi-Civita connection for (M,g).
Proof. We will derive a formula that must be satisfied by V and that can
be used to actually define V. Let X, У, Z, W be arbitrary vector fields on
M. If V exists as stated, then we must have
X(Y,Z) = (VXY,Z) + (Y,VXZ),
Y(Z,X) = (VYZ,X) + (Z,VYX),
Z(X,Y) = (VzX,Y) + {-- Y).
13.1. Levi-Civita Connection
551
Now add the first two equations and subtract the third to get
X(Y,Z) + Y(Z,X)-Z(X,Y)
= (VXY, Z) + (Y, VXZ) + (VYZ, X) + (Z, VyX)
-(VZX,Y)-{X,VZY).
If we assume the torsion zero hypothesis, then this reduces to
X(Y,Z) + Y(Z,X) - Z{X,Y)
= {Y,[X,Z]) + (X,[Y,Z})
-(Z,[X,Y])+2(VXY,Z).
Solving, we see that V χΥ must satisfy
2<V*y, Z) = X(Y, Z) + Y(Z, X) - Z(X, Y)
( ' ' + (Z,[X,Y}) - (Y,[X,Z}) - (X,[Y,Z]).
Since knowing (V^y, Z) for all Ζ is tantamount to knowing V^y, we
conclude that if V exists, then it is unique. On the other hand, the patient
reader can check that if we actually define (V^y, Z) and hence VχΥ by
this equation, then all of the defining properties of a covariant derivative are
satisfied and furthermore Τ will be zero. D
Formula (13.2) above, which serves to determine the Levi-Civita
connection, is called the Koszul formula. It is easy to see that the restriction of a
Levi-Civita connection to any open submanifold is just the Levi-Civita
connection on that open submanifold with the induced metric. It is a
straightforward matter to show that the Christ off el symbols for the Levi-Civita
connection in some chart are given by
Tk = 1 ki (^9ji ддн _ dgij\
ij Τ \дх{ dxi dxl J '
where gjk9kl = #!·· (Recall that gij = (di,dj) and Г^· are given by formula
(12.6).)
We know from the study in the last chapter that we may take the co-
variant derivative of vector fields along maps. The most important cases
for this chapter are fields along curves and fields along maps of the form
h : (a,b) χ (c, d) -» M.
Exercise 13.10. Show that if а : / —> Μ is a smooth curve and Χ, Υ are
vector fields along a, then £t{X,Y) = (Vd/dtX,Y) + (X,Vd/dtY).
Exercise 13.11. If h : (a, b) χ (с, d) —> Μ is smooth, then dh/dt and dh/ds
are vector fields along h. Show that VQ/Qtdh/ds = ^d/ds9h/dt. [Hint: Use
local coordinates and the fact that V is torsion free.]
552
13. Riemannian and Semi-Riemannian Geometry
13.1.1. Covariant differentiation of tensor fields. Let V be any
natural covariant derivative on M. It is a consequence of Proposition 7.37 that
for each X Ε X(U) there is a unique tensor derivation Vx on 7J(/7) such
that Vx commutes with contraction and coincides with the given covariant
derivative on X(U) (also denoted Vx) and with Cxf on C°°(U).
Recall that if Τ Ε Τ}, then
s
(VxT)(Y1,...,Ys) = Vx(r(Yl,...,Ys))-Y/r(...,VxYi,...).
i=l
If Ζ Ε Tq, we apply this to VZ Ε Τχ and get
(VXVZ)(Y) = X(VZ(Y)) - VZ(VXY) = Vx(VyZ) - VVxYZ,
from which we get the following definition:
Definition 13.12. The second covariant derivative of a vector field
Ζ Ε 7? is
V2Z : (Х,У) н> V^?y(Z) = Vx(VyZ) - VVxy^
Definition 13.13. A tensor field Τ is said to be parallel if V^T = 0 for
all ξ Ε ΤΜ. Similarly, if σ : 7 -» Τζ(Μ) is a tensor field along a curve
c: Ι -ϊ Μ that satisfies Vat^ = 0 on 7, then we say that σ is parallel along
c. Just as in the case of a general connection on a vector bundle we then
have a parallel transport map P(c)\0 : Τ£(Μ)φο) -» Trs(M)c{t).
Prom the previous chapter we know that
It is also true that if Τ Ε TJ, and if cx is the curve t ь-> ^ (p), then
VxT(p) = lim ^L(b^(p))-b,f(ri
It is easy to see that the space of parallel tensor fields of type (r, 5) is a
vector space over R.
Exercise 13.14. Show that if Τ is parallel, then for any smooth curve
с : [a,b] —> Μ such that c(a) = ρ and c(b) = q we have P(c)baTp = Tq.
Deduce that if Μ is connected, then the dimension of the space of parallel
tensor fields of type (r, s) has dimension less than or equal to dim 7J(M)p
for any fixed p.
The map Vx : TJM —> TrsM just defined commutes with contraction
by construction. Furthermore, if the connection we are extending is the
Levi-Civita connection for a semi-Riemannian manifold (M, #), then
νξ# = 0 for all ξ Ε ΤΜ.
13.2. Riemann Curvature Tensor
553
To see this, recall that
νξ(# ® У ® W) = V^ ® X ® Υ + g <g> V^X ® У + 5 ® X ® У€У,
which upon contraction yields
vc(ff(x, у)) = (νξίθ(χ, r) + <?(νξχ, у) + ff(x, νξη
ί (X, У) = (VtfXX, У) + (VCX, У) + (X, Vcy>.
We see that V^ = 0 for all ξ if and only if (X, У) = (νξΧ, У) + (Χ, νξΥ) for
all ξ, Χ, У. In other words, the statement that the metric tensor is parallel
(constant) with respect to V is the same as saying that the connection is a
metric connection.
Exercise 13.15. Let V be the Levi-Civita connection for a semi-Riemann-
ian manifold (M,g). Prove the formula
(13.3) (Cx9)(Y, Z) = g(VxY, Z) + g(Y, VXZ)
for vector fields X, У, Ζ e X(M).
13.2. Riemann Curvature Tensor
For (M, g) a Riemannian manifold with associated Levi-Civita connection
V, the associated curvature tensor field is called the Riemann curvature
tensor: For X,Y e X(M) we have the map R(X,Y) : X(M) -» X(M)
defined by
R (X, У) Ζ := RxyZ := VXVYZ - VYVXZ - V[XX]Z.
Exercise 13.16. Show that V2XY(Z) - V^X(Z) = R (X, У) Ζ (recall
Definition 13.12).
By direct calculation, or by appealing to Theorem 12.44 and Exercise
12.45 from the previous chapter, we find that (X, У, Z) ь-> R(X,Y)Z is
C°°(M)-multilinear (tensorial). Appealing to the isomorphism (7.7), we
conclude that R gives a section of T°3(TM; TM). That is, R can be thought
of as defining a TpM-valued tensor field at each p. In other words, if
Xp, Yp, Zp G TpM, then R (Xp, Yp) Zp is a well-defined element of TpM and
(Xp, Yp, Zp) h-> R (Xp, Yp) Zp gives a multilinear map. Here, R (Xp, Yp) Zp
is defined to be (R(X,Y)Z)(p) for any fields Χ,Υ,Ζ such that X(p) =
Xp, Y(p) = Yp, and Z(p) = Zp. Many interpretations of R arise. From the
previous chapter we know that R is also to be thought of as a TM-valued
2-form. From this point on we will freely interpret elements of TJ(M) as
elements of Tr s (X(M)) when convenient.
Notice that we will use both the notation R (X, Y) as well as RXy.
Definition 13.17. A semi-Riemannian manifold (M,g) is called flat if the
curvature tensor is identically zero.
554
13. Riemannian and Semi-Riemannian Geometry
Recall that if / : (M,g) —> (N,h) is a local diffeomorphism between
semi-Riemannian manifolds such that f*h = g, then / is called a local
isometry and we say that the manifolds are locally isometric.
Theorem 13.18. Let (M,g) be α semi-Riemannian manifold of dimension
η and index v. If (M,g) is flat, that is, if the curvature tensor is identically
zero, then (M,g) is locally isometric to the semi-Euclidean space RJJ.
Proof. Let ρ e Μ given. If the curvature tensor vanishes, then by Theorem
12.51 we can find local parallel vector fields defined in a neighborhood of ρ
with prescribed values at p. So we may find parallel fields Xi,..., Xn such
that Xi(p),.. ·, Xn(p) is an orthonormal basis. But since parallel translation
preserves the various scalar products (Χι, Xj), we see that we actually have
an orthonormal frame field in a neighborhood of p. Next we use the fact that
the Levi-Civita connection is symmetric (torsion zero). We have VxtXj —
VxjXi — [X{,Xj] = 0 for all i, j. But since the X{ are parallel, this means
that [Xi, Xj] = 0. Therefore there exist coordinates x1,..., xn on a possibly
smaller open set such that
τ—τ = X{ for all i.
ox1
The result is that these coordinates give a chart which is an isometry of
a neighborhood of ρ with an open subset of the semi-Riemannian space
Щ. D
For another proof of the previous theorem see the online supplement
[Lee, Jeff]. The next theorem exhibits the symmetries of the Riemann
curvature tensor:
Theorem 13.19. The map X,Y,Z,W ь-> {RxyZ,W) is tensorial in all
variables. Furthermore, the following identities hold for all X, Y,Z,W G
X(M):
(i) RXy = -RY,X.
(ii) (RXXZ,W) = -(RXXW,Z).
(iii) RxyZ + RyyzX + Rz,xY = 0 (First Bianchi identity).
(iv) (RXfYZ,W) = (RzfwX,Y).
Proof. Tensorality is immediate from our previous observations. Also, (i)
is immediate from the definition of R and (ii) is just a special case of
Proposition 12.70 from the previous chapter. For (iii) we calculate:
Rx,yZ + Ry,zX + Rz,xY
= VXVYZ - VyVxZ + VYVZX - VzVyX + VzVxr - VXVZY = 0.
13.2. Riemann Curvature Tensor
555
The proof of (iv) is rather unenlightening and is just some combinatorics.
Since R is a tensor, we may assume without loss of generality that [X, Y] = 0.
For any X, У, Z, let {CR)XY Ζ be defined by
(CR)XY Ζ := RxyZ + Ry,zX + Rz,xY.
By (iii) we have ((CR)YZ X, W) = 0 for any W. Summing over all cyclic
permutations of У, Z, X, W, we obtain
0 = ((CR)Y>Z X, W) + {{CR)wy Z, X) + ((CR)XtW У, Z) + <(Ci2)ZiX W, У).
Expand this expression using the definition of СR, and we have twelve terms.
Four pairs of terms cancel due to (i) and (ii) resulting in
2 (Rx,YZ, W) + 2 (Rw,zX, Y) = 0.
Using (i) we obtain the result. D
Theorem 13.20 (Second Bianchi identity). For Χ,Υ,Ζ € 3ί(Μ), we have
(VZR)(X, Y) + Vx R(Y, Z) + VYR(Z, X) = 0.
Proof. This is the Bianchi identity for the Levi-Civita connection and in
this context is also called the second Bianchi identity. We give an
independent proof here. Since this is a tensor equation, we only need to prove it
under the assumption that all brackets among the Χ, Υ, Ζ are zero (recall
Exercise 7.34). First we have
(VZR){X, Y)W = Vz(Rx,yW) - R{VZX, Y)W
- R(X, VZY)W - RxyVzW
= [Vz, Rx,y}W - R(VZX, Y)W - R{X, VZY)W.
Using this, we calculate as follows:
(VZR)(X, Y)W + (VXR)(Y, Z)W + {VYR)(Z, X)W
= [Vz,Rx,y]W+ [Vx,Ry,z]W+ [Vy,Rz,x]W
- R{4ZX, Y)W - R(X, VzY)W
- R(VXY, Z)W- R(Y, VXZ)W
- R(VYZ, X)W - R(Z, VYX)W
= [Vz, Rxy\W + [Vx, RYtZ]W + [Vy, Rz,x]W
+ R([X, Z],Y)W + R([Z, Y},X)W + R([Y, X], Z)W
= [Vz, [Vx, Vy]] + [Vx, [VY,VZ]] + [Vy, [Vz, Vx]] = 0.
The last identity is the Jacobi identity for commutators and is true for purely
algebraic reasons (see the next exercise). D
556
13. Riemannian and Semi-Riemannian Geometry
Note: Given a semi-Riemannian manifold (M, g), the tensor R defined
by R(X, Y, Z, W) := (Rx,yZ, W) for all X, Y, Z, W, is also called the Rie-
mann curvature tensor. Often this tensor is defined with a different
ordering of the slots and one should always check which conventions are in
use. One traditional ordering is R(W, Z, X, Y) := (RxyZ, W).
Exercise 13.21. Show that if Li,i = 1,2,3, are linear operators, and the
commutator is defined as usual ([A, B] = AB — В A), then we always have
the Jacobi identity [L\, [L2, L3]] + [L2, [L3, £1]] + [^3, [^1,^2]] = 0.
We now introduce several objects which hold all or part of the
information in the curvature tensor in different forms. First we mention that the
reader should keep an eye out for expressions of the form (R(v,w)v,w) or
(R(v, w)w, v) for v, w in the tangent space of a point on the semi-Riemannian
manifold (M,g) under study.
It will be convenient to introduce a little linear algebra at this point.
Recall that if (V, (·,·)) is a finite-dimensional scalar product space, then
there is an associated natural scalar product on Д V defined so that for an
orthonormal basis {e^}, the basis {e; Л ej}ij for Д V is also orthonormal.
On simple elements we have
ίΚ* Λ ι»,ι* Λ i,4) = det (<'*''*> (viM\
We will use the angle brackets (·, ·) for this scalar product also. The quantity
(v A w,v Л w) is important and needs special attention in the case of an
indefinite scalar product. If (·, ·) on V is indefinite, then the induced scalar
product is also indefinite and (v Aw,ν Aw) may be zero, even when υ and w
are linearly independent. For the next lemma, recall that a subspace W of
a scalar product space (V, (·, ·)) is called nondegenerate if (·, ·) restricted
to W is nondegenerate.
Lemma 13.22. Let (V, (·, ·)) be α finite-dimensional scalar product space.
Let Ρ be a plane spanned by v,w. Then,
(i) Ρ is nondegenerate if and only if (v Aw,υ Aw) φ 0.
(ii) (г; Aw, υ Aw) > 0 if and only if (·, ·) restricted to Ρ is definite.
(Hi) (v Aw,v Aw) < 0 if and only if (·, ·) restricted to Ρ is indefinite.
Proof. Exercise. D
If υ and w span a nondegenerate plane, then \(v A w,v A w)\ is the
squared area of the parallelogram spanned by ν and w.
Lemma 13.23. Let (V, (·, ·)) be a finite-dimensional scalar product space.
Ifv, w G V are any two vectors, then there exist vectors vf, wf G V arbitrarily
close to ν and w respectively such that vf ,wfspan a nondegenerate plane.
13.2. Riemann Curvature Tensor
557
Proof. Assume that (v Aw,ν Aw) = 0, or there is nothing to prove. Any
pair of vectors is close to a pair of linearly independent vectors, so we may
assume that υ and w are linearly independent. There exists a vector χ
such that (г; Λ χ, ν Α χ) < 0. Indeed, if v is null, then we can pick χ so
that (ν,χ) Φ 0, which means that (г; Λ χ, ν Α χ) < 0. If г; is not null,
then pick χ of opposite causal character. That is, pick χ to be spacelike if
υ is timelike and vice versa. Now if we := w + ex for small e > 0, then
(г; Л гиб, υ Α we) = 2eb + e2(υ Α χ, υ Α χ) for some number b independent of
6. If b = 0, then (г; Л we, ν A we) < 0, and we are done by Lemma 13.22. If
b φ 0, then (г; Л we, ν A we) is nonzero in case e is sufficiently small, and then
г;, we span a nondegenerate plane by Lemma 13.22 again. D
Note that in the previous lemma, closeness is measured in the standard
topology of a finite-dimensional vectors space. One does not try to use an
indefinite scalar product to define the topology!
The symmetry properties for the Riemann curvature tensor allow that
we have a well-defined map
(13.4) <H : /\2(TM) -» /\2(TM),
which is symmetric with respect to the natural extension of g to Д (TM).
The map 9Я is defined implicitly as follows:
s(JK(vi Л г>2), Уз Л υ4) := {R{v\,v2)vA, v3).
Notice the switch in the indices 3 and 4. This hides a sign and must be
remembered to avoid confusion later.
Another commonly used quantity is the sectional curvature K. If ν
and w span a nondegenerate plane in TpM, then define
r,/ ν (R(v,w)w,v)
K(v A w) := \/ / \' N2
{v,v){w,w) — (v,w)z
(JK(v Λ w),v A w)
(v A w, ν A w)
The value K(v A w) only depends on the oriented plane spanned by the
vectors ν and ги; therefore if Ρ = span{t>, ги} is such a nondegenerate plane,
we also write K(P) instead of K(v A w). The set of all planes in TPM is
denoted Grp(2). We remark that if Μ is 2-dimensional, then К is a scalar
function on M. It turns out that this function is exactly the Gauss curvature
introduced in Chapter 4. There we showed that the Gauss curvature is
intrinsic and we found an expression for it in terms of the metric, which is
still valid in this situation.
In the following definition, V is an R-module. The two cases we have in
mind are (1) V is X(M), R = C°°(M), and (2) V is TPM, R = R.
558
13. Riemannian and Semi-Riemannian Geometry
Definition 13.24. A multilinear function F:VxVxVxV—»Ris said
to be curvature-like if it satisfies the symmetries proved for the curvature
R above; namely, if for all x,y,z,w G V we have
(i) F(x,y,z,w) = -F(y,x,z,w);
(ii) F(x,y,z,w) = -F(x,y,w,z);
(iii) F(x, y, z, w) + F(y, z, x, w) + F(z, x, y, w) = 0;
(iv) F(x,y,z,w) = F(w,z,x,y).
Exercise 13.25. Define the tensor Cg by
Cg(X, У, Z, W) := g{Y, Z)g(X, W) - g(X4 Z)g(Y, W).
Show that Cg is curvature-like.
Proposition 13.26. If F is curvature-like and F(v,w,v,w) = 0 for all
v,w e V, then F = 0.
Proof. Prom (iv) it follows that for each i>, the bilinear map (w, z) \->
F(v, w, v, z) is symmetric, and so if F(v, w, г;, w) = 0 for all г;, w G V, then
F(v, w, г;, ζ) = 0 for all v,w,z G V. Now it is a simple matter to show that
(i) and (ii) imply that F = 0. D
Proposition 13.27. If (Rv^wv,w) is known for all v,w G TpM, then R
itself is determined at p. If K(P) is known for all nondegenerate planes in
TPM, then R itself is determined at p.
Proof. Let R2(v,w) := (Rv^wv,w) for v,w G TPM. Using an orthonormal
basis for TPM, we see that К and i?2 contain the same information, so we
will just show that i?2 determines R:
d2 I
(R2(v + tz,w + su) — R2(v + tu,w + sz))
Ιο,ο
dsdt
d2
dsdt
{g(R(v + tz,w + su)v + tz,w + su)
0,0
— g(R(v + tu,w + sz)v + tu,w + sz)}
= 6R(v,w,z,u).
The second part follows by continuity and the fact that (R(v,w)w,v) =
(v Aw,v Λ w)K(v Λ w) for v, w spanning a nondegenerate plane P. D
For each υ G ΓΜ, the tidal operator Ry : TpM -» TpM is defined by
Rv(w) := Rv,wv.
We are now in a position to prove the following important theorem.
13.2. Riemann Curvature Tensor
559
Theorem 13.28. The following assertions are all equivalent (к is a
constant):
(i) K(P) = к for all Ρ G Grp(2), Ρ nondegenerate.
(ii) (RVuv2V3,V4) = K,Cg(vi,V2,V3,V4) for all v\, v2,V3,V4 G TPM.
(iii) —Rv(w) = k(w — (w, v) v) for all w, ν G TpM with \\v\\ = 1.
(iv) И(0 = κξ for all ξ € Λ2ΓΡΜ.
Proof. Let ρ e M. The proof that (ii)=>(iii) and that (iii)=>(i) is left as
an easy exercise. We prove that (i) =>(ii)=>(iv)=>(i).
(i)=>(ii): Let R be defined by R(vi,V2,V3,V4) := (Rvi,v2v3,v4) an(i let
Tp := i? — кСд. Then T^ is curvature-like and Tg(v,w,v,w) = 0 for all
v,w G TpM by assumption. It follows from Proposition 13.26 that 7^ ξ 0.
(ii)=>(iv): Let {ei,...,en} be an orthonormal basis for TPM. Then
{ei A ej}i<j is an orthonormal basis for /\2TpM. Using (ii), we see that
(JK(ei Л ej), e* Л et) = (Rei,ejek , e/)
= {R(ei,ej)ek,ei)
= KCg(vi,v2,v?nvA)
= к {ei A ej, e/c Л e/) for all /c, I.
This implies that 5Η(βΐ Л ej) = ке; Л ej.
(iv)=>(i): This follows because if v,w are orthonormal, then we have
к = (JR(v Aw) ^ υ Aw) = K(v A w). D
Definition 13.29. Let (M,g) be a semi-Riemannian manifold. The Ricci
curvature is the (1, l)-tensor Ric defined by
η
Ric(v,w) := У^б^Д^ег,^),
г=1
where (ei,..., en) is any orthonormal basis of TpM and ei := (e^, e^).
We say that the Ricci curvature Ric is bounded from below by к and
write Ric > k if Ric(v,w) > k(v,w) for all v,w G ГМ. Similar and obvious
definitions can be given for Ric < k and the strict bounds Ric > k and
Ric < k. Actually, it is often the case that the bound on Ricci curvature is
given in the form Ric > к(п — 1), where η = dim(M).
In passing, let us mention that there is a very important and interesting
class of manifolds called Einstein manifolds. A semi-Riemannian manifold
(M,g) is called an Einstein manifold with Einstein constant k if and only
if Ric(i>, w) = k(v, w) for all г;, w G TM. We write this as Ric = kg or even
Ric = k. For example, if (M, g) has constant sectional curvature к, then it is
an Einstein manifold with Einstein constant k = к(п — 1). The effect of this
560
13. Riemannian and Semi-Riemannian Geometry
condition depends on the signature of the metric. Particularly interesting
is the case where the index is 0 (Riemannian) and also the case where the
index is 1 (Lorentz manifold). Perhaps the first question one should ask
is whether there exist any Einstein manifolds that do not have constant
sectional curvature. It turns out that there are many interesting Einstein
manifolds that do not have constant sectional curvature. For manifolds
of dimension > 2 the Einstein manifold condition is natural and fruitful.
Unfortunately, we do not have space to explore this fascinating topic (but
see [Be]).
Exercise 13.30. Show that if Μ is connected and dim(M) > 2, and
Ric(v) = /(·,·), where / e C°°(M), then / = к for some к е Ш (so
(M,g) is Einstein).
13.3. Semi-Riemannian Submanifolds
Let Μ be a d-dimensional submanifold of a semi-Riemannian manifold Μ
of dimension n, where d < n. The metric #(·, ·) = (·, ·) on Μ restricts to a
tensor on M, which we denote by h. Since h is a restriction of g, we shall
also use the notation (·, ·) for h. If the restriction h is nondegenerate on
each space TpM and has the same index for all p, then h is a metric tensor
on Μ and we say that Μ is a semi-Riemannian submanifold of M. If
Μ is Riemannian, then this nondegeneracy condition is automatic and the
metric h is automatically Riemannian.
More generally, if φ : TV —> (M, g) is an immersion, we can consider the
pull-back tensor ф*д defined by
φ*9(Χ,Υ)=9(Τφ·Χ,Τφ-Υ).
If ф*д is nondegenerate on each tangent space, then it is a metric on N
called the pull-back metric and we call φ a semi-Riemannian immersion.
If N is already endowed with a metric #дг, and if ф*д = #дг, then we say
that φ : (N,gw) —> (M,#) is an isometric immersion. Of course, if ф*д
is a metric at all, as it always is if (M,^) is Riemannian, then the map
φ : (N^*g) —> (M,^) is automatically an isometric immersion. Every
immersion restricts locally to an embedding, and for the questions we study
here there is not much loss in focusing on the case of a submanifold Μ С М.
There is an obvious bundle on Μ which is the restriction of Τ Μ to Μ.
This is the bundle TM\M = UpeMTpM. Recalling Lemma 7.47, we see that
each tangent space TPM decomposes as
TpM = TPM Θ (TpM)1-,
where (TpM)1- = {v e TpM : (v,w) = 0 for all w e TPM}. Then TM1 =
U Μ (TpM) , with its natural structure as a smooth vector bundle, is called
13.3. Semi-Riemannian Submanifolds
561
the normal bundle to Μ in Μ. The smooth sections of the normal bundle
will be denoted by Γ (ΓΜ1) or X(M)1-. The orthogonal decomposition
above is globalized as
тм\м = тм®тм±.
A vector field on Μ is always the restriction of some (not unique) vector field
on a neighborhood of M. The same is true of any, not necessarily tangent,
vector field along M. The set of all vector fields along Μ will be denoted by
X(M)\M. Since any function on Μ is also the restriction of some function
on M, we may consider X(M) as a submodule of X(M)\M. If X G X(M),
then we denote its restriction to Μ by X\M or sometimes just X. Notice
that X(M)1- is a submodule of X(M)\M. We have two projection maps,
nor: TpM -» TpM1- and tan: ΤρΉ -» TpM which in turn give module
projections nor: X(M)\M -» X(M)1 and tan: X{M)\M -» X(M). We
also have the pair of naturally related restrictions C°°{M) 1еь!^ю" С°°(М)
and X(M) rest_^ion X(M)\M. Note that X{M) is a C°°(M)-module, while
X(M)\M is a C°°(M)-module. We have an exact sequence of modules:
0 -» X(M)^ -> 3£(M)|M ^ X(M) -> 0.
Now we shall obtain a sort of splitting of the Levi-Civita connection of Μ
along the submanifold M. First we notice that the Levi-Civita connection
V on Μ restricts nicely to a connection on the bundle TM\M —> M. The
reader should be sure to realize that the space of sections of this bundle is
exactly X(M)\M. We wish to obtain a restricted covariant derivative
V\M:X(M)xX(M)\M^X(M)\
Μ
If X e X{M) and W e X(M)|M, then VXW does not seem to be defined
since X and W are not elements of X(M). But we may extend X and W
to elements of X(M), use V, and then restrict again to get an element of
X(M)\M. Then recalling the local properties of a connection we see that
the result does not depend on the extension.
Exercise 13.31. Use local coordinates to prove that VxH^ does not depend
on the extensions used.
It is also important to observe that the restricted covariant derivative
is exactly the covariant derivative obtained by the methods of Section 12.3
of the previous chapter. Namely, it is the covariant derivative along the
inclusion map Μ <-> Μ. Thus, it is defined even without an appeal to the
process of extending fields described above.
562
13. Riemannian and Semi-Riemannian Geometry
We shall write simply V in place of V|M since the context will make it
clear when the latter is meant. Thus for all ρ G M, we have V'xW(p) =
V~xW(p), where X and W are any extensions of X and W respectively.
Clearly we have VxJYi,y2) = (VxYi,y2) + (*ъ Vxy2) and so V is a
metric connection on ΓΜ|Λ/. For fixed I,7g X(M), we have the
decomposition of V χΥ into tangent and normal parts. Similarly, for V G X(M)1-,
we can consider the decomposition of VxV into tangent and normal parts.
Thus we have
VXY = (VxY)tan + (VXF)X,
VXV = (VxV)tan + {VXV)L.
Proposition 13.32. For α semi-Riemannian submanifold Μ С М, we have
Vxy = (Vxy)tan for all Χ, Υ e X(M),
where V is the Levi-Civita covariant derivative on Μ with its induced metric.
Proof. It is straightforward to show that if we extend fields X, У, Ζ to
X, У, Z, then the Koszul formula (13.2) for V-χΥ implies that
2((Vxr)tan, Z) = X(Y, Z) + Y(Z, X) - Z(X, Y)
+ (Z,[X,Y}) - (Y,[X,Z\) - (X,[Y,Z})
for all Z. But the Koszul formula which determines VxY shows that
(vxr)tan = vxy. a
Definition 13.33. Define maps//: X{M)xX(M) -> £(M)X, Tl: X{M)x
3£(M)X -» X{M) and Vх : X(M) χ £(M)X -+ £(M)X according to
II(X,Y) := (V*r)x for all X,Y € X(M),
H(X, V) := (VxV)tan for all X € X(M), V € £(M)X,
Vх-V := (VxF)1 for all X € 3£(M), V € £(M)X.
It is easy to show that Vх defines a metric covariant derivative on the
normal bundle TM1. The map (X, Y) ^ II(X, Y) is clearly C°°(M)-linear
in the first slot. If Χ, Υ € X{M) and V € £(M)X, then 0 = (Y, V) and we
have
о = VX(Y, v) = (VXY, V) + (Y,Vxv)
= ({VxY)±,v) + (Y,{Vxv)tAn)
= (II(X,Y),V) + (Y,Tl(X,V)).
It follows that
(13.6) (II{X,Y),V) = -(Y,H(X,V)).
13.3. Semi-Riemannian Submanifolds
563
Prom this we see that II(X, Y) is not only C°°(M)-linear in X, but also
in Y. This means that 77 is tensorial, and so II(Xp,Yp) is a well-defined
element of TpM1- for each Xp, Yp e TpM. Thus 77 is a TM^-valued tensor
field and for each ρ we have an R-bilinear map 77p : TpM χ TpM —> TpM1-.
(We often suppress the subscript p.)
Proposition 13.34. 77 is symmetric.
Proof. For any Χ, Υ e X(M) we have
II(X,Y) - //(r,Xi) = {VxY-VyX)^
= ([X,Y])± = 0. D
We can also easily deduce that 77 is a symmetric C°°(M)-bilinear form
with values in X(M) and is similarly tensorial. So II(XP, Vp) is a well-
defined element of TpM for each fixed Xp e TpM and Vp e TpM1-. We thus
obtain a bilinear form
77p : TpM χ TpM^ -» TpM.
In summary, 77 and 77 are tensorial, V1- is a metric covariant derivative
and we have the following formulas:
VXY = VXY + II(X,Y),
vxv = vj<v + n(x,v)
for X,Y e X(M) and V e X(M)1-.
Recall that if (V, (·, -)1) and (W, (·, ·)2) are scalar product spaces, then
a linear map A : V —> W has a metric transpose A1 : W —> V uniquely
defined by the requirement that
(Av,w)2 = (v,Atw)1
for all ν e V and w eW.
Definition 13.35. For υ G TpM, we define the linear map Βυ(·) := ΙΙ(υ, ·).
Formula (13.6) shows that the map II(v,·) : TpM1- -» TpM is equal
to -B\ : TpM -» TpM1-. Writing any Υ e X{M)\M as a column vector
(ytan^yi.^ we can write fae map yx . χ(Μ}\^ _+ χ(Μ)|Μ as a matrix
of operators:
I ^ Vx J '
Next we define the shape operator, also called the Weingarten map.
We have already met a special case of the shape operator in Chapter 4. The
shape operator is sometimes defined with the opposite sign.
564
13. Riemannian and Semi-Riemannian Geometry
Definition 13.36. Let ρ G M. For each unit vector и normal to Μ at p,
we have a map called the shape operator Su associated to и defined by
Su(v) := - {VvU)tan ,
where U is any unit normal field defined near ρ such that U(p) = u.
Exercise 13.37. Show that the definition is independent of the choice of
normal field U that extends u.
The family of shape operators {Su : и a unit normal} contains essentially
the same information as the second fundamental tensor II or the associated
map B. This is because for any I,7g X(M) and U G X(M)1-, we have
(SuX,Y) = <(-Vx[/)tan,y> = (U,-VXY)
= ([/,(-VxF)X) = (U,-II(X,Y)).
In the case of a hypersurface, we have (locally) only two choices of unit
normal. Once we have chosen a unit normal w, the shape operator is denoted
simply by S rather than Su-
Theorem 13.38. Let Μ be α semi-Riemannian submanifold of M. For any
V,W,X,Y eX(M), we have
{RVWX,Y)={RVWX,Y)
- (II(V,X),II(W, Y)) + (II(V, У), II{W,X)).
This equation is called the Gauss equation or Gauss curvature
equation.
Proof. Since this is clearly a tensor equation, we may assume that [V, W] =
0 (see Exercise 7.34). We have (RVWX, Y) = (VVVWX, Y)-(VWVVX, Y).
We calculate:
(VVVWX,Y) = (VVVWX,Y) + (VV(II(W,X)),Y)
= (VvVwX,Y) + (Vv(H(W,X)),Y)
= (VVVWX,Y) + V(II(W,X),Y)- (II(W,X),VVY)
= (VVVWX,Y) - (II(W,X),VVY).
Since
(II(W,X),VVY) = (II(W,X), {VyY)1)
= (II(W,X),II(V,Y)),
we have {VVVWX, Y) = (VVVWX, Y) - {II(W, X),II(V, Y)).
Interchanging the roles of V and W and subtracting we get the desired conclusion. D
13.3. Semi-Riemannian Submanifolds
565
The second fundamental form contains information about how the semi-
Riemannian submanifold Μ bends in M.
Definition 13.39. Let Μ be a semi-Riemannian submanifold of Μ and TV
a semi-Riemannian submanifold of N. A pair isometry Φ : (Μ, Μ) —>
(Ν, Ν) consists of an isometry Φ : Μ —> TV such that Φ(Μ) = TV and such
that Φ|Μ : Μ —> TV is an isometry
Proposition 13.40. A pair isometry Φ : (Μ, Μ) —> (JV, Ν) preserves the
second fundamental tensor:
ГрФ · ΙΙ(υ, w) = ΙΙ(ΤΡΦ · ν, ГрФ · ги)
for all г>, w e TPM and all ре М.
Proof. Let ρ e Μ and extend v,w G TpM to smooth vector fields V and
W. Since isometries respect Levi-Civita connections, we have Ф+УуИ^ =
νφ+γΦ+W. Since Φ is a pair isometry we have ТРФ(ТРМ) С ΤΦ^Ν and
ТрФррМ1-) С (Τφ^Ν)1-. This means that Φ* : X(M)\M -» £(]V)|N
preserves normal and tangential components Ф*(Х(М)) С X(N) and
Ф^М)^) С ^(Ν)1-. We have
ГРФ · II(v, w) = Φ*//(ν, И0(Ф (ρ)) = Φ* (V^)1 (Φ (ρ))
= {Φ+VyW)^ (Φ (ρ)) = (νΦ^Φ·^)χ (Φ (ρ))
= //(Φ·ν, Ф*И^)(Ф (ρ)) = //(Φ,V, Ф*И^)(Ф (ρ))
= //(ТрФ-г;,ТрФ·™). D
The following exercise gives a simple but conceptually important
example.
Exercise 13.41. Let Μ be the 2-dimensional strip {(x,y,0) : —π < χ < π}
considered as a submanifold of M3. Let TV be the subset of R3 given by
{(x, j/, \/l - x2) : -1 < χ < 1}. Show that Μ is isometric to N. Show that
there is no pair isometry (Μ3, Μ) -» (M3, JV).
Definition 13.42. Let Μ be a semi-Riemannian submanifold of a semi-
Riemannian manifold M. Then for any V,W,Z e X(M) define (Vyll)
by
(Vyll) (W, Z) := V^ (//(W, Z)) - //(VvX, Y) ~ H(X, Vvy).
Theorem 13.43. №Ш M, M, and V,W,Z <E X(M) as in the previous
definition we have the following identity:
{RVWZY = (VVII) (W, Z) - (VWH) (V, Z) (Codazzi equation).
566
13. Riemannian and Semi-Riemannian Geometry
Proof. Since both sides are tensorial, we may assume that [V, W] = 0.
Then (RvwZY = (yvVwZ)± - (VwVyZ)^. We have
(VvV^Z)1 = (W {VWZ)Y - (W (//(W, Z)))L
= II(V,VWZ)-(VV(II(W,Z)))±.
Now recall the definition of (VyII) (W, Z) and find that
(VvVivZ)1 = //(V, VWZ) + (Vv//) (W, Z) + II(VVW, Z) + II(W, VyZ).
Now compute {VyVwZ) — (VjyVyZ) and use the fact that VyW -
VWV = [V, W} = 0. D
The Gauss equation and the Codazzi equation belong together. If we
have an isometric embedding f : N -* M, then the Gauss and Codazzi
equations on f(N) С М pull back to equations on TV and the resulting
equations are still called the Gauss and Codazzi equations. Obviously, these
two equations simplify if the ambient manifold Μ is a Euclidean space. For
a hypersurface existence theorem featuring these equations as integrability
conditions see [Pe].
13.3.1. Semi-Riemannian hyper surfaces. A semi-Riemannian sub-
manifold of codimension one is called a semi-Riemannian hypersurface.
Let Μ be a semi-Riemannian hypersurface in M. By definition, each
tangent space TpΜ is a nondegenerate subspace of TpM. The complementary
spaces (TpM) are easily seen to be nondegenerate, and ind (TpM1-) is
constant on Μ since we assume that ind (TpM) is constant on M. The number
ind {TpM1-) called the co-index of M.
Exercise 13.44. Show that the co-index of a semi-Riemannian hypersurface
must be either 0 or 1.
Definition 13.45. The sign ε of a hypersurface Μ is defined to be +1 if
the со-index of Μ is 0 and is defined to be —1 if the co-index is 1. We
denote it by sgnM.
Notice that if ε = 1, then ind(M) = ind(M), while if ε = -1, then
ind(M) = ind(M) - 1.
Proposition 13.46. Let f e C°°(M) and Μ := /_1(c) for some с е R.
Suppose that Μ φ § and that grad/ φ 0 on Μ. Then Μ is a semi-
Riemannian submanifold if and only if either (grad/, grad/) > 0 on M, or
(grad/,grad/) < 0 on M. The sign of Μ is the sign of (grad/,grad/),
and grad// ||grad/|| restricts to a unit normal field along M.
13.4. Geodesies
567
Proof. The relation (grad/, grad/) /OonM ensures that df φ 0 on M,
and it follows that Μ is a regular submanifold of codimension one. Now if
υ e TM, then
(grad/, v) = df(v) = v(f)
= "(/Im) = 0,
so grad / is normal to M. Thus for any ρ G Μ the space (TPM) is non-
degenerate, and so the orthogonal complement TpM is also nondegenerate.
The rest is clear. D
We now consider certain exemplary hypersurfaces in R™+1. Let q :
M™+1 —> R be the quadratic form defined by
q(x):=(x,x) = -J2H2+ ЕИ2
г=1 г=1/+1
г=1
where the reader will recall that ει = — 1 or Si = 1 as 1 < i < ν or
l + i/<i<n+l. Hypersurfaces in R™+1 defined by Q(n,r,s) := {x G
MJJ4"1 : g(x) = £r2}, where ε = -1 or ε = 1, are called hyperquadrics.
Exercise 13.47. Let Q(n,r,e) be a hyperquadric as defined above. Let
Ρ := Y^xldi be the position vector field in R™+1. Show that the restriction
of P/r to Q(n, r, ε) is a unit normal field along Q(n, r, ε).
Exercise 13.48. Show that a hyperquadric Q(n,r,e) as defined above is a
semi-Riemannian hypersurface with sign ε.
13.4. Geodesies
In this section, 7 will denote a nonempty interval assumed to be open unless
otherwise indicated by the context. Usually, it would be enough to assume
that I has nonempty interior. We also allow 7 to be infinite or "half-infinite".
Let (M, g) be a semi-Riemannian manifold. Suppose that 7 : 7 —> Μ is a
smooth curve that is self-parallel in the sense that
Vat7 = 0
along 7. We call 7 a geodesic. To be precise, one should distinguish
various cases as follows: If 7 : [a, b] —> Μ is a curve which is the restriction
of a geodesic defined on an open interval containing [a, b], then we call 7 a
(parametrized) closed geodesic segment or just a geodesic for short. If
7 : [a, 00) —> Μ (resp. 7 : (—00, a] —> M) is the restriction of a geodesic
then we call 7 a positive (resp. negative) geodesic ray.
568
13. Riemannian and Semi-Riemannian Geometry
If the domain of a geodesic is R, then we call 7 a complete geodesic. If
Μ is an η-manifold and the image of a geodesic 7 is contained in the domain
of some chart with coordinate functions ж1,..., xn, then the condition for 7
to be a geodesic is
(«•7) i|p(i, + Er.t(7((,)^p((,^(t) = 0
for alH e I and 1 < г < п. This follows from formula (12.7) and this is a
system of η second order equations often abbreviated to ^h + ^ ^)кЧГЧГ
= 0, 1 < i < n. These are the local geodesic equations. Now consider a
smooth curve 7 whose image is not necessarily contained in the domain of a
chart. For every to £ /, there is an e > 0 such that 7L0_e to+e) is contained
in the domain of a chart, and thus it is not hard to see that 7 is a geodesic
if and only if each such restriction satisfies the corresponding local geodesic
equations for each chart which meets the image of 7. We can convert the
local geodesic equations (13.7) into a system of 2n first order equations by
the usual reduction of order trick. We let ν denote a new dependent variable
and then we get
dxl i л
— =v\l<i<n,
^ + ^r^V = 0,l<i<n.
We can think of x1 and vl as coordinates on TM. Once we do this, we
recognize that the first order system above is the local expression of the
equations for the integral curves of a vector field on TM.
Exercise 13.49. Show that there is a vector field G G X(TM) such that
α is an integral curve of G if and only if 7 := птм ° ol is a geodesic. Show
that the local expression for G is
Σ
г j,/c
The vector field G from this exercise is an example of a spray (see
Problems 4-8 from Chapter 12). The flow of G in the manifold Τ Μ is called the
geodesic flow.
Lemma 13.50. For each ν G TPM', there is an open interval I containing 0
and a unique geodesic 7 : / —> Μ, such that 7(0) = ν (and hence 7(0) = p).
Proof. This follows from standard existence and uniqueness results for
differential equations. One may also deduce this result from the facts about
flows since, as the exercise above shows, geodesies are projections of
integral curves of the vector field G. The reader who did not do the problems
13.4. Geodesies
569
on sprays in Chapter 12 would do well to look at those problems at this
time. D
Lemma 13.51. Let 71 and 72 be geodesies I —> M. 7/71 (to) = 72(^0) for
some to G 7, then 71 =72·
Proof. If not, there must be tf G 7 such that 71 (ί;) ^ 72(0· Let us assume
that tf > to since the proof of the other case is similar. The set A = {t e I:
t > to and 71 (i) 7ε 72(0} has an infimum b = inf A. Note that b > to-
Claim: 71(b) = 72(b)· Indeed, if b = to, there is nothing to prove. If
b > to, then 7χ(ί) = 72(i) on the interval (ίο, b). By continuity 71(b) = 72(b).
Now t h-> 71(6 + i) and ί ь-> 72(6 + i) are clearly geodesies with initial
velocity 71(b) = 72(b)· Thus by Lemma 13.50, 71 = 72 for some open
interval containing b. But this contradicts the definition of b as the infimum
of A D
A geodesic 7 : I —> Μ is called maximal if there is no other geodesic
with open interval domain J strictly containing 7 that agrees with 7 on 7.
Theorem 13.52. For any υ G TM, there is a unique maximal geodesic
ην with 7V(0) = v.
Proof. Take the class Qv of all geodesies with initial velocity v. This is not
empty by Lemma 13.50. If α,/З G Qv and the respective domains Ia and
Ιβ have nonempty intersection, then a and β agree on this intersection by
Lemma 13.51. Prom this we see that the geodesies in Qv fit together to form
a manifestly maximal geodesic with domain 7 = \J~eg Ц- Obviously this
geodesic has initial velocity v. D
Definition 13.53. If the domain of every maximal geodesic emanating from
a point ρ G TpM is all of M, then we say that Μ is geodesically complete
at p. A semi-Riemannian manifold is said to be geodesically complete if
and only if it is geodesically complete at each of its points.
Exercise 13.54. Let Ш™ be the semi-Euclidean space of index v. Show that
all geodesies are of the form 11->> xo + tw for w G M™.
Definition 13.55. A continuous curve 7 : [a, b] —> Μ is called a broken
geodesic segment if it is a piecewise smooth curve whose smooth segments
are geodesic segments. If t* is a point of [a, b] where 7 is not smooth, we call
7(£*) a break point. (A smooth geodesic segment is considered a special
case.)
Exercise 13.56. Prove that a semi-Riemannian manifold is connected if
and only if every pair of its points can be joined by a broken geodesic
7 : [a, b] -» M.
570
13. Riemannian and Semi-Riemannian Geometry
Exercise 13.57. Show that if 7 is a geodesic, then a reparametrization
c '·— 7 ° / is a geodesic if and only if f(t) := at + b for some a, b G Μ and
α ^ 0. Show that if 7 is never null, then we may choose a, b so that the
geodesic is unit speed and hence parametrized by arc length.
The existence of geodesies passing through a point ρ G Μ at parameter
value zero with any specified velocity allows us to define a very important
map. Let Vp denote the set of all ν G TpM such that the geodesic ην is
defined at least on the interval [0,1]. The exponential map, expp : Vv ->
M, is defined by
ехррг; :=7ν(1).
Lemma 13.58. If ηυ is the maximal geodesic with 7V(0) = ν G TpM', then
for any c,t GK, we have that jcv{t) is defined if and only ifjv(ct) is defined.
When either side is defined, we have
lcv{t) =7v(ct).
Proof. Let JVjC be the maximal interval for which ην(σί) is defined for all
t G JV}C. Certainly 0 G J. Use the chain rule for covariant derivatives or
calculate locally to see that t ь-> ην(αί) is a geodesic with initial velocity cv.
But then by uniqueness and the maximality of 7сг;, the interval Jv^c must be
contained in the domain of ηον and for t G JVyC we must have
lcv{t) =7v(ci).
In other words, if the right hand side is defined, then so is the left and we
have equality. Now let и = cv, s = ct and b = 1/c. Then we just as well
have that
%u(s) =7u(bs),
where if the right hand side is defined, then so is the left. But this is just
7v(ci) = 7cv(i).
So left and right have reversed and we conclude that if either side is defined,
then so is the other. D
Corollary 13.59. If ην is the maximal geodesic with 7V(0) = ν G TVM,
then
(i) t is in the domain of ην if and only if tv is in the domain of expp;
(ii) 7v(i) = expp(h;) for all t in the domain of ην.
Proof. Suppose that tv is in the domain of expp. Then 7*v(l) is defined.
But 7iV(l) = 7υ(ί) and t is in the domain of ην by the previous lemma. The
converse is proved similarly and so we obtain expp(it;) = 7*v(l) = 7υ(ί)· ^
13.4. Geodesies
571
Now we have a very convenient situation. The maximal geodesic through
ρ with initial velocity υ can always be written in the form t ь-> expptv.
Straight lines through 0P G TPM are mapped by expp onto geodesies which
we sometimes refer to as radial geodesies through p. Similarly, we have
radial geodesic segments and radial geodesic rays emanating from p. The
result of the following exercise is a fundamental observation.
Exercise 13.60. Show that (Ύν(ϊ),Ίν(ϊ)) = (v,v) f°r aU t in the domain of
The exponential map has many uses. For example, it is used in
comparing semi-Riemannian manifolds with each other. Also, it provides special
coordinate charts. The basic theorem is the following:
Theorem 13.61. Let (M,g) be a semi-Riemannian manifold and ρ G M.
There exists an open neighborhood Up С Vp containing 0P such that expp| ~
is a diffeomorphism onto its image Up.
Proof. The tangent space TpM is a vector space, which is isomorphic to
Rn and so has a standard differentiable structure. Using the results about
smooth dependence on initial conditions for differential equations, we can
easily see that expp is well-defined and smooth in some neighborhood of 0P G
TPM. The main point is that the tangent map Texpp : Т0р(ТрМ) -» TPM
is an isomorphism and so the inverse mapping theorem gives the result. To
see that Texpp is an isomorphism, let vqp G Tqp(TpM) be the velocity of the
curve t ь-> tv in TPM. Then, unraveling definitions, we have Texppi>op =
^|0ехрр£г> = v. Thus Texpp is just the canonical map vop ь-> v. Π
Definition 13.62. A subset С of a vector space V that contains 0 is called
star-shaped about 0 if whenever υ G C, tv G С for all t G [0,1].
Definition 13.63. If U С Vp is a star-shaped open set about 0P in TPM
such that exppU is a diffeomorphism as in the theorem above, then the
image expp(/7) = U is called a normal neighborhood of p. In this case,
U is also referred to as star-shaped.
Theorem 13.64. If U С Μ is a normal neighborhood about ρ with
corresponding preimage U С TPM, then for every point q G U there is a unique
geodesic 7 : [0,1] -> U С Μ such that 7(0) = p, 7(1) = q, 7(0) G U and
expp7(0) = q. (Note that uniqueness here means unique among geodesies
with image in U.)
Proof. The preimage U corresponds diffeomorphically to U under expp. Let
ν = exppU (q) so that υ G U. By assumption U is star-shaped and so the
map ρ : [0,1] —> TpM given by t ^ tv has image in U. But then, the
572
13. Riemannian and Semi-Riemannian Geometry
geodesic segment 7 : i н-> ехрр£г>, t G [0,1] has its image inside U. Clearly,
7(0) = ρ and 7(1) = q. Since ρ = г>, we get
7(0) = Texppp(0) = Техррг; = υ
under the usual identifications in TPM.
Now assume that 71 : [0,1] —> U С Μ is some geodesic with 71 (0) = ρ
and 71(1) = q. If 71(0) = w, then 71 (i) = expptw.
Claim: The ray pi : t ^ tw (t e [0,1]) stays inside U. If not, then
the set A = {t : tw £ U} is nonempty. Let i* = inf A and consider the set
С := {tw :t e (0,i*)}. Then С CU and U\C is contractible (check this).
But its image expp|^ (U\C) is /7\C, where С is the image of (0,i*) under
71. Since U\C is certainly not contractible, we have contradiction. Thus
the claim is true, and in particular w = pi(l) G U. Therefore, both w and
υ are in U. On the other hand,
exppw = 71(1) = g = ехррг;.
Thus, since exppU is a diffeomorphism and hence injective, we conclude
that w = v. By the basic uniqueness theorem for geodesies, the segments 7
and 71 are equal and both given by t ь-> expp tv. D
Let (M, g) be a semi-Riemannian manifold of dimension n. Let po £ ^
and pick any orthonormal basis (ei,... ,en) for the semi-Euclidean scalar
product space (TpoM, (·, ·)ρο). This basis induces an isometry I : M™ —У
TP0M by (хг) ь-> Σ x%ei· If f is a normal neighborhood centered at po G M,
then Xnorm := /o exppo |^ : U —> M^ = Mn is a coordinate chart with domain
U. These coordinates are referred to as normal coordinates centered at
Po· Normal coordinates have some very nice properties:
Theorem 13.65. //xnorm = (x1,... ,xn) are normal coordinates defined on
U and centered atpo, then
9ij(Po) = (-Q-, -Q-) = eAj for all г, j,
rj/c(Po) = 0 for alli,j,k.
(When using normal coordinates, it should not be forgotten that the Г^ are
only guaranteed to vanish at po.)
Proof. Let ν eTP0M and let {e*} be the basis of T*QM dual to {ej. We
write ν = Σα%βΐ' We have that ег о ехрро|^ = хг. Now jv(t) = exppotv
and so
χ\Ίν(ϊ)) = e\tv) = te\v) = ta{.
13.4. Geodesies
573
So we see that ν = 7V(0) = Х)аг ^| .In particular, if аг = £j-, then e* =
^-|ρο and (^, ^j)po = e^ij- Since 7^ is a geodesic and хг(ъ(*)) = №,
the coordinate expression for the geodesic equations reduces to
£r*fc(7,(i)Kafc = 0
for all i. In particular, this holds at po = 7V(0). But υ is arbitrary, and
hence the η-tuple (аг) is arbitrary. Thus the quadratic form defined on
Rn by Ql(u) = J2^)k(Po)u^uk ls identically zero, and by polarization, the
bilinear form Ql : (u,v) ь-> ^Tl-k(po)u^vk is identically zero. Of course this
means that Γ^,(ρο) = 0 for all j, к and arbitrary i. D
Notation 13.66. Prom now on, whenever we write exp"1, we must have
in mind an open set U and an open set U such that exppU : U —> U is
a diffeomorphism. Thus, exp"1 is an abbreviation for expp|^ . Usually, U
will be star-shaped and thereby U is a normal neighborhood of the point p.
Definition 13.67. Let U be a normal neighborhood of a point po in a
semi-Riemannian manifold M. The radius function r : U —> Ш is defined
by
rP0(p) '= ||exp;^(p)||
for ρ G U. We often write simply r if the central point po is understood.
If (x1,..., xn) are normal coordinates defined on U and centered at po>
then the radius function is given by
ι ν η 11/2
r = rpo= -Dx<)2+ Σ (j,):
The radius function is smooth except on the set where it is zero. This zero
set is called the local nullcone and is the image of the intersection of the
nullcone in TpoM with U = exp~^(U). In the Riemannian case, where the
metric is definite, the radius function r is smooth except at the center point
Po· Note that in this case, r2 is smooth even at po·
Now suppose that 7 : [0,1] —> U С Μ is the geodesic with 7(0) = po,
7(1) = ρ and 7(0) = v. Then we have the useful and often used fact that
L(7) =r(p).
To see this, first note that υ = exp~^(p). Then, since Ц7Ц is constant (by
Exercise 13.60), we have
L{l)= I \\i\\dt= [ \\v\\dt=\\v\\=r(p).
Jo Jo
574
13. Riemannian and Semi-Riemannian Geometry
More generally, if r > 0 is such that ην : t ь-> expp tv is defined for 0 < t < r,
then
Г \(ъ^Ыт1/2 dt = r\(v,v)\1/2 = r\\v\\.
Jo
In particular, if г; is a unit vector, then the length of the geodesic 7^|r0ri is
equal to r.
Let V := UpDp. We can gather the maps expp : Vp С TPM —> Μ
together to get a map exp : V —> Μ defined by exp(v) := ехр^^г;). The
set V is the set of ν G ΓΜ such that the geodesic ην is defined at least on
[0,1].
Proposition 13.68. V is open and for each ρ G M, Vp is open and star-
shaped. Thus Vv := expp(X>p) is a (maximal) normal neighborhood of p.
Proof. Let Wclx Τ Μ be the domain of the geodesic flow (5, г;) ь-> 7v(s).
This is the flow of a vector field on ГМ, and so W is open. W is also
the domain of the map (s,v) ь-> π ο 7v(s) = 7^(5). The map (1,г;) ь-> г>
is a diffeomorphism {1} χ ΓΜ —> TM. Under this diffeomorphism, V
corresponds to the set W Π ({1} χ ΓΜ), and so it must be open in ΓΜ. It
also follows that Vp = V Π ΓΡΜ is open in ГРМ and Vp is open in M.
To see that Vv is star-shaped, let υ G Vp. Then ην is defined for all
t G [0,1]. On the other hand, 7*ν(1) is defined and equal to 7v(i) for all
t G [0,1] by Lemma 13.58. Thus, by definition, tv G Vp for all ί G [0,1]. D
Let Δ = {(ρ,ρ) : ρ G Μ} be the diagonal subset of Μ χ Μ. Let
EXP : V С ГМ -» Μ χ Μ be defined by
EXP : г> ь-> (π(ν),βχρρν),
where π : ΓΜ —> Μ is the tangent bundle projection.
Theorem 13.69. If 0P is the zero element ofTpM, then there is a
neighborhood W of 0P in Τ Μ such that EXP|^ is a diffeomorphism onto a
neighborhood of(p,p) G Δ С Μ χ Μ.
Proof. We first show that if Tx expp is nonsingular for some χ G Vp С ГРМ,
then Tx EXP is also nonsingular at x. So assume that Tx expp is nonsingular
and suppose that Tx ΕΧΡ(^) = 0. We have π = prx о EXP and so Τπ(υχ) =
Грг1(ГгЕХР(г;а;)) = 0. This means that vx is tangent to Vp С ГрМ. But
13.4. Geodesies
575
the restricted map EXP|p is related to expp by trivial diffeomorphisms:
EXpl«D
;v
VO > px Μ
ΡΓ2
id
~ exp
vp —^м
Thus Txexpp(vx) = 0 and hence vx = 0.
Since Top expp is nonsingular at each point 0P of the zero section, we see
that the same is true for Top EXP, and the result follows from the inverse
mapping theorem. D
Definition 13.70. An open subset /7 of a semi-Riemannian manifold will
be said to be totally star-shaped if it is a normal neighborhood of each
of its points.
Notice that /7 being totally star-shaped according to the above
definition implies that for any two points p,g G t/, there is a geodesic segment
7 : [0,1] —> U such that 7(0) = ρ and 7(1) = g, and this is the unique
such geodesic with image in /7. (One may always make an affine change of
parameter, but then we have a different interval as the domain.) Thinking
about the sphere makes it clear that even if /7 is totally star-shaped, there
may be geodesic segments connecting ρ and q whose images do not lie in /7.
Theorem 13.71. Every ρ G Μ has a totally star-shaped neighborhood.
Proof. Let ρ e Μ and choose a neighborhood W of 0P in TM such that
EXP^ is a diffeomorphism onto a neighborhood of (p,p) G Μ χ Μ. By a
simple continuity argument we may assume that EXP|^ (W) is of the form
U(S) χ U(S) for U(δ) := {q : EiUO^fa))2 < δ) and x = (ж1,···,^) is a
normal coordinate system. Now consider the tensor b on U(6) whose
components with respect to χ are Ъц = δ^ ~Σΐς ^ijxk· This is clearly symmetric
and positive definite at p, and so by choosing δ smaller if necessary we may
assume that this tensor is positive definite on υ(δ). Let us show that ΙΙ(δ)
is a normal neighborhood of each of its points q. Let Wq := W Π TqM.
We know that ЕХР|^ is a diffeomorphism onto {q} χ /7(5), and it is easy
to see that this means that exp9|„, is a diffeomorphism onto U(6). We
now show that Wq is star-shaped about 0q. Let q' G /7(5), qf Φ g, and
υ = EXP|^ (g, qf). This means that 7^ : [0,1] —> Μ is a geodesic from q to
qf. If 7v([0,1]) С /7(5), then tv <E Wq for all t <E [0,1] and so we could
conclude that Wq is star-shaped. Let us assume that 7V([0,1]) is not contained
in /7(5) and work for a contradiction.
576
13. Riemannian and Semi-Riemannian Geometry
If in fact ην leaves /7(5), then the function /:ίπ ΣΓ=ι(χΖ(7υ(^)))2 has
a maximum at some to G (0,1). Thus the second derivative of / cannot be
positive at to. We have
<й2/-2гд dt +*°ъ dt2 j.
But ηυ is a geodesic, and so using the geodesic equations we get
d^-^L·^ L·1^ dt dt ■
i,j \ к /
Plugging in to we get
d2
^f(to) = 2b(iv(to)nv(to))>0,
which contradicts / having a maximum at to- Π
Note: It follows from the proof that given ρ G M, there is a δ > 0 such
that exp({i>p G TVM : ||г;р|| < ε}) is totally star-shaped for all ε < δ.
Warning: Clearly a totally star-shaped open set is, in some sense,
"convex". Indeed, some authors define convexity in this manner. However,
notice that on the circle, open intervals are totally star-shaped while the
intersection of two such intervals need not even be connected. For proper
Riemannian manifolds, there is another definition of convexity that does not
suffer from this defect (see Problem 2). Actually, there are several notions
of convexity on manifolds and the terminology does not seem to be quite
standardized.
Theorem 13.72 (Gauss lemma). Let ρ G Μ, χ G TpM, with χ φ 0Ρ in
the domain of expp . Choose vx,wx G Tx(TpM), where vx,wx correspond to
v,w G TpM under the canonical isomorphism between TX(TPM) and TPM.
If vx is radial, i.e. if ν is a scalar multiple of x, then
(Tx expp vx , Tx expp wx) = (vx, wx).
Proof. Clearly we may suppose that χ = v. For small e > 0, we define
h : [0,1 + e) χ (-6, б) -» TPM by
h(t,s) = t(v + sw).
If we take e sufficiently small, then on the same domain we may define h by
h(t, s) := expp(t(v + sw)). Then
dh, д
Ж(М):=Г(м)/>·-,
dh, N m , д
13.4. Geodesies
577
Figure 13.2. Gauss lemma
dhi
We have that ^(1,0) = vv and $£(1,0) = wv, so that
dh ,Λ ^Ν m
— (1,0) = rt,exppvt„
— (1,0) = Tvexpp wv.
We wish to show that (^,|j)(l,0) = (vv,wv) = (v,w). Since the curve
ί ь-> expp(t(v + sw)) is a geodesic with initial velocity ν + sw, we have
/dh dh\/t . /dh dh\ . ч . .
/ \(t,5) = (—— )(0,5) = (ν + 5«;,ν + 5«;).
\dt* at
We have
d /dh dh\
\at' at
dh dh\ /dh dh\
di dt' ds I \dV di ds I
/dh _ dh\ , . , , N . , . ч
= ( —, V_a_ — ) (since £ ь-> h{t, s) is a geodesic)
\dt'
= /
dh
\dt
,V_
ds/
dh\
■at/
(by Exercise 13.11)
1 d /dh dh\
h dh\(
t' 0s Μ
t(v,w). The result follows by letting ί = 1.
2 ds \ ^ ' ^ /
Since /i(0, 5) = ρ for all s, we have (§, f:)(0,0) = 0 and so (§£, g)(t,0)
D
If i^ is not assumed to be radial, then the above equality does not always
hold. We need to have the dimension of the manifold greater than 3 in order
to see what can go wrong. Figure 13.3 shows a unit sphere in the tangent
space of a Riemannian manifold and a pair of orthogonal vectors tangent to
this sphere. Under the exponential map these vectors map to vectors which
need not be orthogonal.
578
13. Riemannian and Semi-Riemannian Geometry
TpM,
Μ
e^5(l)
Figure 13.3. Distortions under the exponential map
We now introduce the position vector fields associated with a normal
neighborhood of a point p. First, let us consider a vector space V with scalar
product (·,·). Then V is a semi-Riemannian manifold with metric defined
by (vx,wx) := (v,w). Let V denote the associated Levi-Civita connection.
Because of the canonical isomorphisms TVY = V, every vector field У on V
can be identified with a map Υ : V —> V. Under this identification, VxY
is just the directional derivative of Υ in the X direction. The position
vector field on V is defined by Ρ : υ м> vv, and it is easy to check that
VxP = X for any vector field X. Now consider the quadratic form q defined
by q(v) = (i>, v). Unraveling the definitions we see that q = (Ρ, Ρ). We have
for any X
(gradq.X) = Xq = X(P,P) = 2(VXP,P) = (X,2P),
and we conclude that
gradq = 2P.
It follows that Ρ is normal to every hyperquadric q_1(c), с G К, с / 0.
Now we want a similar result in a normal neighborhood U of a point ρ on
a semi-Riemannian manifold M. We consider TPM as a semi-Riemannian
manifold in its own right. We have the position vector field
Ρ : ν ь-* vv,
and we have the quadratic form q defined by q(v) = (v,v). Let U =
exp~1({7) and for each с φ 0 let Qc := q_1(c) and Qc := expp(C/ Π Qc).
Corresponding to q we have a function q defined on U by
q:=qoexpi
so Qc = q λ(ο).
13.4. Geodesies
579
Definition 13.73. For с φ 0, the sets Qc = q_1(c) are called local hy-
perquadrics associated to the normal neighborhood centered at p. The set
Л = q_1(0) = qo ехр~х(0) is called the local nullcone at p.
On the normal neighborhood U there is a unique vector field Ρ that is
expp-related to P. We refer to Ρ as the local position vector field for
the normal neighborhood at p,
Ρ = T expp oP о ехр"1.
Proposition 13.74. Let U be a normal neighborhood of ρ and let q, Qc,
Qc, P, and Ρ be as above. Then
(i) Ρ is normal to each Qc;
(ii) (P,P)oexVp = (P,P);
(iii) gradq = 2P.
Proof, (i) follows from the Gauss lemma (Lemma 13.72) and the
corresponding fact that Ρ is normal to each Qc in TpM. The Gauss lemma also
immediately gives (ii).
(iii) For any v, let υ be such that Texppv = v. Then
(gradq,г;) = v(q) = (Texppv) q
= υ (q о ехрр) = v(q)
= (gradq, v) = 2(P,v) = 2(P,i;>,
where we used the Gauss lemma in the last step. Since υ was arbitrary, we
obtain (iii). D
Consider the unit sphere 5η_1 and the map Rn —> (0, oo) χ 5η_1 given
Ьухи (||x|| ,x/ ||x||). Now put coordinates on the sphere, say 01,... ,ίΡ-1.
Composing, we obtain coordinates (г,01,... ,θη~ι) on an open subset of
Mn\{0}, where r gives the distance to the origin and the θ directions are
normal to the r direction. A standard method of choosing the angle functions
01,... ,θη~ι leads to what is sometimes called hyperspherical coordinates.
If (M,p) is Riemannian, and if (г, 01,... ,#n_1) are "spherical" coordinates
580
13. Riemannian and Semi-Riemannian Geometry
on Rn as above, then we can compose with normal coordinates centered at ρ
to obtain coordinate functions on our normal neighborhood, which we again
denote by (r, 01,..., 0n~~1). These coordinates are called geodesic
spherical coordinates or geodesic polar coordinates. As usual, the function
r is extended to be zero at the center, and in the case of hyperspherical
coordinates, the angle functions are extended to be multivalued. The resulting
"coordinates" are not really proper coordinates on the normal neighborhood
since they suffer from the usual defects. For example, r is not smooth at the
center point where it is zero, and the angles 01,..., 0n_1 become ambiguous
when r = 0. Thus a little care is need when using spherical coordinates. No
matter how we choose the 01,..., 0n~~1, the function r is the radial function
introduced earlier. Whether or not angle functions are introduced, one often
uses the notation J^ to denote the unit vector field defined as follows: If ν
is a unit vector in TPM, then
d_\ a_ d_
drL '~ ~dt
expp(iv),
t=t0
where q = expp(tov) (and ρ is the center point of the normal coordinates).
In fact, it is not hard to see that ■&- = P/ \\P\\. One might use this last
equation to define ^ in the case of an indefinite metric but note that P/ \\P\\
is undefined when ||P|| = 0 and so is undefined on the local nullcone. We
refer to -gp as the unit radial vector field. If (r, 01,..., θη~ι) are geodesic
spherical coordinates centered at some point po of a Riemannian manifold,
then by the Gauss lemma
= 0fori = l,2,...,n-l.
Exercise 13.75. Show that if a geodesic 7 : [a, b) —> Μ is extendable
to a continuous map 7 : [a, 6] —> M, then there is an ε > 0 such that
7 : [a, b) —> Μ is extendable further to a geodesic 7 : [a, b + ε) —> Μ with
%а,Ъ] =7·
Under certain conditions, geodesies can help us draw conclusions about
maps. The following result is an example and a main ingredient in the proof
of the Hadamard theorem to be given later.
Theorem 13.76. Let f : {M,g) —> (N,h) be α local isometry of serni-
Riemannian manifolds with N connected. Suppose that f has the property
that given any geodesic 7 : [0,1] —> N and ρ G Μ with f(p) = 7(0), there
is a curve 7 : [0,1] —> Μ such that ρ = 7(0) and 7 = / о 7. Then φ is a
semi-Riemannian covering.
Proof. Since any two points of TV can be joined by a broken geodesic, it is
easy to see that the hypotheses imply that / is onto.
13.4. Geodesies
581
Let U be a normal neighborhood of an arbitrary point q e N and let U С
TqN be the open set such that expq(U) = U. We will show that U is evenly
covered by /. Choose ρ G /_1(^)· Observe that Tpf : TPM -» TqN is a linear
isometry (the metrics on TpM and TqN are given by the scalar products g(p)
and h(q)). Thus Vp := Tpf~l(U) is star-shaped about 0P G TPM. Now if ν G
Vp, then by hypothesis, the geodesic *y(t) := expq(t (Tpf (v))) has a lift to a
curve 7 : [0,1] —> Μ with 7(0) = p. But since / is a local isometry, this curve
must be a geodesic. It is also easy to see that Tp/(7;(0)) = 7;(0) = Tpf (v).
It follows that υ = 7;(0) and then expp(i>) = 7(1). Thus expp is defined on
all of V. In fact, it is clear that f(exppv) = ехрдр)(Т/ {ν)) and so we see
that / maps Vp := expp(T^,) onto the set expq(U) = U. We show that Vp is a
normal neighborhood of p. Prom /oexpp = exp^p) о Tf we see that /oexpp
is a diffeomorphism on V. But then expp : Vp —> Vp is bijective. Combining
this with the fact that Tf о Гехрр is a linear isomorphism at each ν e Vp
and the fact that Tf is a linear isomorphism, it follows that Tvexpp is a
linear isomorphism. It follows that Vp is open and expp : Vp —> Vp is a
diffeomorphism. Composing, we obtain f\v = exp^(p) \ц oTf о expp\y ,
which is a diffeomorphism taking V^onto U.
Now we show that if Pi,Pj G f~l{q) and pi ^ p^ then the sets VPt and
Vp (obtained for these points as we did for a generic ρ above) are disjoint.
Suppose to the contrary that m G VPt Π VPj and let ηΡιΎΐι and ^Pjm be the
reverse radial geodesies from m to pi and pj respectively. Then f ojPim and
f °lpjm are both reversed radial geodesies from f(x) to g, and so they must
be equal. But then 7РгГП and 7Pjm are equal since they are both lifts of the
same curve and start at the same point. It follows that pi = pj after all. It
remains to prove that f~l{U) С Upef-i(q)Vp since the reverse inclusion is
obvious. Let χ G /_1(i7) and let a : [0,1] —> U be the reverse radial geodesic
from f(x) to the center point q. Now let 7 be the lift of a starting at χ and
let ρ = 7(1). Then f(p) = a(l) = g, which means that ρ G f~1(q)- On the
other hand, the image of 7 must lie in Vp and so χ G Vp. D
13.4.1. Geodesies on submanifolds. Let Μ be a semi-Riemannian sub-
manifold of M. For a smooth curve 7 : I —> M, it is easy to show using
Proposition 13.32 that VQtY = (V^y) and that we have
VdtY = VdtY + II(i,Y)
for any vector field У along 7. If У is a vector field in X(M)\M or in
3£(M), then У о 7 is a vector field along 7. In this case we shall still write
VdtY = уdtγ+7/(7, Υ) rather than Vdt (У о 7) = Vdt (У о 7)+77(7, У07).
582 13. Riemannian and Semi-Riemannian Geometry
Figure 13.4. Semi-Riemannian covering
Recall that 7 is a vector field along 7. We also have V#f7, which in this
context will be called the extrinsic acceleration (or acceleration in M).
The intrinsic acceleration (acceleration in M) is Vat7- Thus we have
Vft7 = Vft7 + //(7,7)·
Since 11(η, η) is the normal part of V^t7, we immediately obtain the
following:
Proposition 13.77. If η : I —> Μ is α smooth curve where Μ is α semi-
Riemannian submanifold of M, then η is a geodesic in Μ if and only if
V<9t7 is normal to Μ for every t £ I.
Exercise 13.78. A constant speed parametrization of a great circle in Sn(r)
is a geodesic. Every geodesic in Sn(r) is of this form.
Definition 13.79. A semi-Riemannian submanifold Μ С М is called
totally geodesic if every geodesic in Μ is a geodesic in M.
Theorem 13.80. For a semi-Riemannian submanifold Μ С М, the
following conditions are equivalent:
(i) Μ is totally geodesic;
(ii) // = 0;
(iii) For all ν G TM, the geodesic ην in Μ with initial velocity ν is such
that ην ([0, б]) С Μ for e > 0 sufficiently small;
(iv) For any curve a : I —> M, parallel translation along a induced by
V in Μ is equal to parallel translation along a induced by V in M.
Proof. (i)=>(iii) follows from the uniqueness of geodesies with a given
initial velocity.
(iii)=>(ii): Let ν G TM. Applying Proposition 13.77 to jv we see that
II(v,v) = 0. Since ν was arbitrary, we conclude that 77 = 0.
13.4. Geodesies
583
(ii)=>(iv): Suppose ν G TpM. If V is a parallel vector field with respect
to V that is defined near ρ such that V(p) = v, then VdtV = VdtV +
77(7, V) = 0 for any 7 with 7(0) = ρ so that V is a parallel vector field with
respect to V.
(iv)=>(i): Assume (iv). If 7 is a geodesic in M, then 7 is parallel along
7 with respect to V. Then by assumption 7 is parallel along 7 with respect
to V. Thus 7 is also an Μ geodesic. D
Prom Proposition 13.46, it follows that if Μ = /_1(c) is a semi-
Riemannian hypersurface, then U = V//||V/|| is a unit normal for Μ
and (/7, U) = ε = sgnM. Notice that this implies that Μ = /_1(c) is ori-
entable if Μ is orientable. Thus not every semi-Riemannian hypersurface is
of the form /_1(c). On the other hand every hypersurface is locally of this
form.
We are already familiar with the sphere £n(r), which is /_1(r2) where
Ηχ) = (χ,χ) = Σΐ=ιΗ2-
Definition 13.81. For η > 1 and 0 < υ < η, we define
S:(r) = {x€K+1:(x,x)„ = r2}.
S™(r) is called the pseudo-sphere of radius r and index v.
Definition 13.82. For η > 1 and 0 < ν < η, we define
Н?(г) = {хеЖ%{:(х,х)„ = -г2}.
77™(r) is called the pseudo-hyperbolic space of radius r and index v.
If Π is a two-dimensional plane through the origin in M^+1, and if С С П
is a conic section (ellipse, straight line, hyperbola, etc.), then we shall say
that С is a conic section in M™+1. If Q С M^+1 is a hyperquadric, then it is
easy to show that IlflQ is a conic section in Π and hence in R™+1. Problems
4 and 5 show that geodesies in hyperquadrics can be understood once we
understand the case of 5^(1). With this in mind, we have
Proposition 13.83. All geodesies in S™(r) are parametrizations of the
connected components of sets of the form Π Π S™(r), where Π is a plane.
a) If у is a timelike geodesic in S™(r), then it is a parametrization of
one branch of a hyperbola.
b) If 7 is a null geodesic in S™(r), then it is a parametrization of a
straight line.
c) If j is a spacelike geodesic in S™(r), then it is a parametrization of
an ellipse (and hence periodic).
584
13. Riemannian and Semi-Riemannian Geometry
Timelike geodesic
Spacelike geodesic
Lightlike geodesic
Figure 13.5. Geodesies in 5?(1)
Proof. We follow [ONI]. Let ρ G SJ}(r) be given and let Π be a plane in
M™+1 containing 0 and p. We will show that the conic section Π Π S™(r) can
be parametrized as a geodesic. We identify the type of conic section and we
argue that these account for all geodesies on S™(r). We restrict the scalar
product g on M™+1 to Π. Since ρ is spacelike from the definition of 5™, we
only have three possibilities for g\u. We handle these in turn:
(1) g\u is positive definite. Choose an orthonormal basis e\,e2 for Π.
Then a point ae\ +be2 of Π is also on S? only if a2 + b2 = r2. Thus S%(r) ПП
is a circle in Π and hence an ellipse in RJJ+1. Now j(t) := r cost e\ +r sint e2
is a parametrization of S™(r) Π Π, and since (7,7)^ = r2, it is a constant
speed spacelike curve. But also Vat7 = — P07 so Vat7 is normal to S™ and
thus 7 is a geodesic.
(2) g\u is nondegenerate with index 1. Choose an orthonormal basis
eo,ei for Π such that (eo,eo) = —1 and re\ = ρ for some r. Observe that
a point aeo + be 1 of Π is also on S™ if and only if -a2 + b2 = r2. Thus
S™(r) Π Π is both branches of a hyperbola. We can parametrize the branch
through ρ as
7(i) := (r sinh t) eo + (r cosh t) e\.
This time (7,7)^ = —r2, so 7(2) is timelike. Furthermore, Vat7 = Ρ °7, so
Vat7 is normal to S™ and thus 7 is a geodesic.
(3) g\u is degenerate. In this case, the null space οι g\u must be of
dimension 1. We choose a nonzero null vector υ so that ρ, υ is a basis for Π.
Then a point ap + bv of Π is also on S™ only if α = ±1 which means that
S™(r) Π Π is a pair of lines. The line through ρ is parametrized as t ь-> p + tv
and is a geodesic of M^+1 contained in S™ and so is certainly a geodesic of
13.5. Riemannian Manifolds and Distance
585
Finally, we argue that, up to reparametrization, this accounts for all
geodesies in S™. Indeed, if 7 is such a geodesic, then υ = 7(0) is based at
ρ = 7(0) and there is a unique plane Π through the origin containing ρ and
v. By uniqueness, 7 must be a reparametrization of one of the geodesies
already discovered above. D
13.5. Riemannian Manifolds and Distance
In this section we consider only Riemannian manifolds (definite metrics).
Then we have the notion of the length of a curve (Definition 13.5). Using
this we can then define a distance function (a metric in the sense of "metric
space") as follows: Let p,q G M. Consider the set Path(p,q) consisting of
all piecewise smooth1 curves that begin at ρ and end at q. We define the
Riemannian distance from ρ to q as
(13.8) dist(p, q) = inf{L(c) : с е Path(p, q)}.
On a general Riemannian manifold, dist(p, q) = r does not necessarily
mean that there must be a curve connecting ρ to q having length r. To
see this, just consider the points (—1,0) and (1,0) on the punctured plane
M2\{0}.
Definition 13.84. If ρ e Μ is a point in a Riemannian manifold and R > 0,
then the set Вц(р) (also denoted B(p,R)) defined by Вц(р) = {q e Μ :
dist(p, q) < R} is called an open geodesic ball centered at ρ with radius R.
It is important to notice that unless R is small enough, Вц(р) may not
be homeomorphic to a ball in a Euclidean space. To see this just consider a
ball of large radius on a circular cylinder of small diameter.
Proposition 13.85. Let U be α normal neighborhood of a point ρ in a
Riemannian manifold (M,g). If q G U and if η : [0,1] —> Μ is the radial
geodesic such that 7(0) = ρ and 7(1) = q, then 7 is the unique shortest curve
in U (up to reparametrization) connecting ρ to q.
Proof. Let α be a curve connecting ρ to q (refer to Figure 13.6).Without
loss of generality we may take the domain of a to be [0,6]. Let ^ be the
radial unit vector field in U. Then if we define the vector field R along a
by ί ι-> §p\a(t), we may write ά = (/?, a)R + N for some field TV normal to
1 Recall that by our conventions, a piecewise smooth curve is assumed to be continuous.
586
13. Riemannian and Semi-Riemannian Geometry
R (but note that N(0) = 0). We now have
L(a)= f {a,a)l'2dt= [ [(R,a)2 + (N,N)]1/2 dt
Jo Jo
> [ \(R,a)\dt> [ (R,a)dt = [ ^ (г о a) dt
Jo Jo Jo dt
= r(a(b)) = r(q).
On the other hand, if υ = 7(0), then r(q) = J0 \\v\\ dt = JQ (7,7)ly/2 dt so
L(a) > 1/(7). Now we show that if L(a) = 1/(7), then α is a reparametriza-
tion of 7. Indeed, if L(a) = £/(7), then all of the above inequalities must
be equalities so that TV must be identically zero and ^ (г о a) = (/?, a) =
|(i?,ά)|. It follows that ά = (R,a)R = (|(гоа))й, and so a travels
radially from ρ to q and must be a reparametrization of 7. D
Figure 13.6. Normal neighborhood of ρ
It is important to notice that the uniqueness assertion of Theorem 13.85
only refers to curves with image in U. This is in contrast to the proposition
below.
Proposition 13.86. Letpo be a point in a Riemannian manifold M. There
exists a number £o(p) > 0 such that for all ε, 0 < ε < εο(ρ) we have the
following:
(i) The open geodesic ball Β(ρο,ε) is normal and has the form
Β(ρο,ε) = exppo{v e TP0M : \v\ < ε} .
(ii) For any ρ 6 Β(ρο,ε), the radial geodesic segment connecting po to
ρ is the shortest curve in M, up to parametrization, from po to p.
(Note carefully that we now mean the shortest curve among curves
into Μ rather than just the shortest among curves with image in
B(po,e)J
13.5. Riemannian Manifolds and Distance
587
Proof. Let U С TP0M be chosen so that U = exppo U is a normal
neighborhood of pq. Then for sufficiently small ε > 0 the ball
β(0,ε) := {ν eTP0M : ||v|| < ε}
is a starshaped open set in /7, and so Αρο,ε = βχρρο(5(0,ε)) is a normal
neighborhood of po· From Proposition 13.85 we know that the radial
geodesic segment σ from po to ρ is the shortest curve in Αρο^ε from po to p.
This curve has length less than ε. We claim that any curve from po to ρ
whose image leaves APOy£ must have length greater than ε. Once this claim
is proved, it is easy to see that
APQ,e = Β(ρ0,ε) = {peM : dist(po,p) < ε}
and that (ii) holds. Now suppose that α : [α, Ь] —>· Μ is a curve from po
to ρ which leaves Αρο,ε. Then for any r > 0 with r < ε, the curve α must
meet the set S(r) := exppo({i> G TP0M : ||i>|| = r}) at some first parameter
value ίχ G [a, 6]. Then ai := a|[a,ti] ^es *n Αρο,ε, and Proposition 13.85 tells
us that L(a) > L(a|rati) > r. Since this is true for all r < ε, we have
L(a) > ε, which is what was claimed. D
Theorem 13.87 (Distance topology). Given a Riemannian manifold,
define the distance function dist as before. Then (M, dist) is a metric space,
and the metric topology coincides with the manifold topology on M.
Proof. To show that dist is a true distance function (metric) we must prove
that
(1) dist is symmetric; dist(p, q) = dist(g,p);
(2) dist satisfies the triangle inequality dist(g,p) < dist(p, x) + dist(x, q);
(3) dist(p,g) >0; and
(4) dist(p, q) = 0 if and only if ρ = q.
Now, (1) is obvious, and (2) and (3) are clear from the properties of
the integral and the metric tensor. For (4) suppose that ρ φ q. Then since
Μ is Hausdorff, we can find a normal neighborhood U of ρ that does not
contain q. In fact, by the previous proposition, we may take U to be of the
form Б(р, ε). Since (by the proof of the previous proposition) every curve
starting at ρ and leaving Β(ρ,ε) must have length at least ε/2, we see that
dist(p,g) >ε/2. D
By definition a curve segment in a Riemannian manifold, say с : [α, b] —>
Μ, is a shortest curve if L(c) = dist(c(a), c(b)). We say that such a curve is
(absolutely) length minimizing. Such curves must be geodesies.
Proposition 13.88. Let Μ be a Riemannian manifold. A length
minimizing curve с : [α, b] —> Μ must be an (unbroken) geodesic.
588
13. Riemannian and Semi-Riemannian Geometry
Proof. There exist numbers U with а = to < t\ < · · - < t^ = b such that
for each subinterval [^,^+1], the restricted curve c\u t 1 has image in a
totally star-shaped open set. Thus since cL t 1 is minimizing, it must be
a reparametrization of a unit speed geodesic (use the uniqueness part of
Proposition 13.85). Thus there is a reparametrization of с that is a broken
geodesic. But this new reparametrized curve is also length minimizing, and
so by Problem 1 it is smooth. D
13.6. Lorentz Geometry
In this section we define and discuss a few aspects of Lorentz manifolds.
Lorentz manifolds play a prominent role in physics and are often singled
out for special study. We discuss the local length maximizing property of
timelike geodesies in a Lorentz manifold and derive the Lorentzian analogue
of Proposition 13.85.
Definition 13.89. A Lorentz vector space is a scalar product space with
index equal to one and dimension greater than or equal to 2. A Lorentz
manifold is a semi-Riemannian manifold such that each tangent space is
a Lorentz space with the scalar product given by the metric tensor.
Under our conventions, the signature of a Lorentz manifold is of the form
(-1,1,,..., 1,1).2
Each tangent space of a Lorentz manifold is a Lorentz vector space, and
so we first take a closer look at some of the distinctive features of Lorentz
vector spaces. Let us now agree to classify the zero vector in a Lorentz
space as spacelike. For Lorentz spaces, we may classify subspaces into three
categories:
Definition 13.90. Let V be a Lorentz vector space such as a tangent space
of a Lorentz manifold. A subspace W С V is called
(1) spacelike if <?|w is positive definite (or if W is the zero subspace);
(2) timelike if <?|w nondegenerate with index 1;
(3) lightlike if g\w is degenerate.
Thus a subspace falls into one of the three types, which we refer
to as its causal character.
If we take a timelike vector г; in a Lorentz space V, then Rv, the space
spanned by г;, is nondegenerate and has index 1. By Lemma 7.47, v1- is
nondegenerate and V = Rv Θ v^. Since 1 = ind (V) = ind (Rv) + ind (f1-),
it follows that ind (г;-1) = 0, so that v1- is spacelike. This little observation
is useful enough to set out as a proposition.
2Some authors use (1,-1,...,—1,-1), but this does not really change the geometry.
13.6. Lorentz Geometry
589
Figure 13.7. Causal character of a subspace
Proposition 13.91. If У is a Lorentz vector space and ν is a timelike
element, then v1- is spacelike, and we have the orthogonal direct sum V =
Exercise 13.92. Show that if W is a subspace of a Lorentz space, then W
is timelike if and only if W1- is spacelike.
Exercise 13.93. Suppose that v,w are linearly independent null vectors
in a Lorentz space V. Show that (v,w) Φ 0. [Hint: Use an orthonormal
basis to orthogonally decompose; V = Meo Θ Ρ, where eo is timelike and
where (·, ·) is positive definite on P. Suppose (v,w) = 0; write υ = aeo + p\
and w = ββο + P2- Then show that (рьрг) — ^/3, {pi,Pi) = a2 = β2 and
1(рьР2>| = IIpiII Hp2||.]
Lemma 13.94. Let W be a subspace of a Lorentz space. Then the following
conditions are equivalent:
(i) W is timelike and so a Lorentz space in its own right.
(ii) There exist null vectors v,w G W that are linearly independent.
(iii) W contains a timelike vector.
Proof. Suppose (i) holds. Let ei,...,em be an orthonormal basis for W
with e\ timelike. Then e\ +β2 and e\ — β2 are both null and, taken together,
are a linearly independent pair so that (ii) holds. Now suppose that (ii) holds
and let г>, it; be a linearly independent pair of null vectors. By Exercise 13.93
above, either v+w οτν-w must be timelike so we have (iii). Finally, suppose
(iii) holds and υ G W is timelike. Since v1- is spacelike and W1- С ν1, we
590
13. Riemannian and Semi-Riemannian Geometry
see that W1- is spacelike. But then W is timelike by Exercise 13.92 so that
(i) holds. D
Exercise 13.95. Use the above lemma to prove that if W is a nontrivial
subspace of a Lorentz space, then the following three conditions are
equivalent:
(i) W contains a nonzero null vector but no timelike vector.
(ii) W is lightlike,
(iii) The intersection of W with the nullcone is one-dimensional.
Definition 13.96. The timecone determined by a timelike vector ν is the
set C(v) := {w e V : (v,w) < 0}.
In Problem 6 we ask the reader to show that timelike vectors ν and w
in a Lorentz space V are in the same timecone if and only if (v,w) < 0.
Exercise 13.97. Show that there are exactly two timecones in a Lorentz
vector space whose union is the set of all nonzero timelike vectors. Describe
the relation of the nullcone to the timecones.
Now we come to an aspect of Lorentz spaces that underlies the twins
paradox of special relativity.
Proposition 13.98. Ifv,w are timelike elements of a Lorentz vector space
then we have the backward Schwartz inequality
\(v,w)\ > \\v\\ \\w\\,
with equality only if ν is a scalar multiple of w. Also, if ν and w are in the
same timecone, then there is a uniquely determined number α > 0, called
the hyperbolic angle between ν and w, such that
(v,w) = — \\v\\ \\w\\ cosha.
Note: The minus sign appears because of our convention that (г;, г;) =
— 1 for timelike vectors.
Proof. We may write w = av + ζ where ζ e v1-. We have
а2 (г>, v) + (z, z) = (w, w) < 0.
Using this and recalling that (г;, г;) < 0, we have
(v,w)2 = a2(v,v) = {(w,w) - (z,z))(v,v)
> (v,v) (w,w) = \\v\\ \\w\\.
13.6. Lorentz Geometry
591
Equality holds exactly when (z,z) = 0, but since ζ G гЛ, this implies that
ζ = 0 so w = av. Using Problem 6, we see that since ν and w are in the
same timecone, we have (v,w) < 0, and hence
IMI IIHI ~~
The properties of the function cosh now give a unique number a > 0 such
that (v,w) = — \\v\\ \\w\\ cosh α as required. D
Corollary 13.99. If v,w are timelike elements of a Lorentz vector space
which are in the same timecone, then we have the backward triangle
inequality:
\\v\\ + \\w\\ < \\v + w\\.
Equality holds only if ν is a scalar multiple ofw.
Proof. Since (v,w) < 0 by hypothesis, we have — (v,w) > \\v\\ \\w\\ by the
proposition. Then
(IMI + IHI)2 = \\v\\2 + 2 \\w\\ \\v\\ + \\w\\2
< \\v\\ —2(v,w) + \\w\\ =||г; + гу|| .
Equality happens only if — (v,w) = \\v\\ \\w\\, which, by the previous
proposition, means that г; is a scalar multiple of w. D
In relativity theory, spacetime (the set of all "idealized" possible events)
is modeled as a 4-dimensional Lorentz manifold and the paths of massive
bodies are to be timelike curves. But we have yet to talk about what
distinguishes the past from the future! In each tangent space we have two
timecones and we could arbitrarily choose one of them to be the future
timecone. But what we really want is a smooth way of choosing a future
timecone in each tangent space. This leads to the notion of time orientabil-
ity. First we say that a vector field X on a Lorentz manifold Μ is timelike
if X(p) is timelike for each ρ G M.
Definition 13.100. A Lorentz manifold Μ is said to be time-orientable
if and only if there exists a timelike vector field X G X(M). A time
orientation of Μ is a choice of timecone C(p) G TVM for each ρ such that there
exists a timelike X G X(M) with Xp G C(p) for each p. In the latter case,
C(p) is referred to as the positive or future timecone at p. The other
timecone at TVM is called the negative timecone. Timelike vectors in the
positive timecone are said to be future pointing (and those in the negative
timecone are past pointing).
Definition 13.101. A lightlike vector in a tangent space TpM of a time-
oriented Lorentz manifold Μ is said to be future pointing if it is the limit of
592
13. Riemannian and Semi-Riemannian Geometry
a sequence of future pointing timelike vectors in TPM. Thus the lightcone
(nullcone) in TpM is partitioned into future lightcone and past lightcone.
Based on these definitions we can speak of timelike or lightlike curves
as being either future pointing or "past pointing". Time orientability is
certainly a global condition since we need a choice of timecone in every tangent
space and this choice must be made smoothly. However, the following
exercise shows that the smoothness condition can be described in terms of local
vector fields:
Exercise 13.102. Suppose that it is possible to choose a timecone C(p) in
every tangent space TPM of a Lorentz manifold Μ in such a way that in a
neighborhood of each point there is a local smooth vector field with values
in these timecones. Show that this implies that Μ is orientable. [Hint: Use
a partition of unity argument.]
Exercise 13.103. Show that S?(r) := {p G Щ : (p,p) = r2} is a time-
orientable Lorentz manifold. [Hint: consider the restriction to S™ (r) of the
first coordinate vector field from Щ.]
Now let us consider timelike curves in a Lorentz manifold M. Since we
want to include piecewise smooth curves, we have to decide what timelike
should mean. If 7 : 7 —> Μ is a piecewise smooth curve and U G I is a
parameter value at which 7 is not smooth, then what condition is appropriate
if we are to refer to the curve as a timelike curve? We consider the one-sided
limits
Ί(4) := lim7(ti + ε) and j{t~) := limj{U - ε).
For many purposes, the following definition is appropriate:
Definition 13.104. Let Μ be a Lorentz manifold. A piecewise smooth
curve 7 : I —> Μ is called timelike if
(i) 7 is timelike where it is smooth, and
(ii) for every U where 7 is not smooth, we have that 7(t/") and 7(£~)
are timelike and the following further condition holds at each such
U:
(i(tt)Mt-))<o.
Thus, for timelike curves, 7(t^") are 7(t~) are in the same timecone.
Following [ONI], we next prove a useful technical lemma.
Lemma 13.105. Let ρ be a point in a Lorentz manifold Μ and U a normal
neighborhood of p. Let U be the corresponding starshaped open set in TpM
with U = expp(/7). Let 7 : [0,b] —> U С ТРМ be a piecewise smooth curve
such that a := exppo~/ : [0,b] —> Μ is timelike (in the sense of Definition
13.6. Lorentz Geometry
593
13.104/ Then the image ofj is contained in a single timecone ofTpM and
(ά,Ρ) <0.
Proof. Let us first handle the case where 7 is smooth. Then since 7(0)
is timelike, 7 is initially in one of the timecones which we denote by C(p).
Here and below, "initially" is taken to mean "for all sufficiently small positive
parameter values t". Let Ρ be the position vector field on TPM and let Ρ be
the local position vector field which is expp-related to Ρ and defined on the
normal neighborhood U. Note that Ρ is timelike and outward radial at each
point of UnC(p). Thus (7, P) is initially negative. Letting q(x) := (я, я) in
TpM and considering TpM as a Lorentz manifold itself we have gradq = 2P
and hence ^q 07 = 2(7, P). The Gauss lemma (Lemma 13.72) gives
(7,Ρ> = (ά,Ρ>,
which implies that (ά, Ρ) and hence ^q ο η are initially negative. For any
t > 0 such that η(ί) is in C(p), the vector P(7(i)), and hence P(a(£)), must
be timelike. For such t, (ά, Ρ) < 0 which implies that (7, P) < 0 and hence
^q о 7 < 0. So q о 7 starts out negative and goes down hill as long as j(t)
is in C(p). Since 7 can only exit C(p) by reaching the nullcone (or 0) where
q vanishes, we see that 7 must remain inside C(p).
Now we consider what happens if 7 is timelike but merely piecewise
smooth. The first segment remains in C(p), and at the first parameter value
t\ where 7 fails to be smooth, we must have (7(iJ~), P) < 0. But then by the
Gauss lemma again (ά(£["),Ρ) < 0. The technical restriction of Definition
13.104 forces (ά(^),Ρ) < 0 so that a(tj") G C{p). Applying the Gauss
lemma gives (7(^1"), P) < 0 and so ^q°7 cannot change sign at t\. We are
now set up to repeat the argument for the next segment. The result follows
inductively. D
The following proposition for Lorentz manifolds should be compared to
Proposition 13.85 proved for Riemannian manifolds. In this proposition we
find that the geodesies are locally longer than nearby curves.
Proposition 13.106. Let U be a normal neighborhood of a point ρ in a
Lorentz manifold. If the radial geodesic 7 connecting ρ to q EU is timelike,
then it is the unique longest geodesic segment in U that connects ρ to q.
Once again, uniqueness is up to reparametrization.
Proof. Let U be related to U as usual. Take any timelike curve a : [0,6] —>
U segment in U that connects ρ to q. By the previous lemma, β := exp"1 о а
stays inside a single timecone C(p) and so also inside C(p)C\U. Thus a stays
inside expp(C(p)DU) where it is timelike and where the field R = (P/r)oa is
594
13. Riemannian and Semi-Riemannian Geometry
a unit timelike field along a. We now seek to imitate the proof of Proposition
13.85. We may decompose ά as
a = -(R,a)R + N,
where TV is a spacelike field along a that is orthogonal to R. We have
|H| = yf^aj = yf(R,a)2-(N,N) < \(R,a)\.
Recall that q(·) = (·, ·) and q:=qo exp"1. Since r = y/11^ and so gradr =
—P/r, we have (gradr)oq = —R. By the previous lemma (ά, R) is negative.
Then
ι/, г^м /T^ .4 d(roa)
|(а,Д>| = -(Д,а) = -^-^.
Thus we have
L(a)= [ ||a(t)||dt<r(g) = L(7).
Jo
If L(a) = £(7), then N = 0 and we argue as in the Riemannian case to
conclude that a is the same as 7 up to reparametrization. D
Recall that the arc length of a timelike curve is often called the curve's
proper time and is thought of as the intrinsic duration of the curve. We
may reparametrize a timelike geodesic to have unit speed. The parameter
is then an arc length parameter, which is often referred to as a proper
time parameter. We may restate the previous theorem to say that the
unit speed geodesic connecting ρ to q in U is the unique curve of maximum
proper time among curves connecting ρ to q in U.
13.7. Jacobi Fields
Once again we consider a semi-Riemannian manifold {M,g) of arbitrary
index. We shall be dealing with smooth two-parameter maps h : (—e,e) x
[a,b] —> M. The partial maps t н-> hs(t) = h(s,t) are called the longitudinal
curves, and the curves s ь-> h(s, i) are called the transverse curves. Let a be
the center longitudinal curve t \-> ho{t). The vector field along a defined by
V(t) = 2^| =Qhs(t) is called the variation vector field along a. We will
use the following important result more than once:
Lemma 13.107. Let Υ be a vector field along the smooth map h : (—e,e) x
[a, 6] -» M. Then
VasVdtY - VdtVdsY = R(dsh, dth)Y.
Proof. If one computes in a local chart, the result falls out after a mildly
tedious computation, which we leave to the curious reader. D
13.7. Jacobi Fields
595
Suppose we have the special situation that, for each s, the partial maps
t h-> hs(t) are geodesies. In this case, let us denote the center geodesic
t h-> ho(t) by 7. We call h a variation of 7 through geodesies. Let h be such
a special variation and V the variation vector field. Using Lemma 13.107
and the result of Exercise 13.11 we compute
VdtVdtV = VdtVdtdsh = VdtVdsdth
= Vdydtdth + R(dth,dsh)dth
= R(dth,dsh)dth
and evaluating at s = 0 we get V^ V^ V(i) = i?(7(i), V(i))7(i). This
equation is important and shows that V is a Jacobi field as per the folowing
definition:
Definition 13.108. Let 7 : [a, 6] —> Μ be a geodesic and let J G 3£7(M)
be a vector field along 7. The field J is called a Jacobi field if
VftVAJ = .R(7(t),J(t))7(i)
for all i G [a, 6].
In local coordinates, we recognize the above as a second order system of
linear differential equations and we easily arrive at the following
Theorem 13.109. Let {M,g) and the geodesic 7 : [a,b] —> Μ be as above.
Given wi,W2 G ΤΊ^Μ, there is a unique Jacobi field JWl^W2 g X7(M) such
that J (a) = w\ and V#f.J (a) = vj^- The set Jac (7) of all Jacobi fields along
η is a vector space isomorphic to Γ7(α)Μ χ ΤΊ^Μ.
We now examine the more general case of a Jacobi field JWl }W2 along a
geodesic 7 : [a, b] —> M. First notice that for any curve a : [a, b] —> Μ with
|(ά(ί), ά(ί))| > 0 for all t G [a, 6], any vector field Υ along a decomposes into
an orthogonal sum YT+ Y±. This means that У is a multiple of ά and
that Y1- is normal to ά. If 7 : [a, b] —> Μ is a geodesic, then Vdt Ух is also
normal to 7 since 0 = |(УХ,7> = ΦκΥ\ί) + (Υ\^Ί) = <^АУХ,7>-
Similarly, У&УТ is parallel to 7 all along 7.
Theorem 13.110. Let 7 : [a, b] —> Μ be a geodesic segment.
(i) If Υ G X7(M) zs tangent to 7, then Υ is a Jacobi field if and only
ifV2dY = 0 a/ong 7. In this case, Y(t) = (at + 6)7(i).
(ii) If J is a Jacobi field along 7 and there are some distinct £1, £2 G [a, b]
той J(ii)±7(ii) and J(i2)±7(i2), iuen J(i)±7(i) /or a//1 G [a, 6].
(iii) If J is a Jacobi field along 7 and there is some to G [a, b] wii/i
J(i0)±7(io) and Vfc J(t0)-L7(to), i^en J(i)±7(i) /or a//1 G [a, 6].
(iv) // 7 zs no£ a null geodesic, then Υ is a Jacobi field if and only if
both YT and Y1- are Jacobi fields.
596
13. Riemannian and Semi-Riemannian Geometry
Proof, (i) Let Υ = /7. Then the Jacobi equation reads
Vlt/7(i) = #(7(i),/7(i))7(i) = 0.
Since 7 is a geodesic, this implies that /" = 0 and (i) follows.
(ii) and (iii) We have ^(J,7) = (R{>y{t),J{t))j{t),j{t)) = 0 (from the
symmetries of the curvature tensor). Thus (J(i),7(i)) = at + b for some
a, 6 GK. The reader can now easily deduce both (ii) and (iii).
(iv) The operator V|f preserves the normal and tangential parts of
Υ. We now show that the same is true of the map Υ ь-> i?(7(i),y)7(i).
Since we assume that 7 is not null, we have YT = /7 for some 7. Thus
R{>y{t),YT)>y(t) = /2(7(t),/7(t))7(t) = 0, which is trivially tangent to 7(4).
On the other hand, (i?(7(i), Y'J-(£))7(£),7(£)) = 0 by symmetries of the
curvature tensor. We have
(vity)T + (vity)x = v%y = Rm),Y(tm(t)
= R№),YT(m(t)+R(mYHm(t)
= o+Rm),Y±(m(t).
So the Jacobi equation V|f Y(t) = i?(7(i), Y(t))"y(t) splits into two equations
vltrT(i) = 0,
and the result follows from this. D
Corollary 13.111. Let 7 = 7^ and J7'™ &e a5 a&oi;e. Γ/ien J^rv(t) =
rt%(t). Ifw±v, then (J°>w{t),<yv(t)) = 0 for all t G [0,6]. Furthermore,
every Jacobi field J°>w along expv tv with J°>w(0) = 0 has the form J°'w :=
rt% + J0'™1, where w = VdtJ{0) = rv + гиь wi_U; and J°>Wl(t)±.<yv(t) for
allte [0,6].
The proof of the last result shows that a Jacobi field decomposes into a
parallel vector field along 7, which is just a multiple of the velocity 7, and
a "normal Jacobi field" J1-, which is normal to 7 at each of its points. Of
course, the important part is the normal part since the tangential part is
merely the infinitesimal model for a variation through geodesies which are
merely reparametrizations of the 5 = 0 geodesic. Thus we focus attention
on the Jacobi fields that are normal to the geodesies along which they are
defined. Thus we consider the Jacobi equation V|f J(t) = i?(7(i), J(i))7(i)
with initial conditions such as in (ii) or (iii) of Theorem 13.110.
Exercise 13.112. For υ G TpM, let v1- := {w G TPM : (ги,г;) = 0}. Prove
that the tidal operator Ry : w \-> Rv,wv maps v1- to itself.
In light of this exercise, we make the following definition.
13.7. Jacobi Fields
597
Definition 13.113. For υ G TPM, the (restricted) tidal force operator
Fv : ν1- —Ϊ v1- is the restriction of Ry to v1- С TpM.
Notice that in terms of the tidal force operator the Jacobi equation for
normal Jacobi fields is
V%J(t) = FW)(J(t)) for alii.
If J is the variation vector field of a geodesic variation, then it is an
infinitesimal model of the separation of nearby geodesies. In general relativity,
one thinks of a one-parameter family of freely falling particles. Then V<9t J
is the relative velocity field and V|f J is the relative acceleration. Thus the
Jacobi equation can be thought of as a version of Newton's second law with
the curvature term playing the role of a force.
Proposition 13.114. For υ G TpM, the tidal force operator Fv : v1- —> v1-
is self-adjoint and Trace (Fv) = -Ric(t;,v).
Proof. First, (Fvwi,w2) = (Ftv,Wlv,w2) = (Ftv,W2v,wi) = (Fvw2,wi) by
(iv) of Theorem 13.19. The proof that Trace Fv = -Ric(v,v) is easy for
definite metrics but for indefinite metrics the possibility that ν may be a null
vector involves a little extra work. If ν is not null, then letting e2, - · ·, en be
an orthonormal basis for Vх- we have
Ric(t>, v) = -^6i (Rv,etv, ei) = - ^ ει {Fvei, e*) = - Trace Fv.
If υ is null, then we can find a vector w such that (ги, г;) = — 1 and ги, ν span
a Lorentz plane L in TpM. Define e\ := (v + w) /л/2 and e2 := (v — w) /y/2.
One checks that e\ is timelike while e2 is spacelike. Now choose an
orthonormal basis ез,..., en for lA С ν1- so that ei,..., en is an orthonormal basis
for TpM. Then we have
Ric(v, г;) = {Rv^v, ei> - (i^,e2^ e2> - ^ e{ (Rv,eiv, e»).
г>2
But (Rv.eiV^i) — \ {Rv,wV,w) = (Rv,e2vie2) and so we are left with
Ric(v,v) = ~Y^ei (Rv,eiv,ei) = ~Σει (FvCi,ei).
г>2 г>2
Since ν, ез,..., en is an orthonormal basis for v1- and Fvv = 0, we have
Ric(v, v) = - ^2 £г (Fvei, ^)
г>2
= - (Fvv, v) - ^2 £г (Fvei, e^ = - Trace Fv. D
г>2
Definition 13.115. Let 7 : [a, 6] —> Μ be a geodesic. Let Jb(7,tt,b) denote
the set of all Jacobi fields J such that J(a) = J(b) = 0.
598
13. Riemannian and Semi-Riemannian Geometry
Definition 13.116. Let 7 : [a,b] —> Μ be a geodesic. If there exists a
nonzero Jacobi field J G Jb(7?a>b)? then we say that 7(a) is conjugate to
7(b) along 7.
Prom standard considerations in the theory of linear differential
equations it follows that the set *7о(7?а?Ь) is a vector space. The dimension of
the vector space Jo (7? a,b) is the order of the conjugacy. Since the Jacobi
fields in Jo (7, a> b) vanish twice, and, as we have seen, this means that such
fields are normal to 7 all along 7, it follows that the dimension of Jo(7> a, b)
is at most η — 1, where η = dim Μ. We have seen that a variation through
geodesies is a Jacobi field; so if we can find a nontrivial variation h of a
geodesic 7 such that all of the longitudinal curves t \-> hs(t) begin and end
at the same points 7(a) and 7(b), then the variation vector field will be
a nonzero element of JO(7,a, b). Thus we conclude that 7(a) is conjugate
to 7(b). We will see that we may obtain a Jacobi field by more general
variations, where the endpoints of the curves meet at time b only to first
order.
Let us bring the exponential map into play. Let 7 : [0,6] —> Μ be a
geodesic as above. Let υ = 7(0) G TpM. Then 7 : t ь-> expptv is exactly
our geodesic 7 which begins at ρ and ends at q at t = b. Now we create a
variation of 7 by
/1(5, t) = expp t(v + sw),
where w G TPM and s ranges in (—€,c) for some sufficiently small e. We
know that J(t) = g^ I 0 /1(5, t) is a Jacobi field, and it is clear that J(0) :=
J^|5=0Ms,0) — 0. If Wbv is the vector tangent in Tbv{TpM) which canoni-
cally corresponds to w, in other words, if w^ is the velocity vector at s = 0
for the curve s \-> b (v + sw) in TpM, then
r\ I
expp b(v + sw) = Тьу expp(wbv).
15=0
™=£
h(s,b) =
s=0 ds
(We have just calculated the tangent map of expp at χ = bv\) Also,
VftJ(O) = Vdt — exppt{v + sw)
5=0,i=0
exppt(v + sw).
ρ
t=o
But X(s) := §i\t=0 expp t(v+sw) = v+sw is a vector field along the constant
curve t \-+ p, and so by Exercise 12.42 we have V<9s|5=0.X(s) = X'{0) = w.
The equality J(b) = Г^ехрр(г;^) is important because it shows that if
T^expp : Tbv(TpM) —> ΤΊ^Μ is not an isomorphism, then we can find a
vector w^ € Tbv(TpM) such that Г^ехрр(г^^) = О. But then if w is the
vector in TpM which corresponds to it;^ as above, then for this choice of w,
13.8. First and Second Variation of Arc Length
599
the Jacobi field constructed above is such that J(0) = J(b) = 0 so that 7(0)
is conjugate to 7(b) along 7. Also, if J is a Jacobi field with J(0) = 0 and
V<9t J(0) = w, then this uniquely determines J and it must have the form
Jj I 0 expp t(v + sw) as above.
Theorem 13.117. Let 7 : [0,6] -^ Μ be α geodesic. Then the following are
equivalent:
(i) 7(0) is conjugate to 7(b) a/ong 7.
(ii) Γ/iere zs a nontrivial variation h of 7 through geodesies that all
start atp = 7(0) sucft iftai J(b) := ψ8 (0, Ь) = 0.
(iii) //г; = 7(0), ί/ien 7&vexpp zs singular.
Proof. (ii)=>(i): We have already seen that a variation through geodesies
is a Jacobi field J and that if (ii) holds, then by assumption J(0) = J(b) = 0,
and so we have (i).
(i)=>(iii): If (i) is true, then there is a nonzero Jacobi field J with
J(0) = J(b) = 0. Now let w = VdtJ(0) and h(s,t) = exppt(v + sw). Then
/1(5, i) is a variation through geodesies and 0 = J(b) = §^\0exPpKv + sw)
= Tbv expp(wbv) so that T&v expp is singular.
(iii) ==>(ii): Let υ = 7(0). If T^expp is singular, then there is a w
with TbvexppWbv = 0. Thus the variation h(s,t) = exppt(v + sw) does the
job. D
13.8. First and Second Variation of Arc Length
Let us restrict attention to the case where a is either spacelike or timelike
(not necessarily geodesic). This is just the condition that |(ά(ί),ά(ί))| > 0.
Let ε = +1 if a is spacelike and ε = —1 if a is timelike. We call ε the sign
of a and write ε = sgna. Consider the arc length functional defined by
L(a)= [ {ε(ά{ή,ά{ή))1/2 dt= f \(a{t),a{t))\1/2 dt.
J a J a
If h : (—6, e) χ [a, b] —> Μ is a variation of a as above with variation vector
field V, then formally V is a tangent vector at a in the space of curves
[a, b] —> M. By a simple continuity and compactness argument we may
choose a real number e > 0 small enough that (/is(t),/is(£)) > 0 for all
s G (—6, e). Then we have the variation of the arc length functional defined
by
ЩАУ):=1
L(h3) := ±
s=0 ds
[ (e(hs{t),hs(t)))1 2 dt.
s=0 J а Ч '
600
13. Riemannian and Semi-Riemannian Geometry
Thus, we are interested in studying the critical points of L(s) := L(hs), and
so we need to find 1/(0) and I/'(0). For the proof of the following proposition
we use the result of Exercise 13.11 to the effect that Vdsdth = Vdtdsh.
Proposition 13.118. Let h : (—e, e) χ [α, b] —ϊ Μ be α variation of a curve
a := ho such that \(hs{t),hs{t))\ > 0 for all s G (—€,c). Then
rb
Lf{s) = / e(Vd&h{s^),dth{s,t)){e(dth{s,t),dth{s,t)))-l/2dt.
J a
Proof. We have
rb ,
L'(s)=l/ \hs{t)
-/'
J a
dt
f
J a
(e(ha(t),h8(t))) dt
ίΰ · · 1 / · · Ч-1/2
= J 2е(ЧдМ*)МЩ {e(hs(t)Mt))) dt
rb
s(Vdsdth(s, t),dth{s, t)) {e(dth(s, t),dth{s, t)))'1'2 dt
rb
= ε (Vdtdsh{s, t),dth{s, t)) {e(dth{s, t), dth{s, t)))'1'2 dt. D
J a
Corollary 13.119. We have
5L\a(V) = L'(0)=e i\vdtV(t),a(t))(£(a(t),a(t))r^ dt.
J a
Let us now consider a more general situation where a : [a, b] —> Μ is
only piecewise smooth (but still continuous). Let us be specific by saying
that there is a partition α = to < t\ < · · · < tk < ifc+i = b so that a
is smooth on each [i^ij+i]. A variation appropriate to this situation is a
continuous map h : (—e, e) x [a,b] —> Μ with /i(0,t) = a(t) such that h is
smooth on each set of the form (—6, e) χ [i^, ii+i]. This is what we mean by
a piecewise smooth variation of a piecewise smooth curve. The velocity a
and the variation vector field V(t) := ^ ^ are only piecewise smooth. At
each "kink" point ij we have the jump vector Δά(ίί) := a(U+) — ά(^~),
which measures the discontinuity of ά at U. Using this notation, we have
the following theorem which gives the first variation formula:
Theorem 13.120. Let h : (—6, e) χ [α, b] —> Μ be a piecewise smooth
variation of a piecewise smooth curve a : [a, b] —> Μ with variation vector field
V. If a has constant speed с = (ε(ά,ά))1/2, then
5L\JV) = L'(0) = -- f (Vdta,V)dt--J2(Aa(tl),V(U))+i{a,V)\b
С Ja С С la
13.8. First and Second Variation of Arc Length
601
Proof. Since с = (ε(ά,ά)) ' , Proposition 13.119 gives
C Jti
Since we have (ά, V^ V) = jg (ά, V) - (V^a, V), we can employ integration
by parts: On each interval [ii,£i+i] we have
- (VdtV,a)dt=-(a,V)\ --/ (Vdta,V)dt.
с Jti с \ti cjt.
We sum from г — 0 to г = к to get
L'(0) = i(a,V)\b -£-J2(&a(U),V(U))-- [ {Vdta,V)dt,
С Ια C^ Cja
which is the required result. D
A variation h : (—e, e) x [a,b] ч Μ of a is called a fixed endpoint
variation if h(s,a) = a (a) and /1(5,6) = a(b) for all 5 G (—€, e). In this
situation, the variation vector field V is zero at α and b.
Corollary 13.121. A piecewise smooth curve a : [a, b] —> Μ with constant
speed с > 0 on each subinterval where a is smooth is a (nonnull) geodesic if
and only if SL\a (V) = 0 for all fixed endpoint variations of a. In particular,
if Μ is a Riemannian manifold and a : [a,b] —> Μ minimizes length among
nearby curves, then a is an (unbroken) geodesic.
Proof. If a is a geodesic, then it is smooth and so Δά(ίΐ) = 0 for all U (even
though a is smooth, the variation still only needs to be piecewise smooth).
It follows that 1/(0) = 0.
Now if we suppose that α is a piecewise smooth curve and that 1/(0) = 0
for any variation, then we can conclude that α is a geodesic by picking some
clever variations. As a first step we show that aL·^ л is a geodesic for
each segment [t^+i]. Let t G (£i?ii+i) be arbitrary and let ν be any
nonzero vector in Ta^M. Let β be a cut-off function on [a, b] with support
in (i — <J,i + δ) and δ chosen sufficiently small. Then let V(t) := /3(£)У(£),
where Υ is the parallel translation of у along a. We can now easily produce
a fixed endpoint variation with variation vector field V by the formula
h(s,t) :=expa{t)sV{t).
With this variation the last theorem gives
ε fb ε Гш
L'(0) = — / (Vdta,V)dt = — / <VAa,/3(t)y(t))dt,
which must hold no matter what our choice of у and for any δ > 0. Prom this
it is straightforward to show that V^a(i) = 0, and since t was an arbitrary
602
13. Riemannian and Semi-Riemannian Geometry-
element of (ij, ii+i), we conclude that aL t ι is a geodesic. All that is left
is to show that there can be no discontinuities of ά. Once again we choose a
vector y, but this time у G Γα^.)Μ, where t% is a potential kink point. Take
another cut-off function β with supp/3 С [ii-i?ii+i] = [U-i,U] U [ii,ii+i],
/3(ii) = 1, and г a fixed but arbitrary element of {1, 2,..., k}. Extend у to
a field Υ as before and let V = βΥ. Since we now have that α is a geodesic
on each segment, and we are assuming that the variation is zero, the first
variation formula for any variation with variation vector field V reduces to
0 = L'(0) = --c{Aa(ti),y)
for all y. This means that Δά(ίΐ) = 0, and since i was arbitrary, we are
done. D
We now see that, for fixed endpoint variations, 1/(0) = 0 implies that
α is a geodesic. The geodesies are the critical "points" (or curves) of the
arc length functional restricted to all curves with fixed endpoints. In order
to classify the critical curves, we look at the second variation but we only
need the formula for variations of geodesies. For a variation h of a geodesic
7, we have the variation vector field V as before, but now we also consider
the transverse acceleration vector field A(t) := V<9s<9s/i(0,£). Recall that
for a curve 7 with |(7,7)| > 0, a vector field Υ along 7 has an orthogonal
decomposition Υ = ΥΎ + Υ1- (tangent and normal to 7). Also we have
{VdtY) = Vft Y1-, and so we can use У^У1- to denote either of these
without ambiguity.
We now have the second variation formula of Synge:
Theorem 13.122. Let 7 : [a,b] —> Μ be α (nonnull) geodesic of speed
с > 0. Let ε be the sign of 7 as before. If h : (—e,e) χ [a,b] is a variation
of 7 with variation vector field V and acceleration vector field A, then the
second variation of L(s) := L(h3(i)) at s = 0 is
L"(0) = E- f ((V9tFx,Vatrx} + (ЩуъУ)) dt+-(<y,A)
С J a \ / С
Proof. Let tf(M) := |<Й(М),Й(M)>|1/2 = (е<й(*,*),й(М)»1/2· We
have L'(s) = Ja -^H(s}t) dt. Computing as before, we see that
dH(s,t) ε /dh, ч „ dh,
13.8. First and Second Variation of Arc Length
603
Taking another derivative, we have
d2H(s,t) _ ε f d /dh dh\_/dh dh\dH\
ds2 -H2[ Hds\dt,Wdsdt/ \dt,Wdsdt/ ds)
ε (ι dh dh\ /dh 2 dh\ 1 /dh η dh\dH
ε (ι„ dh „ dh\ /dh 2 ^\ £ /^h dhy
η \\Va*¥'Va* ft /+ \ ft'v^ ft / - #nft'Va» ft /
Using Vdtdsh = Vdsdth and Lemma 13.107, we obtain
dh
and then
д2н
„ „ dh _ _ dh Jdh dh\dh _ _
ds1
ε (,_ <9/ι _ fti. /fti ^idh dh\dh\
,dh „ - dh. ε /dh _ fti\2]
+ (^,V9tV9s^)--p^,Vat^; j.
Now we let s = 0 and get
|J(0,i) = ^{(VftV, VdtV) + (7, H(V,7)V>
+ (7,VatA)-^(7,VatF}2}.
Before we integrate the above expression, we use the fact that (7, V^-A) =
^ (7, A) (7 is a geodesic) and the fact that the orthogonal decomposition of
VdtV is
so that {VdtV,VdtV) = ^(7,VatF)2 + (VaV-S Va^). Plugging these
identities in, observing the cancellation, and integrating, we get
L"(0) = |60(Ο,ί)ώ = -Jb ((V^.V^) + (7,Д(У,7)^}) dt
+ -{ЪА)\Ь. D
С Ια
The right hand side of the main equation of the second variation formula
just proved depends only on V except for the last term. But if the variation
is a fixed endpoint variation, then this dependence drops out.
It is traditional to think of the set Ωα^(_ρ, q) of all piecewise smooth
curves а : [α, b] —> Μ from ρ to q as an infinite-dimensional manifold. Then
a variation vector field V along a curve α G Ω(ρ, q) which is zero at the
endpoints is the "tangent vector" at α to the curve in Ωα?ί,(ρ, q) given by
the corresponding fixed endpoint variation h. Thus the "tangent space"
604
13. Riemannian and Semi-Riemannian Geometry
ΤαΩ = Γα (Ωα^(ρ, q)) at α is the set of all piecewise smooth vector fields
V along α such that V(a) = V(b) = 0. We then think of L as being a
function on Ωα^(ρ, q) whose constant speed and nonnull critical points we
have discovered to be nonnull geodesies beginning at ρ and ending at q
at times α and 6 respectively. Further thinking along these lines leads to
the idea of the index form. Let us abbreviate Ωα^(ρ, q) to Ωα^ or even to
Ω. For our present purposes, we will not lose anything by assuming that
α = 0 whenever convenient. On the other hand, it will pay to refrain from
assuming that 6=1.
Definition 13.123. For a given nonnull geodesic 7: [0, 6] —> M, the index
form 77: Γ7Ω χ Γ7Ω -> Ш is defined by J7(V, V) = L"(0), where L7(s) =
f!l\(hs(t),hs(t))\1/2dt and Vdsh(0,t) = V.
Of course this definition makes sense because £"(0) only depends on
V and not on h itself. Also, we have defined the quadratic form /7(У,У),
but not directly J7(V, W). Of course, polarization gives J7(V, W), but if
V, W G Γ7Ω, then it is not hard to see from the second variation formula
that
(13.9) ΙΊ(ν,Ψ) = £-Ι {(ν^\ν9(^χ) + <β(7,^)7,^}}^.
It is important to notice that the right hand side of the above equation is
in fact symmetric in V and W.
It is important to remember that the variations and variation vector
fields we are dealing with are allowed to be only piecewise smooth even if
the center curve is smooth. So let 0 = ίο < ii < · · · < ί/e < ί/e+i = 6 as
before and let V and W be vector fields along a geodesic 7. We now derive
another formula for 77(У,И/Г). Rewrite formula (13.9) as
ЗДИО = -Σ Γ"" {<VftV\VAWX> + (R(<y,V)<y,W)} dt.
C г=0 Jti
On each interval [^,^+1] we have
<V5(F\VdtWL) = Vdt{VdtVL,WL) - (V%tVL,WL),
and substituting this into the above formula we obtain
к +.
ιΊ(ν,\ν) = -Σ / 1+1{v^v^\w±>-<v1(f\wx>
C г=0 Jt*
+(R(i,v)>y,w)}dt.
As for the last term, we use symmetries of the curvature tensor to see that
13.8. First and Second Variation of Arc Length
605
Substituting we get
к
i=0 ^*«
+(Д(7,^Х)7,^Х}}^.
Using the fundamental theorem of calculus on each interval [t^+i], an(i
the fact that W vanishes at α and b, we obtain the following alternative
formula:
Proposition 13.124 (Formula for index form). Let 7 : [0, b] —> Μ be а
nonnull geodesic. Then for V, W G Τ7Ωαι&,
rb
' dt
J7(V, W) = -- [ (V\V^ + Д(<у, У^)ъ WL) .
c Jo
к
ci=i
)>,
«Лете AVatFx(ii) = VdtVL(U+) - VatFx(ii-).
Letting V = W we have
C JV ° г=1
and the presence of the term (i?(7, ^J")7, ΐ^1") indicates a connection with
Jacobi fields.
Definition 13.125. A geodesic segment 7 : [a,b] —> Μ is said to be
relatively length minimizing (resp. relatively length maximizing)
if for all piecewise smooth fixed endpoint variations h of 7 the function
L(s) := Ja \(hs(t),h3(t))\1/2 dt has a local minimum (resp. local maximum)
at s = 0 (where 7 = h0(t) := /ι(0, ί))·
If 7 : [a, b] —> Μ is a relatively length minimizing nonnull geodesic,
then L"(0) = 0, which means that Iy(V,V) = 0 for any V G Τ7Ωα,6. The
adverb "relatively" is included in the terminology because of the possibility
that there may be curves in Ωα?5 which are "far away" from 7 and which
have smaller length than 7. A simple example of this is depicted in Figure
13.8, where 72 has greater length than 7 even though 72 is relatively length
minimizing. We assume that the metric on (0,1) χ Sl is the usual definite
metric dx2+dy2 induced from that on Rx (0,1), where we identify S1 x (0,1)
with the quotient R χ (0, l)/((x, у) ~ (χ + 2π, y)). On the other hand, one
sometimes hears the statement that geodesies in a Riemannian manifold
are locally length minimizing. This means that for any geodesic 7 :
606
13. Riemannian and Semi-Riemannian Geometry-
Figure 13.8. Geodesic segments on a cylinder
[a, b] —> M, the restrictions to small intervals are always relatively length
minimizing. But note that this is only true for Riemannian manifolds. For a
semi-Riemannian manifold with indefinite metric, a small geodesic segment
can have nearby curves that decrease the length. For example, consider the
metric — dx2 + dy2 on R χ (0,1) and the induced metric on the quotient
S1 x (0,1) = R χ (0, l)/~. In this case, the geodesic 7 in the figure has
length greater than all nearby geodesies; the index form 77 is now negative
semidefinite.
Exercise 13.126. Prove the above statement concerning 77 for S1 x (0,1)
with the index 1 metric —dx2 + dy2.
It is not hard to see that if even one of V or W is tangent to 7, then
J7(V, W) = 0 and so /7(V, W) = /7(^-\ W1-). Thus, we may as well restrict
77 to
Γ^Ω = {Ve Γ7Ω : VL^}.
Notation 13.127. The restriction of ΙΊ to Τ^Ω will be called the
restricted index and will be denoted by 1^. The nullspace N(I^) is then
defined by
M(I^) := \V G Γ^Ω : I$(V, W) = 0 for all W G Γ^ω} .
Theorem 13.128. Let 7 : [0,6] —l· Μ be a nonnull geodesic. The nullspace
Af{ljr) of 1^ · Tj-Q —> R is exactly the space Jo(7,0,6) of Jacobi fields
vanishing at 7(0) and 7(6).
Proof. The formula of Proposition 13.124 makes it clear that Jo(7>0, 6) С
лед.
Suppose that V G λί(Ι^τ). Let t G (ii,ii+i), where the U determine a
partition of [0, 6] such that V is potentially nonsmooth at the U as before.
Pick an arbitrary nonzero element у G (7(£)) С ΤΊ^Μ and let Υ be the
unique parallel field along y\u.ti л such that Y(t) = y. Picking a cut-off
13.8. First and Second Variation of Arc Length
607
function β with support in [t + δ, t - δ] С (U, U+ι) as before we extend βΥ
to a field W along 7 with W(i) = y. Now V is normal to the geodesic and
so ΙΊ(ν,\¥) = Ij-(V,W) and
ε ft+S
ιΊ(ν,π) = — / (v2dtv + R(<y,v)j^Y)dt.
c Jt—S
For small £, /ЗУ1 is approximately the arbitrary nonzero у and it follows
that V|f V + i?(7, V)7 is zero at t. Since t was arbitrary, V^ V + i?("y, V)7
is identically zero on (ii,ii+i)· Thus V is a Jacobi field on each interval
(ii,ii+i), and since V is continuous on [0,6], it follows from the standard
theory of differential equations that V is a smooth Jacobi field along all of
7. Since V e Γ7Ω, we already have V(0) = V(b) = 0. We conclude that
Proposition 13.129. Let (M,p) be α semi-Riemannian manifold of index
ind(g) and 7 : [a, 6] —> Μ a nonnull geodesic. If the index form ΙΊ is positive
semidefinite, then ind(p) = 0 or η (thus the metric is definite and so, up
to the sign of g, the manifold is Riemannian). On the other hand, if ΙΊ
is negative semidefinite, then 'md(g) = 1 or η — 1 (so that up to the sign
convention, Μ is a Lorentz manifold).
Proof. For simplicity we assume that a = 0 so that 7 : [0, 6] —> M. Let
77 be positive semi-definite and assume that 0 < ν < η {у = ind(M)).
In this case, there must be a unit vector и in ΤΊ^Μ which is normal to
7(0) and has the opposite causal character of 7(0). This means that if
ε = (7(0),7(0))/||7(0)||, then e(u,u) = -1. Let U be the field along 7
which is the parallel translation of u. By choosing δ > 0 appropriately we
can arrange that δ is as small as we like and simultaneously that sin(i/5)
is zero at t = 0 and t = b. Let V := Ssm(t/S)U and make the harmless
assumption that Ц7Ц = 1. Notice that by construction VU/y. We compute:
ΙΊ(ν,ν) = ε ί {(ν9ίν,ν9ίν) + (ϋ(4,ν)4,ν)}(1ί
Jo
= ε f {(V9tV,V9tV) - (R(i,V)V,i)}dt
Jo
= e [ {(V9tV,V9tV) - K(V Aj)(V Ai,V Aj)}dt
Jo
= ε ί {(VdtV,V9tV) - K(V Ai){V,V)e}dt,
Jo
where
K(V Aj) :=
(V A 7, V A 7} e{V, V)
608
13. Riemannian and Semi-Riemannian Geometry
as defined earlier. Continuing the computation we have
ΙΊ(ν,ν) = ε / {(u,u)cos2(t/S) + K(V A j)52 sin2 (t/δ)} dt
Jo
b
{-cos2(t/S) + eKiV ΐ\η)δ2sm2{t/5)} dt
rb
= -6/2 + δ2 / eK(V Λ 7) sm2(t/6) dt
Jo
Now as we said, we can choose δ as small as we like, and since K(V(t) Λ
7(ί)) is bounded on the (compact) interval [0,6], this clearly means that
J7(V, V) < 0, which contradicts the fact that 77 is positive semidefinite.
Thus our assumption that 0 < ν < η is impossible.
Now let 77 be negative semidefinite. Suppose that we assume that
contrary to what we wish to show, ν is not 1 or η — 1. In this case, one can
find a unit vector и G T7(0)M normal to 7(0) such that e{u,u) = +1. The
same sort of calculation as we just did shows that 77 cannot be semidefinite;
again a contradiction. D
By changing the sign of the metric the cases handled by this last theorem
boil down to the two important cases: 1) where (M, g) is Riemannian, 7 is
arbitrary, and 2) where (M, g) is Lorentz and 7 is timelike. We consolidate
these two cases by a definition:
Definition 13.130. A geodesic 7 : [a, 6] —> Μ is cospacelike if the sub-
space 7(5) -1 С Г7(5)М is spacelike for some (and consequently all) s G [a, 6].
Exercise 13.131. Show that if 7 : [a, 6] —>· Μ is cospacelike, then 7 is
nonnull, 7(5)1 С ΤΊ(3)Μ is spacelike for all s G [a, 6], and also show that
(M, g) is either Riemannian or Lorentz.
A useful observation about Jacobi fields along a geodesic is the following:
Lemma 13.132. If we have two Jacobi fields J\ and J2 along a geodesic 7,
then (V<9t Ji, J2) - («/1, Vft J2) is constant along 7.
Proof. To see this, we note that
VA(Vft Ji, J2> = (V|t Ji, J2) + (VdtJuVdtJ2)
= (Д(7, Ji)7, J2) + (VdtJuVdtJ2)
= (Д(7, J2)7, Ά> + <Vft J2, Vft Ji) = VA(Vft J2, Λ).
Similarly, we compute V& («Λ> ν^«/2) and subtract the result from the above
to obtain the conclusion. D
-L
13.8. First and Second Variation of Arc Length
609
In particular, if (V^Ji, J2) = (Ji^dth) at ί = 0, then (V^Ji, J2) -
(JbVatJ2) = 0 for all*.
We need another simple technical lemma:
Lemma 13.133. If Ji,..., J^ are Jacobi fields along a geodesic 7 such that
(VdtJi, Jj) — (Ji, ^dtJj) for all h J' £ {1? · · · j ^}> ^en any field Υ which can
be written as Υ = ^,4)lJi ^as ^e property that
(VdtY,VdtY) + (R(Yn)Yn) = ({W) λ, (V) Ji) +9г(У,^г^Л)}.
Proof. We have VdtY = (9ίψι) Ji + y>r (V<9t Jr) and so using the summation
convention,
dt(Y, <pr (v* jr)> = <(vAy), </ (v* jr)> + <y, v* br (vft Jr)])
= <($^) Ji, φΓ {Vd.Jr)) + (<pr (VdtJr), / (VftJfc)>
The last term (У, </?rV| Jr) equals (Д(У, 7)У, 7) by the Jacobi equation.
Using this and the fact that (У, ^(//V^ Jr) — ({dtp1) Ji, ψτ (V^ Jr)), which
follows from a short calculation using the hypotheses on the J;, we arrive at
dt(Y, φΓ (VdtJr)) = 2((drf) Ju <// (VA Jr)> + (<pr (VdtJr), ψτ {VdtJr))
+ (Д(У,7)У,7>·
Using the last equation together with VdtY = (dtp1) Ji + ψτ (V#t Jr) gives
the result (check it!). D
Exercise 13.134. Work through the details of the proof of the lemma
above.
Throughout the following discussion, 7 : [0,6] —> Μ will be a cospacelike
geodesic with sign ε and speed c.
Suppose that there are no conjugate points of ρ = 7(0) along 7. There
exist Jacobi fields Ji,..., Jn-\ along 7 which vanish at t = 0 and are such
that the vectors V#t Ji(0),..., V#t Jn_i(0) G TpM are a basis for the space
7(0)x С Г7(0)М.
Claim: Ji(i),..., Jn-i(t) are linearly independent for each t > 0.
Indeed, suppose that ci«/i(i) + · · · + C2«/n-i(i) = 0 for some t. Then,
Ζ := ΣΊ=ι ciJi ls a normal Jacobi field with Z(0) = Z(i) = 0. But then,
since there are no conjugate points, Ζ = 0 identically and so 0 = V#tZ(0) :=
ΣΓ^ι1 CiVdtJi(0). Since the V$«Λ(Ο) are linearly independent, we conclude
that Ci = 0 for all i and the claim is proved.
It follows that at each t with 0 < t < b the vectors Ji(i),..., «Λι-ι(ί)
form a basis of ^(t)1 С ΤΊ^Μ. Now let У G Γ7(Ω) be a piecewise smooth
610
13. Riemannian and Semi-Riemannian Geometry
variation vector field along 7 and write Υ = Y^y>lJi f°r some piecewise
smooth functions φ1 on (0, b], which can be shown to extend continuously to
[0,6] (see Problem 3). Since (V<9t J;, Jj) = (J;, V<9t Jj) = 0 at t = 0, we have
(VdtJi, Jj) — (Ji, V&J;) = 0 for all t by Lemma 13.132. This allows for the
use of Lemma 13.133 to arrive at
(VdtY,VdtY) + (R(Yn)Y,j)
= (Σ (ΜJ» Σ ΜJ*)+^(y> Σ *r (νβ, jr))
and then
(13.10)
εΙΊ(Υ,Υ) = - ί (Σ&φ1) 1τ,Σ(8ιΨ*) Ji)dt + - (Υ,φ1· {VdtJr))\l.
On the other hand, Υ is zero at α and b and so the last term above
vanishes. Now we notice that since 7 is cospacelike and the Ji are normal to
the geodesic, we must have that the integrand in equation (13.10) above
is nonnegative. We conclude that εΙΊ(Υ, Υ) > 0. On the other hand,
if ΙΊ(Υ,Υ) = 0 identically, then /06(Σ (δίφί) Λ, Σ ($¥>') Λ) dt = 0 and
(Σ {9ίψι) Ji, Σ {9ίψι) Ji) = 0. In turn, this implies that Σ {^ψι) J% = 0
and that each φ1 is constant, in fact zero, and finally that Υ itself is
identically zero along 7. All we have assumed about Υ is that it is in the domain
of the restricted index 1^ and so we have proved the following:
Proposition 13.135. If 7 G Ω is cospacelike and there is no conjugate
points to ρ = 7(0) along η, then εΙ^(Υ,Υ) > 0 and Υ = 0 along 7 if and
onlyiflJr(Y,Y) = 0.
We may paraphrase the above result as follows: For a cospacelike
geodesic 7 without conjugate points, the restricted index ij- is definite; it is
positive definite if ε = +1 and negative definite if ε = —1. The first case
(ε = +1) is exactly the case where (M,g) is Riemannian (Exercise 13.131).
Next we consider the situation where the cospacelike geodesic 7 : [0, b] -¥
Μ is such that 7(b) is the only point conjugate to ρ = 7(0) along 7. In this
case, Theorem 13.128 tells us that ijr has a nontrivial nullspace and so 77
cannot be definite.
Claim: 77 is semidefinite. To see this, let Υ G Γ7Ω and write Υ in
the form (b — t)Z(t) for some (continuous) piecewise smooth Z. Let bi —> b
and define Y{ to be (bi — t)Z(t) on [0,6$]. Our last proposition applied to
7г := 7l[o,bi] shows that εΙΊί(Υι, Yi) > 0. Now εΙΊί(Υι, Υι) -» εΙΊ(Υ, Υ) (some
uninteresting details are omitted) and so the claim is true.
Now we consider the case where there is a conjugate point to ρ before
7(b). Suppose that J is a nonzero Jacobi field along 7Lri with 0 < r < b
13.8. First and Second Variation of Arc Length
611
such that J(0) = J(r) = 0. We can extend J to a field Jext on [0, b] by
defining it to be 0 on [r, b]. Notice that S7dtJext{r—) is equal to V<9(J(r),
which is not 0 since otherwise J would be identically zero (over
determination). On the other hand, V dtJext(r+) = 0 and so the "kink" AJext(r) :=
V^ Jext(r+) — VdtJext(r—) is not zero. Notice that AJext(r) is normal to 7
(why?). We will now show that if W e Γ7(Ω) is such that W(r) = AJext(r)
(and there are plenty of such fields), then εΙΊ{ Jext + 5W, Jext + $W) < 0 for
small enough δ > 0. This will allow us to conclude that 77 cannot be definite
since by Proposition 13.135 we can always find a Z with ε/7(Ζ, Ζ) > 0. We
have
εΙΊ( Jext + SW, Jext + SW) = εΙΊμ€Χί, Jext) + 25еЦ^ехи W) + εδ2ΙΊ(\¥, W).
It is not hard to see from the formula of Theorem 13.124 that I<y(Jext, Jext)
is zero since it is piecewise Jacobi and is zero at the single kink point r. But
using the formula again, εΙΊ( Jext(r), W(r)) reduces to
--(AJ'ext(r),W(r)) = -- \AJ'ext(r)\2 < 0,
С С
and so taking δ small enough gives the desired conclusion.
Summarizing the conclusions of the above discussion (together with the
result of Proposition 13.135) yields the following nice theorem:
Theorem 13.136. If η : [0, b] —> Μ is a cospacelike geodesic of sign ε, then
(M, g) is either Riemannian or Lorentz and we have the following three
cases:
(i) If there are no points conjugate to 7(0) along 7, then εΐ^τ is positive
definite.
(ii) // 7(b) is the only conjugate point to 7(0) along η, then ΙΊ is not
definite, but must be semidefinite.
(iii) If there is a point 7(7*) conjugate to 7(0) with 0 < r < b, then ΙΊ is
not semidefinite (or definite).
As we mentioned the Jacobi equation can be written in terms of the
tidal force operator Ry : TpM —> TpM as
V2dtJ(t) = Rm(J(t)).
The meaning of the term force here is that Щ^) controls the way nearby
families of geodesies attract or repel each other. Attraction tends to create
conjugate points, while repulsion tends to prevent conjugate points. If 7 is
cospacelike, then we take any unit vector и normal to j(t) and look at the
component of Щ^(и) in the и direction. Up to sign this is
(Щ(Ь)(и),и)и = (%t)iU(7(i)),u)u = -(9t(7(t) Au),7(i) Au)u.
612
13. Riemannian and Semi-Riemannian Geometry
In terms of sectional curvature,
(Rm(u),u)u = K(j(t)Au) <<y(i),7(*)> ·
It follows from the Jacobi equation that if (Щ^(и),и) > 0, i.e., if
Κ{η(ΐ) Λ ν)(η(ί),η(ί)) < 0, then we have repulsion, and if this always
happens anywhere along 7, we expect that 7(0) has no conjugate point along
7. This intuition is indeed correct.
Proposition 13.137. Let 7 : [0,6] —> Μ be α cospacelike geodesic. If
for every t and every vector ν G 7(i)^ we have (R^(v)^v) > 0 (i.e. if
Κ(η(ΐ) Λ ϋ)(7(ί)>7(0) — ^Л ^еп 7(0) has no conjugate point along 7.
In particular, a Riemannian manifold with sectional curvature К < 0
has no conjugate pairs of points. Similarly, a Lorentz manifold with sectional
curvature К > 0 has no conjugate pairs along any timelike geodesies.
Proof. Take J to be a Jacobi field along 7 such that J(0) = 0 and J-L7.
We have |(J, J) = 2(Vdt J, J) and
^(J,J) = 2(VatJ,VatJ) + 2(VltJ,J)
= 2(VAJ,VAJ> + 2(Av(t)fJ(7(i)),^>
= 2(VatJ,VatJ) + 2(^(i)(J),J),
and by the hypotheses J^(J, J) > 0. On the other hand, (J(0), J(0)> = 0
and -3t\0(J,J) — 0. It follows that since (J, J) is not identically zero we
must have (J, J) > 0 for all t G (0, b] and the result follows. D
13.9. More Riemannian Geometry
Recall that a manifold is geodesically complete at ρ if and only if expp is
defined on all of TpM. The following lemma is the essential ingredient in
the proof of the Hopf-Rinow theorem stated and proved below. Note that
this is a theorem about Riemannian manifolds.
Lemma 13.138. Let (M,g) be a connected Riemannian manifold. Suppose
that expp is defined on the ball of radius ρ > 0 centered at 0 G TPM. Let
Bp(p) := {x : dist(p, x) < p}. Then each point q G Bp(p) can be connected to
ρ by an absolutely minimizing geodesic. In particular, if Μ is geodesically
complete at ρ Ε Μ, then each point q G Μ can be connected to ρ by an
absolutely minimizing geodesic.
Proof. Let q G Bp(p) with ρ φ q and let R = dist(p, q). Choose e > 0
small enough that B2€(p) is the domain of a normal coordinate system.
(Refer to Figure 13.9.) By Lemma 13.85, we already know the theorem is
true if Bp(p) С Be(p), so we will assume that с < R < p. Because dBe(p) is
13.9. More Riemannian Geometry
613
diffeomorphic to Sn λ С Mn, it is compact and so there is a point pe G dBe(p)
such that χ \-> dist(x,g) achieves its minimum at pe. This means that
dist(p, q) = dist (p,pc) + dist(pe, q) = e + dist(pe, q).
Let 7 : [0, p] —> Μ be the geodesic with |^y| = 1, 7(0) = p, and 7(e) = pe.
Figure 13.9
It is not difficult to see that the set
T = {te [0, R] : dist(p,7(0) + dist(7(i), g) = dist(p,q)}
is closed in [0,-R] and is nonempty since e G T. Let tsup = sup Γ > 0. We
will show that tsup = R from which it will follow that 7L щ is a minimizing
geodesic from ρ to q since then dist(j(R)^q) = 0 and so 7(i?) = q. With
an eye toward a contradiction, assume that tsup < R. Let χ := ^(tsup) and
choose 61 with 0 < c\ < R — tsup and small enough that i?2ei(x) С Вр(р) is
the domain of normal coordinates about x. Arguing as before we see that
there must be a point xei G dBei (x) such that
dist (ж, g) = dist(x,xei) +dist(xei,g) = 61 + dist(xei,g).
Now let 71 be the unit speed geodesic such that 71 (0) = χ and 71 (ei) = xei.
But since tsup G Τ and χ = 7(iSup)> we also have
dist(p, x) + dist (ж, q) = dist(p, q).
Combining, we now have
dist(p, q) = dist(p, x) + dist(x, x€l) + dist(xCl, q).
By the triangle inequality, dist(p,q) < dist(p,xei) + dist(xei,g) and so
dist(p,x) +dist(x,xCl) < dist(p,x€l).
614
13. Riemannian and Semi-Riemannian Geometry
But also dist(p, x€l) < dist(p, x) + dist(x, xei) and so
dist(p, xei) = dist(p, x) + dist(x,xei).
Examining the implications of this last equality, we see that the
concatenation of 7L igu 1 with 71 forms a curve from ρ to x€1 of length dist(p, xei),
which must therefore be a minimizing curve. By Problem 1, this potentially
broken geodesic must in fact be smooth and so must actually be the geodesic
^l[(Hsu +€il* Then, tsup + ^i G Γ which contradicts the definition of tgup- This
contradiction forces us to conclude that tsup = R and we are done. D
Theorem 13.139 (Hopf-Rinow). If(M,g) is α connected Riemannian
manifold, then the following statements are equivalent:
(i) The metric space (M, dist) is complete. That is, every Cauchy
sequence is convergent.
(ii) There is a point ρ G Μ such that Μ is geodesically complete at p.
(iii) Μ is geodesically complete.
(iv) Every closed and bounded subset of Μ is compact.
Proof. (iv)=>(i): The set of points of a Cauchy sequence is bounded and
so has compact closure. Thus there is a subsequence converging to some
point. Since the original sequence was Cauchy, it must converge to this
point.
(i)=>(iii): Let ρ be arbitrary and let 7v(i) be the geodesic with ηυ{ϋ) = ν
and J its maximal domain of definition. We can assume without loss of
generality that (v, v) = 1 so that L(7VL ЬЛ = t<i — t\ for all relevant £1, £2·
We want to show that there can be no upper bound for the set J. We argue
by contradiction: Assume that i+ = sup J is finite. Let {tn} С J be a
Cauchy sequence such that tn —> i+ < 00. Since dist(7v(i),7v(s)) < |i — s|,
it follows that 7v(in) is a Cauchy sequence in M, which by assumption must
converge. Let q := limn_>oc7i;(in) and choose a small ball Be(q) which is
small enough to be a normal neighborhood. Take ίχ with 0 < t+ — t\ < e/2
and let 71 be the (maximal) geodesic with initial velocity 7v(ii). Then in
fact 71 (ί) = 7υ(ίι + t) and so 71 is defined for ii + e/2 > i+ and this is a
contradiction.
(iii)=>(ii) is a trivial implication.
(ii)==>(iv): Let К be a closed and bounded subset of M. For χ G M,
Lemma 13.138 tells us that there is a minimizing geodesic ax : [0,1] —>· Μ
connecting ρ to x. Then ||аж(0)|| = dist (ρ, χ) and ехр(аж(0)) = χ. Using
the triangle inequality, one sees that
зир{||^ж(0)||} <r
хек
13.9. More Riemannian Geometry
615
for some r < oo. Prom this we obtain {άχ(0) : χ G X} С Br := {ν G ΓρΜ :
IMI — r}· The set Br is compact. Now exp(f?r) is compact and contains
the closed set if, so К is also compact.
(ii)=^(i): Suppose Μ is geodesically complete at p. Now let {xn} be any
Cauchy sequence in M. For each xn, there is (by assumption) a minimizing
geodesic from ρ to xn, which we denote by 7рЖп. We may assume that
each ηρΧη is unit speed. It is easy to see that the sequence {/n}, where
ln := i(7Pxn) = dist(p, xn), is a Cauchy sequence in Ш with some limit, say
/. The key fact is that the vectors ηρΧη are all unit vectors in TPM and so
form a sequence in the (compact) unit sphere in TpM. Replacing {%Xn} by
a subsequence if necessary we have ηρΧη 4uG TpM for some unit vector u.
Continuous dependence on initial velocities implies that {xn} = {^Ρχη(Ιη)}
has the limit ju(l). D
Let (M, g) be a complete connected Riemannian manifold with sectional
curvature К < 0. By Proposition 13.137, for each point ρ G M, the geodesies
emanating from ρ have no conjugate points and so TVp expp : TVpTpM —>
Μ is nonsingular for each vp G TpM. This means that expp is a local
diffeomorphism. If we give TpM the metric exp*(p), then expp is a local
isometry. It now follows from Theorem 13.76 that expp : TpM —> Μ is a
Riemannian covering. Thus we arrive at the Hadamard theorem.
Theorem 13.140 (Hadamard). If (M,g) is α complete simply connected
Riemannian manifold with sectional curvature К < 0, then expp : TPM —>
Μ is a diffeomorphism and each two points of Μ can be connected by a
unique geodesic segment.
Definition 13.141. If (M, g) is a Riemannian manifold, then the diameter
of Μ is defined to be
diam(M) := sup{dist(p,q) :p,gG M}.
The injectivity radius at ρ G M, denoted inj(p), is the supremum over
all б > 0 such that expp : Б(0р,б) —> B(p,e) is a diffeomorphism. The
injectivity radius of Mis inj(M) := infpGM{inj(p)}.
The Hadamard theorem above has as a hypothesis that the sectional
curvature is nonpositive. A bound on the sectional curvature is stronger
than a bound on Ricci curvature since the latter is a sort of average sectional
curvature. In the sequel, statements like Ric > С should be interpreted to
mean Ric(i;,i;) > C(v,v) for all ν G TM.
Lemma 13.142. Let (M,g) be an η-dimensional Riemannian manifold and
let 7 : [0, L] —> Μ be a unit speed geodesic. Suppose that Ric > (n — 1) к > 0
for some constant к > 0 (at least along η). If the length L of η is greater
than or equal to π/^/ϊϊ, then there is a point conjugate to 7 (0) along 7.
616
13. Riemannian and Semi-Riemannian Geometry
Proof. Suppose 0 < π/у/к < L. If we can show that ijr is not positive
definite, then Theorem 13.136 implies the result. To show that ijr is not
positive definite, we find an appropriate vector field V Φ 0 along 7 such that
1(У, У) < 0. Choose orthonormal fields E2,... ,En so that 7, £2, · · · ·>Εη is
an orthonormal frame along 7. For a function / : [Ο,π/у/к] —> R that
vanishes at endpoints, we form the fields fE{. Using (13.9), we have
fTT/у/к
ΙΊ{
(fEj,fEj)= Γ Κ{ί'(8)2 + /(8)2(ϋΕ]ή(Ει(8))Μ*))}(ΐ8,
Jo
and then
£/7(/^,/я,)= Г *{(n-i){f')2-f2mc(<y,<y)}d8
3=2 J°
<(n-l)jf ((f)2 - Kf2) ds.
Letting f(s) = sin(-v/«s), we get
y^IifEj, fEj) < (n - 1) / к (cos2(y/Ks) - sin2(VKs)) ds = 0,
j=2 Jo
and so I (fEj, fEj) < 0 for some j. D
The next theorem also assumes only a bound on the Ricci curvature and
is one of the most celebrated theorems of Riemannian geometry. A weaker
version involving sectional curvature was first proved by Ossian Bonnet (see
[Hicks], page 165).
Theorem 13.143 (Myers). Let (M,g) be α complete connected Riemannian
manifold of dimension n. 7/Ric > (n — 1) к > 0, then
(i) diam(M) < π/y/H, Μ is compact, and
(ii) πι (Μ) is finite.
Proof. Since Μ is complete, there is always a shortest geodesic ην4 between
any two given points ρ and q. We can assume that ηΡ4 is parametrized by
arc length:
7W: [0,dist(p,g)] -» M.
It follows that 7p<?Lai is arc length minimizing for all α G [0, dist(p, q)}.
Prom Proposition 13.129 we see that the only possible conjugate to ρ along
jpq is q. The preceding lemma shows that π /у/к > dist(p, q) is impossible.
Since the points ρ and q were arbitrary, we must have diam(M) < π/у/к.
It follows from the Hopf-Rinow theorem that Μ is compact.
For (ii) we consider the simply connected covering ρ : Μ —> Μ (which is
a local isometry). Since ρ is a local diffeomorphism, it follows that p~l(p)
13.10. Cut Locus
617
has no accumulation points for any ρ e M. But also, because Μ is complete
and has the same Ricci curvature bound as M, it is compact. It follows that
p~l(p) is finite for any ρ e M, which implies (ii). D
The reader may check that if Sn(R) is a sphere of radius R in Rn+1, then
Sn(R) has constant sectional curvature к = 1/R2 and the distance from any
point to its cut locus (defined below) is π/у/к. A result of S. Y. Cheng states
that with the curvature bound of the theorem above, if diam(M) = π/у/к,
then Μ is a sphere of constant sectional curvature к. See [Cheng].
13.10. Cut Locus
In this section we consider Riemannian manifolds. Related to the notion
of conjugate point is the notion of a cut point. For a point ρ G Μ and a
geodesic 7 emanating from ρ = 7(0), a cut point of ρ along 7 is the first
point q = ~f(tf) along 7 such that for any point r = ^(t") beyond ρ (i.e.
t" > t1) there is a geodesic shorter than 7L $/] which connects ρ with r. To
see the difference between this notion and that of a point conjugate to p, it
suffices to consider the example of a cylinder S1 χ R with the obvious flat
metric. If ρ = (1,0) G S1 χ R, then for any xGl, the point (еш,х) is a
cut point of ρ along the geodesic j(t) := (elt7T,tx). We know that beyond
a conjugate point, a geodesic is not (locally) minimizing. In our cylinder
example, for any e > 0, the point q = 7(1 + e) can be reached by the
geodesic segment 72 : [0,1 — б] —> S1 χ R given by 7(4) := (e2i7r,ax), where
a = (1 + б)/(1 - б). It сап be checked that 72 is shorter than 7I [0,1 + б].
However, the last example shows that a cut point need not be a conjugate
point. In fact, S1 χ R has no conjugate points along any geodesic. Let us
agree that all geodesies referred to in this section are parametrized by arc
length unless otherwise indicated.
Definition 13.144. Let (M, g) be a complete Riemannian manifold and let
ρ e M. The set C(p) of all cut points to ρ along geodesies emanating from
ρ is called the cut locus of p.
For a point ρ G M, the situation is summarized by the fact that if
Q — 7(0 is a cut point of ρ along a geodesic 7, then for any t" > i! there is
a geodesic connecting ρ with q which is shorter than 7L ^i, while if t" < t1',
then not only is there no geodesic connecting ρ and ^({t") with shorter length
but there is no geodesic connecting ρ and η(ίη) whose length is even equal
to that of 7|[0ft//]. (Why?)
Consider the following two conditions:
(CI): 7(io) is the first conjugate point of ρ = 7(0) along 7.
618
13. Riemannian and Semi-Riemannian Geometry
(C2): There is a unit speed geodesic α from 7(0) to 7(^0) that is
different from 7|[o£0i such that L(ct) = L(7|r0i 1).
Proposition 13.145. Let Μ be α complete Riemannian manifold.
(i) If for a given unit speed geodesic 7, either condition (CI) or (C2)
holds, then there is a t\ G (0, ίο] such that 7(ii) is the cut point of
ρ along 7.
(ii) 7/7(^0) is the cut point of ρ — 7(0) along the unit speed geodesic
ray 7, then either condition (CI) or (C2) holds.
Proof, (i) This is already clear from our discussion: For suppose (CI) holds,
then 7|r0i/i cannot minimize for t' > to and so the cut point must be 7(^1)
for some t\ G (0, ίο]. Now if (C2) holds, then choose с > 0 small enough
that α (ίο — б) and 7(ίο + б) are both contained in a convex neighborhood of
7(io). The concatenation of ок|г0 ^ ι and 7L ίο+€ι is a curve, say c, that has
a kink at 7(io)· But there is a unique minimizing geodesic r joining α(ίο — б)
to 7(ίο + б), and we can concatenate the geodesic aL io_€i with r to get a
curve with arc length strictly less than L(c) = to + e. It follows that the cut
point to ρ along 7 must occur at 7(i') for some t' < ίο + б. But e can be
taken arbitrarily small and so the result (i) follows.
(ii) Suppose that 7(io) is the cut point of ρ = 7(0) along a unit speed
geodesic ray 7. We let t{ —> 0 and consider a sequence {щ} of minimizing
geodesies with щ connecting ρ to 7(ίο + €*). We have a corresponding
sequence of initial velocities щ := ά;(0) Ε S1 С ТрМ. The unit sphere in
TpM is compact, so replacing щ by a subsequence we may assume that щ —>
и G S1 С TpM. Let a be the unit speed segment joining ρ to 7(io + ti) with
initial velocity u. Arguing from continuity, we see that a is also minimizing
and L(a) = L(j\[oto]). If α φ 7l[0,t0]> then we are done· If a = ^[ο,*0]'
then since 7|[ог0] is minimizing, it will suffice to show that Γίο^(0) expp is
singular because that would imply that condition (CI) holds. The proof
of this last statement is by contradiction: Suppose that a = tL·^ 1 (so
that 7(0) = u) and that Tio^(0) expp is not singular. Take U to be an open
neighborhood of io7(0) in TpM such that exppL· is a diffeomorphism. Now
&i(to + e'j) = 7(io + бг) for 0 < б^ < бг since the oti are minimizing. We
now restrict attention to г such that 6; is small enough that (ίο + с[)щ and
(ίο + ei)u are in U. Then we have
expp(io + €i)u = 7(io + €i)
= α;(ί0 + б·) = expp(i0 + е[)щ,
and so (ίο + Ci)u = (ίο + е[)щ, and then, since ei —> 0 and both и and щ
are unit vectors, we have 7(0) = и = щ for sufficiently large i. But then
13.11. Rauch's Comparison Theorem
619
for such г, we have ol{ = 7 on [0, £0 + €г], which contradicts the fact that
71 [0*0+6*1 *s no* т1п1т™пё· П
Exercise 13.146. Show that if q is the cut point of ρ along 7, then ρ is the
cut point of q along 7^ (where 7*~(i) := j(L — t) and L = 1/(7)).
It follows from the development so far that if q G M\C(p), then there is
a unique minimizing geodesic joining ρ to g, and that if Б(р, i?) is the ball of
radius R centered at p, then expp is a diffeomorphism on B(p,R) provided
R < d(p, C(p)). In fact, an alternative definition of the injectivity radius at
ρ is d(p, C(p)) and the injectivity radius of Μ is
inj(M)= mi{d(p,C(p))}-
рем
Intuitively, the complexities of the topology of Μ begin at the cut locus of
a given point.
Let TlM denote the unit tangent bundle of the Riemannian manifold:
T1M = {ueTM : \\u\\ = 1}.
Define a function см · Τ1 Μ —> (0, oo] by
-{
, ν . ^o if 7и(*о) is the cut point of тттмЫ) along 7^,
' 00 if there is no cut point in the direction u.
Recall that the topology on (0, 00] is such that a sequence tk converges to the
point 00 if lim/^oo t^ = 00 in the usual sense. It can be shown that if (M, g)
is a complete Riemannian manifold, then the function cm '.TlM —> (0,oo]
is continuous (see [Kob]).
13.11. Rauch's Comparison Theorem
In this section we deal strictly with Riemannian manifolds.
Definition 13.147. Let 7 : [a, 6] -+ Μ be a smooth curve. For Χ, Υ
piecewise smooth vector fields along 7, define
Zy(X,Y) := / (VdtX,VdtY) + (Щ*ъУ)
J а
dt.
The map Χ, Υ ι-+ Χγ(Χ, Υ) is symmetric and bilinear. In defining I7 we have
used a formula for the index 77 valid for fields which vanish at the endpoints
and with 7 a nonnull geodesic. Thus when 7 is a nonnull geodesic, the
restriction of I7 to variation vector fields which vanish at endpoint is the
index 77. Thus I7 is a sort of extended index form.
620
13. Riemannian and Semi-Riemannian Geometry-
Corollary 13.148. Let 7 : [0,6] —> Μ be α cospacelike geodesic of sign ε
with no points conjugate to 7(0) along 7. Suppose that Υ is a piecewise
smooth vector field along 7 and that J is a Jacobi field along 7 such that
У(0) = J(0), Y(b) = J(b), and (У - J) !_7.
Then eZy(J,J) < eZy (Y,Y).
Proof of the corollary. Prom Theorem 13.136, we have 0 < sI^(Y — J,
У -J)= ε!Ί(Υ -J,Y -J) and so
0 < eZy(Y,Y) - 2sl1(J,Y)+sZy(J,J).
Integrating by parts, we have
el1(J,Y) = e(VdtJ,Y)\b0- / (Vly,y)-(iV7,^>
Jo
= s(VdtJ,Y)\b0=s(VdtJ,J)\b0
dt
= εΧΊ{ J, J) (since J is a Jacobi field).
Thus 0 < ε!Ί(Υ, У) - 2εΙγ(J, У) + εΖγ( J, J) = ^(У, У) - εΙ7(J, J). D
Recall that for a Riemannian manifold M, the sectional curvature
KM{P) of a 2-plane Ρ С ТрМ is
(9г(е1Ле2),е1Ле2)
for any orthonormal pair ei, e2 that spans P.
Definition 13.149. Let M, g and iV, Л, be Riemannian manifolds and let
7M : [a, 6] —> Μ and 7^ : [a, 6] —> TV be unit speed geodesies defined on
the same interval [a,6]. We say that KM > KN along the pair (7м,7^)
if Km(Q1m^) > ΚΝ(ΡΊΝφ) for all t G [a,6] and every pair of 2-planes
Qt e τίμ{1)μ, pt e TjN{t)N.
We develop some notation to be used in the proof of Rauch's theorem.
Let Μ be a given Riemannian manifold. If У is a piecewise smooth vector
field along a unit speed geodesic 7м such that Y(a) = 0, then let
1™(Y,Y) := f\vdtY(t),VdtY(t)) + {R^M^M,Y)(t)dt
J a
= Г -(V2dtY(t),Y(t)) + (R^MiYjM,Y)(t)dt + (VdtY, Y)(s).
J a
If У is an orthogonal Jacobi field, then
I^(Y,Y) = (VdtY,Y)(s).
13.11. Rauch's Comparison Theorem
621
Theorem 13.150 (Rauch). Let M,g and N,h be Riemannian manifolds
of the same dimension and let ηΜ : [α, b] —> Μ and 7^ : [α, b] —> N be
unit speed geodesies defined on the same interval [a, b]. Let JM and JN be
Jacobi fields along 7м and 7^ respectively and orthogonal to their respective
curves. Suppose that the following four conditions hold:
(i) jM(a) = jN(a) = о and neither of JM(t) or JN(t) is zero for
t e (a, 6].
(ii) \\VdtJM(a)\\ = \\VdtJN(a)\\.
(iii) L(7M) = dist(7M(a),7M(b)).
(iv) KM > KN along the pair (7M, 7N).
Then \\JM(t)\\ < \\JN(t)\\ for all te [a,6].
Proof. Let /m be defined by /m(s) := ||^M(5)|| and ^м by 1im(s) "·=
ISM(JM, JM)/\\JM(s)\\2 for s e (a, 6]. Define fN and hN analogously. We
have
fM(s) = 21™(JM, JM) and fM/fM = 2hM
and the analogous equalities for Д/· and h^. If с € (α, b), then
ln(|| JM(s)\\2) = ln(|| JM(c)\\2) + 2 Γ hM(s'W
J с
with the analogous equation for N. Thus
Prom the assumptions (i) and (ii) and L'Hopital's rule, we have
PM(c)\
c^a+ ||J/V(C)
|2
lim ' _>r, , 2 = 1,
and so
In ( HJ HI ) = 2 lim fS[hM(s') - hN(s')}ds'.
If we can show that Iim(s) — hw(s) < 0 for s G (a,b], then the result will
follow. So fix r G (a,6] and let ZM (s) := JM(s)/ || JM(r)|| and ZN(s) :=
JN{s)/ || JN(r)||. Notice the r in the denominator; ZN(s) is not necessarily
of unit length for 5 φ r. We now define a parametrized families of sub-
tangent spaces along 7м by Wm(s) := 7M(5)1 С Τίμ^Μ and similarly
for W)v(s). We can choose a linear isometry Lr : W]y(r) —> Wm(x) such
that Lr(ZN(r)) = ZM(r). We now want to extend Lr to a family of linear
isometries Ls : Wn(s) —> Wm(s). We do this using parallel transport by
Ls:=PbMyroLroP^Nys.
622
13. Riemannian and Semi-Riemannian Geometry
Define a vector field Υ along ΊΜ by Y(s) := Ls(ZN(s)). Check that
Υ (α) = ZM(a) = 0,
Y(r) = ZM(r),
\\Y\\2 = \\ZN\\2
\\VdtY\\2 = \\VdtZN\\2.
The last equality is a result of Exercise 12.43 where in the notation of that
exercise /3(i) := P{^M)rtoY{t). Since (iii) holds, there can be no conjugates
along 7M up to r. Now Υ — ZM is orthogonal to the geodesic 7м and so by
Corollary 13.148 we have IrM(ZM,ZM) < 1ГМ(У,У) and in fact, using (iv)
and the list of equations above, we have
IrM(ZM,ZM)<IrM(y,y)= Γ \\ν9ιΥ\\2 + ΙΙΜ{ηΜ,Υ,ηΜ,Υ)
J a
= Γ liv5tr||2 - κ^,υ,ϋ1, Υ) ||7M л υ\\2
J a
= [r\\VdtY\\2-K(>yM,Y,>yM,Y)\\iM\\2\\Y\\2
J a
< ί \\^ZN\\2 -K(jM,ZN,jM,ZN)\\jM\\2\\ZN\\2 (by(iv))
J a
= Γ \\VdtZN\\2 + RN(jN, ZN,jN, ZN) = lrN(ZN, ZN).
J a
Recalling the definition of ZM and ZN we obtain
irM(jM, jm)/ II JM(r)f < irN(jN, jn)/ \\JN(r)\\2,
and so /ιμ(γ) — hw(r) < 0. But r was arbitrary, and so we are done. D
Corollary 13.151. Let M, g and N, h be Riemannian manifolds of the same
dimension and let 7м : [α, b] —> Μ and 7^ : [a,b] -^ N be unit speed
geodesies defined on the same interval [a,b]. Assume that KM > KN along
the pair (7м, 7^). Then if 7м (a) has no conjugate point along 7м, then
ΎΝ (a) has no conjugate point along ~/N.
The above corollary is easily deduced from the Rauch comparison
theorem above and we invite the reader to prove it. The following famous
theorem is also proved using the Rauch comparison theorem but the proof
is quite difficult and also uses Morse theory. The proof may be found in
[C-E].
13.12. Weitzenbock Formulas 623
Theorem 13.152 (The sphere theorem). Let M,g be α complete simply
connected Riemannian manifold with sectional curvature satisfying the
condition
11 „ 1
for some R > 0. Then Μ is homeomorphic to the sphere Sn.
13.12. Weitzenbock Formulas
The divergence of a vector field in terms of Levi-Civita connection is given
by
div(X) = trace(VX).
Thus if (ei,..., en) and (01,..., θη) are dual bases for TpM and T*M, then
(divX)p = ^^(VeiX).
i=i
Exercise 13.153. Show that this definition is compatible with our previous
definition by showing that, with the above definition, we have Lx vol =
(divX) vol, where vol is the metric volume element for M. Hint: Use Lx =
dix +ίχ d.
Definition 13.154. Let V be a torsion free covariant derivative on M. The
divergence of a (k,£) tensor field A is а (к — 1,£) tensor field is defined by
η
(div A)p (αϊ..., afe_b vu ..., ve) = ]Π (Vej A) (0J, αχ..., ak-i,vi, · · ·, vi),
where (ei,..., en) and (01,..., θη) are dual as above.
Notice that the above definition depends on the choice of covariant
derivative but if Μ is semi-Riemannian, then we will use the Levi-Civita
connection. In index notation the definition is quite simple in appearance. For
example, if AlJkl are the components of a (2,2) tensor field, then we have
№АУЫ=ЧГАГ>Ы,
where VaA 3kl are the components of VA Also, the definition given is really
for the divergence with respect to the first contravariant slot, but we could
use other slots (the last slot being popular). Thus VrAjrkl is also a
divergence. If we are to define a divergence with respect to a covariant slot (i.e.,
a lower index), then we must use a metric to raise the index. This leads to
the following definition appropriate in the presence of a metric.
624
13. Riemannian and Semi-Riemannian Geometry
Definition 13Л55. Let V be the Levi-Civita covariant derivative on a
manifold M. The (metric) divergence of a function is defined to be zero and
the divergence of a (0, i) tensor field A is a (0, ί — 1) tensor field defined at
any ρ G Μ by
η
(div A)p (vi,..., ve-i) = J^ (ei? ej) (Ve. A) (ei? гл,..., v*_i),
i=i
where (ei,..., en) is an orthonormal basis for TpM. Note that the factors
(e^, ej) are equal to 1 in the case of a definite metric.
On may check that if А^.^ц are the components of A in some chart,
then the components of div A are given by
Recall that the formal adjoint δ : Ω(Μ) —> Ω(Μ) of the exterior derivative
on a Riemannian manifold Μ is given on Clk(M) by δ := (—i)n(fc+1)+1 * d*.
In Problem 8 we ask the reader to show that the restriction of div to Qk(M)
is —δ:
(13.11) div = -δ on Ω*(Μ) for each k.
In the same problem the reader is asked to show that if μ G Q,k(M) is parallel
(\7μ = 0), then it is harmonic (recall Definition 9.46).
In what follows we simplify calculations by the use of a special kind of
orthonormal frame field. If (M, g) is a Riemannian manifold and ρ G M,
choose an orthonormal basis (ei,..., en) in the tangent space TPM and
parallel translate each e\ along the radial geodesies t ь-> ехрр(£г>) for ν G TPM.
This results in an orthonormal frame field (£χ,... ,£n) on some normal
neighborhood centered at p. Smoothness is easy to prove. The resulting
fields are radially parallel and satisfy Ei(p) = ei and (VJ^) (p) = 0 for
every г. Furthermore we have
[Ei, Ej] (p) = VEiEj(p) - VEjEt(p) = 0 for all г J.
We refer to an orthonormal frame field with these properties as an adapted
orthonormal frame field centered at p.
Before proceeding we need an exercise to set things up.
Exercise 13.156. Describe how a connection V (say the Levi-Civita
connection) on Μ extends to a connection on the bundle ДГ*М in such a way
that
к
Vx (α1 Λ · · · Λ ak \ = Σ α1 Λ · · · Λ Vxa{ Λ···Λ^
2 = 1
13.12. Weitzenbock Formulas
625
for аг G Ω1(Μ). Show that the curvature of the extended connection is
given by
R(X, Υ)μ = νχνγμ - Vy νχμ - V[x?y]M.
Relate this curvature operator to the curvature operator on X(M) and show-
that the extended connection is flat if the original connection on Μ is flat.
Let Μ be Riemannian and μ G £lk(M). Define /?μ by
η к
ϋμ(νΐ, · · · ,Vk) = Σ Σ (R(e*> Уз)^ (Vb · · · , Vj-i,ei, Vj+u . . . , Vfe).
t=i i=i
Theorem 13.157 (Weitzenbock formulas). Lei Μ be Riemannian and μ G
Ω*(Μ). T/ien we /mve
(Δμ|μ) = ΐΔ||μ||2 + ||νμ||2 + (ϋ!μ|μ),
Δμ = -(ϋννμ + Λμ,
where \\Vμ\\ (ρ) := £^ (νβίμ|νβϊμ) /or any orthonormal basis (ei,..., en)
/or TPM.
Proof. Using the formula of Theorem 12.56, we see that for C7, Vi,..., Vk G
X(M), we have
With this in mind, we fix ρ G Μ and i>i,... ,ι^ G TpM. We may choose
V\,..., Vk G X(M) such that V*(p) = i>; and may assume that (VVJ) (p) = 0.
Now choose an adapted orthonormal frame field E\,..., £n centered at ρ so
that (V2?i) (p) = 0 for all i and [JSj, J5j] (p) = 0 for all i,j. In the following
calculation, several steps may appear at first to be wrong. However, if one
begins to write out the missing terms, one sees that they vanish because of
how the fields were chosen to behave at p. Using equation (13.11) we have
(div \7μ + δάμ) (υι, · · ·, vk) = div (\7μ -άμ){ν\,..., vk)
η
2 = 1
η
= £ei[(Vjx-<^)(£i,Vi,...,Vb)]
»=1
626
13. Riemannian and Semi-Riemannian Geometry
and using what we know about the fields at p, this is equal to
η Ik
ΣβΜ Σ (v^) w> · · · > ^-ь Е>> ^+ь · · ·' ук)
i=l \j=l
η Α:
= ΣΣ^((ν^Μ)(^-..,^-ι,^·,κ+1,...,ΐ4))
г=1 j=l
η к
= EE(vftN)^--b^%,.,vfc)(p)
г=1 j=l
η Α:
= Σ Σ (V<4 Vv-μ) (г;ь · · ·, Vi-ι, ej, Щ+ъ ...,vk).
г=1 i=l
We also have
fe
άδμ (vi,..., ν*) = ^(-1)J+1 (νν/μ) (vi,..., v},..., vk)
j=i
к / η \
j=l \i=l J
к п
Since [Ι?*, V?·] = V^I^ - Vy^z = 0, we can add the above results to obtain
Δμ + divVμ = ^гμ.
For the second part we again take advantage of our arrangement VEi =
0, VVj = 0 at p; we have
(div νμ) (г>ь ..., vfe) = Σ (Ve*V^) (e^^b · · · ,vk)
η
η
i=l
13.13. Structure of General Relativity
627
Prom this we obtain
η
(- div νμ Ι μ) (ρ) = - Σ (Vei νΕζμ | μ(ρ)>
г=1
η
= - Σ (е* <ν*^ Ι ^ ~ <Ve^ |vei μ))
г=1
= -1-ΣβίΕί\\μ\\2 + \\νμ\\2(ρ)
г=1
= Qa|hi2 + ||vm||2)(p). d
We give only one application (but see [Pe] or [Poor]).
Proposition 13.158. // (M,g) is α flat connected compact Riemannian
manifold (without boundary), then a form μ G Qk(M) is parallel if and only
if it is harmonic.
Proof. We have observed the inclusion {parallel /c-forms} С {harmonic k-
forms} (Problem 8). On the other hand, if (M,g) is flat, then /?μ = 0 and
so by Theorem 13.157 we have (Δμ|μ) = ^Δ ||μ||2 + || νμ||2. So if Δμ = 0,
we have
\\νμ\\2άν = -4 Δ||μ||2^ = 0
Δ J Μ
by Stokes' theorem. Thus \7μ = 0. D
Corollary 13.159. Let Μ be a connected compact η-manifold. If Μ admits
a flat Riemannian metric g, then dim Hk(M) < (^).
Proof. Pick any ρ e M. Prom the Hodge theorem, and the previous
proposition, we have
dimi/^M) = dim{harmonic /c-forms} = dim{parallel k-forms}
<dimA4*M=Q·
The inequality follows from Exercise 13.14. D
13.13. Structure of General Relativity
The reader is now in a position to appreciate the basic structure of
Einstein's general theory of relativity. We can only say a few words about this
wonderful part of physics. General Relativity is a theory of gravity based
on the mathematics of semi-Riemannian geometry. In a nutshell, the theory
models spacetime as a four dimensional Lorentz manifold M4 that is usually
assumed to be time oriented. The points of spacetime are idealized events.
/
Jm
628
13. Riemannian and Semi-Riemannian Geometry
The motion of a test particle subject only to gravity is along a geodesic in
spacetime and the metric is subject to a nonlinear tensor differential
equation that involves the curvature and a physical tensor Τ that describes the
local flow of energy-momentum. We consider the following equations to be
central:
(13.12) Ric —-Rg = 8πκΓ (Einstein's equation),
(13.13) divT = 0 (continuity),
(13.14) ν<*ά = 0 (geodesic equation),
(13.15) VdVd^ + R(V, ά)ά = 0 (Jacobi equation),
where in the geodesic equation and Jacobi equation, λ ь-> α (λ) is a
parametrized curve that represents the career of a test particle subject only to
gravitation. In the Jacobi equation we are to imagine a smooth family of
geodesic curves t \-> hs(t) = h(t,s) such that ho = a. Then V = jj* is
the variation vector field. The first equation, Einstein's equation, is the
centerpiece of Einstein's theory. On the right hand side of this equation we
see a tensor Τ called the stress-energy-momentum tensor; it represents the
matter and energy that generate the gravitational field. The constant к is
Newton's gravitational constant, which is also often denoted by the letter
G. The left hand side is the Einstein curvature tensor and is built from the
Riemann curvature tensor and so ultimately from the metric tensor. There,
"Ric" is the Ricci curvature whose index form is written RμV\ R is the scalar
curvature defined by contraction R := gμvRμV^l and g is the metric tensor.
Einstein's equation can be seen as an equation for the metric of spacetime;
it shows how the distribution of matter and energy influences the metric and
the resulting curvature of spacetime. We have already studied the last two
equations, but we will say something below about their role in gravitational
theory.
We will base our explanations on the following two-pronged incantation:
Matter tells spacetime how to curve.
Spacetime tells matter how to move.
Newtonian gravity. To fully appreciate Einstein's theory of gravity
one must compare it to Newton's theory. In Newton's theory, the equations
of motion of a test particle moving in (flat Euclidean) space and subject to
a gravitational field g is described by
(13.16) m7^^=mGg(x(i)).
Here x(i) is a vector-valued function of t that gives the location of the test
particle relative to a fixed inertial frame (which entails a choice of origin and
13.13. Structure of General Relativity
629
system of rectangular coordinates). The field g is a vector-valued function
of position in space. Here m/ is the inertial mass of the particle, which is the
m in Newton's F = ma. The constant mc is the gravitational mass, which
plays the role of gravitational charge. As demonstrated by Galileo and later
by Eotvos, we actually have equality mj = txiq. This equality is in fact one
of the key influences on Einstein's thinking, and it led him to assert that,
for sufficiently small regions of spacetime, gravitational forces and inertial
forces (as perceived in an accelerating frame) are indistinguishable.
If ρ represents the mass density in a region of space, then the
gravitational potential φ produced by this matter is given by
V20 = 4πκρ.
The gravitational field is then
g = - grad φ.
Thus we have the pair of equations
(13.17) ν20 = 4πκρ,
(13.18) ^ = _grad0,
where V20 = div (grad φ) and where the right hand side is evaluated at x(i).
The first equation tells how matter creates the Newtonian gravity field, and
the second describes how the field tells matter how to move. This is the
Newtonian analogue of the two-pronged incantation above. In Newton's
picture, gravity is unambiguously treated as a field created by mass that
induces a force on test particles.
Free fall. Let α be a curve parametrized by proper time r that
represents the path of a test particle. In Einstein's theory, what corresponds
to equation (13.18) is the geodesic equation а := ν<*ά = 0. According to
Einstein, if the particle is subject only to gravity, then α is a geodesic. If we
choose a coordinate system (x°, x1, x2, x3), then the geodesic equation gives
four differential equations (the geodesic equations), which can be written as
ά2χμ _ 1 μδ fdg0S { дд6а дда0\ dxa άχβ
(13.19, _ = --r^ + ^-#J__, „ = 0,1,2,3,
where we have written out the formula for T^o explicitly. (We use the
common convention that the Greek indices run over 0,1,2,3, while the Latin
indices run over 1,2,3.) The above equations have a similarity to (13.18)
when the latter are written in the form
£*---**- i-123
630
13. Riemannian and Semi-Riemannian Geometry
From this point of view, (13.19) looks like a force law and the metric
components ga$ play a role analogous to the potential φ in the Newtonian theory.
Prom the point of view of the chosen coordinate system, these equations
appear to tell the particle how to accelerate with respect to the coordinate
system. However, if we choose normal coordinates at an event ρ G M4,
then at ρ the left hand side of (13.19) is zero! In fact, from the intrinsic
point of view, the law is simply that the career of the particle in spacetime
is a geodesic and therefore represents a state of zero intrinsic acceleration.
Rephrasing the second prong of our incantation, we say that spacetime tells
free test particles how to curve or accelerate. Namely, not at all. This
is a wonderfully simple geometric law of motion. Freely falling bodies are
described by geodesies in spacetime.
Tidal forces. Imagine yourself in free fall in a uniform gravitational
field. Imagine that you are surrounded by a spherical array of apples in
free fall which are initially stationary with respect to you. All the apples
appear motionless against the starry background of space. If the field was
truly uniform, the spherical swarm would remain spherical. However, this
situation corresponds to no spacetime curvature, and so from the intrinsic
point of view, is no gravitational field at all. A realistic gravitational field
such as that produced by the Earth is not uniform, and our sphere of apples
would deform becoming elongated along a line passing through the center
of mass of the gravitating body (the Earth, say) and passing through the
center of the array. If you were in free fall at the center of the spherical
array with your feet toward the earth, then apples that are roughly in a
plane perpendicular to the axis of your body would be seen to accelerate
towards you, while those below your feet and above your head would be
seen to recede. Each apple follows a geodesic in spacetime (not in space),
and so we have a family of geodesies. This situation, properly idealized,
is described by the Jacobi equation (13.15). The vector field V can be
thought of as describing the separation of nearby geodesies, and the Jacobi
equation describes the relative acceleration of nearby geodesies. Curvature is
the "force" behind this relative acceleration. It is this relative acceleration,
positive in some directions and negative in others, that is responsible for
the distortion of our initially spherical array of free falling apples. If we
neglect the attraction that the apples have for each other, then the volume
of the array remains constant. This is because we are in a region where the
Einstein tensor (and hence the Ricci tensor) vanishes.
Energy-momentum tensor. Fields and particles carry 4-momentum.
Let α be a unit speed timelike curve giving the career of a particle of
(rest) mass m. The 4-momentum of the particle is ρ = ma. In a flat
spacetime we may choose Lorentz coordinates (x0,^1,^2,^3) so that the
corresponding coordinate frame field (<9μ) is oriented orthonormal with <9q
13.13. Structure of General Relativity
631
timelike. We choose units so that the speed of light is unity; с = 1 so
that ж0 = t is coordinate time. Let ν := (^>^§->^|г) be the
"ordinary" spatial velocity or "3-velocity" as viewed in this Lorentz frame and
let ν := |v| = [^2i=i(dxl/dt)2] . Then the 4-momentum has components
(22,7p), where Ε is the relativistic energy of the particle, 7 = (1 — г;2)-1/2,
and we shall refer to 7p as the relativistic 3-momentum. Notice that if we
change to a different Lorentz coordinate, then we will have a new energy
and 3-momentum, but the geometric 4-momentum vector та is an
"invariant" notion defined without reference to a specific coordinate frame. Thus
4-momentum unifies the notions of momentum and energy. Furthermore,
this relativistic energy is really mass-energy. Indeed, if we choose a frame
in which the particle is (momentarily) at rest, then the 4-momentum has
components (m, 0).
Now when we consider a region in spacetime filled with particles and
fields, it is appropriate to go to the continuum approximation. The right
hand side of Einstein's equation features the (0,2)-tensor Τ that keeps track
of the flow of 4-momentum produced by all the matter and non-gravitational
energy. It will pay to first think about charge and then consider the meaning
of Τ in the setting of special relativity. In this setting of special relativity
we are assuming that Τ does not produce curvature (contrary to fact). In
standard Lorentz coordinates (ж0, ж1, ж2, ж3), the tensor Г has 16
components Τμν. Let us do a type change, Τμν := g^Tav. Then (Τ00,Τ°νΤ02ιΤ°3)
represents the density of 4-momentum and {Т\,Т\,Тг2,Тг^) represents the
flux of 4-momentum in the spatial direction г (so г = 1,2, or 3). By this
we mean that the result of a flux integral should be a 4-vector quantity,
while the flux of a vector field in the usual calculus sense is a scalar (such
as charge or mass per unit time).
In electrodynamics we describe the flow of a charge by a time dependent
vector field J, and if ρ is the charge density, then local conservation of charge
is given by the continuity equation
^ = -dlvJ-
We can combine ρ and J into a unified notion of 4-current J that has
components in a Lorentz frame given by (J°,..., J3) = (p, J) and the
corresponding covector field J has components (Jo,..., J3) = (—p, J). The continuity
equation can then be written as
d* J = 0,
632
13. Riemannian and Semi-Riemannian Geometry
and follows from Maxwell's equations. The corresponding integral version
of conservation of charge is simply
/ *J = 0,
JdR
where dR is the boundary of a region of spacetime R. For a general volume
Ω that does not necessarily bound an open region of spacetime, the integral
Jq *J gives the total charge "crossing" Ω. It is the fact that charge is a
scalar quantity that allows us to do a continuous sum over Ω.
In the Newtonian theory, mass is the analogue of charge and we have
a similar continuity equation. However, in relativity, rest mass is not a
conserved quantity and energy is not a scalar. Somehow charge is to be
replaced by energy-momentum as we cross the bridge of analogy to the land
of gravity. If spacetime is Minkowski space, then there is a quantity we can
integrate to get a total energy-momentum. The integral implies a sum, and
we can add tensors located at different points if we take advantage of distant
parallelism (see [M-T-W]). If we express Γ in a Lorentz frame, then we
can integrate to get a total energy-momentum ptot crossing a "3-volume" V.
The covariant components of iptot in the Lorentz frame are given by
ρ^ = ί ГДО",
Jv
where dE" = *dxv = ^εναβ dxa Λ dx13 Λ dx1. Then, the conservation law
says that Jv Т„аТУ = 0 if V = <9Ω is the boundary of a spacetime 4-volume
Ω. The differential form of the conservation law is divT = 0, which in a
Lorentz frame is just <9μΤ£ί = 0.
Now, for general relativity we retain the local version of conservation
by assuming that div Τ = 0, which is now defined in terms of the covariant
derivative. In general coordinates we have
νμΓ£ = 0.
However, we must give up the integral version, although it would still hold
approximately for sufficiently small regions of spacetime since the latter
would be approximately flat. We shall find that this continuity equation is
automatically satisfied if Einstein's equation holds since it turns out that
the divergence of the left hand side is zero for purely geometric reasons!
The Einstein tensor. The left hand side of Einstein's equation
features the tensor Ric— ^Rg given in index notation as Rμv — \Rgμv. Here,
Rμv — R^au and R — R^ (sum). This tensor is denoted by the letter G
and is called the Einstein tensor. Let us show that divG = 0. This fact
shows that the conservation law is forced by the geometry (assuming that
Einstein's equation holds). In this sense we may take this to be another
manifestation of the second prong of our incantation in that the geometry
13.13. Structure of General Relativity
633
tells matter to behave in accordance with local conservation. We have not
seen many examples of tensor calculations using index notation, so we take
this opportunity to do a calculation. In index notation, what we wish to
show is that V μϋμν = νμϋμν = 0. We start with the Bianchi identity, make
a switch in the first two indices of the first term and then raise indices and
contract:
0 = νμϋαβΊδ + να/2/?μ7<5 + VβRμαΊδ,
0 = -WRilS + VaR% + Ψϋ»αΊ&,
0 = - V Д*^ + VaR% + VsR\lS,
0 = -VRsaiS - VaR^s + VsRllS,
0 = V7i?a7 - VaR + V5Ra$.
This last equation gives V7.Ra7 — ^VaR = 0. But since νμρμι, = 0, we have
WGV = V"(iV - l-R9tlu) = (ViV - ^V (%«,))
= (V%„ - \9μανα (%,„)) = (ν"ΛμΙ/ - \g»a (g^VaR))
= V^Q7 - \stVaR = V^Rai - ^VaR = 0.
The Schwarzschild metric. If we contract both sides of Einstein's
equation, we obtain R = -8πκΤμ. Plugging this back into Einstein's
equation and rearranging we obtain an equivalent form of Einstein's equation
Rμv — 8πκ (Τμι/ — т}Т£д^). Thus in case the tensor Τ vanishes in the
region of interest, we obtain the vacuum field equation
Rμv — 0.
There is a famous metric defined in terms of spherical coordinates (i, r, 0,0),
given by
ds2 = -(i- ^M.) dt2 +(l- IHM.)'1 άν2 + ν2((ΙΘ2+3[η2θάφ2),
where θ is the polar angle 0 < θ < π. Notice that this expression for our
metric is undefined at both r = 0 and r = 2κΜ. But this is only the
coordinate expression for a metric that intrinsically may be quite nice at
r = 2кМ and/or r = 0. It can be shown that there is a metric perfectly
well-defined on r = 2кМ whose expression in (i, r, θ,φ) just happens to be
the above away from r = 2κΜ and r = 0. The fact that the above expression
blows up as we approach r = 2кМ is a failure of the coordinates and not
a feature of the intrinsic metric. On the other hand, if one calculated the
scalar curvature, then it can be seen to blow up as r —> 0, which makes
634 13. Riemannian and Semi-Riemannian Geometry
r = 0 a true singularity. This metric is called the Schwarzschild metric
and describes a spherically symmetric solution of Einstein's equation that
is taken to be due to a spherical distribution of matter concentrated near
r = 0 (such as a star). We should mention that this metric only satisfies
Einstein's equation in the vacuum away from the star. In most cases the
radius of the star is larger than 2κΜ, and then the Schwarzschild metric
is not a solution inside the star anyway. If the radius of the star is less
than 2кМ, then we have a nonrotating black hole, and r = 2кМ defines the
famous event horizon.
Problems
(1) Show that in a Riemannian manifold, a length minimizing piecewise
smooth curve must be a smooth geodesic. [Hint: Each potential kink
point has a totally star-shaped neighborhood. Use Proposition 13.85.]
(2) A set U in a Riemannian manifold Μ is said to be geodesically convex
if for each pair of points p, q G /7, there is a unique length minimizing
geodesic segment connecting them, and this unique geodesic segment
lies completely in U.
(a) Show that the intersection of geodesically convex sets is geodesically
convex.
(b) Show that given ρ e Μ there is a δ > 0 such that exp({i>p G TpM :
\\vp\\ < £}) is geodesically convex for all ε < δ.
(3) Referring to the discussion leading up to Proposition 13.135, let Υ G
Γ7(Ω) be a piecewise smooth variation vector field along 7 and write
Υ = Σ Ψ%^ι f°r some piecewise smooth functions φ1 on (0, b]. Show that
the ψ1 can be extended continuously to [0, b].
(4) A smooth map / : (M, g) —> (JV, h) of semi-Riemannian manifolds is
called a homothety if f*h = eg for some constant с ф 0. The case
of с = —1 is called an anti-isometry. Show that an anti-isometry
preserves covariant derivatives and geodesies.
(5) Show that the mapping σ : M%+1 -> Μ£ΐ*+1 given by
σ(αι,..., αη+ι) := (α„+ι,..., αη+ι, аь ..., α„)
is an anti-isometry (see above) and that its restriction to 5"(r) is an
anti-isometry from 5"(r) onto Н™-„{г).
(6) Show that timelike vectors ν and w in a Lorentz space V are in the same
timecone if and only if (v, w) < 0.
(7) Construct examples sufficient to make the point that time orientability
(of Lorentz manifolds) and orientability are unrelated.
Problems
635
(8) Show that the restriction of div to Qk(M) is -δ. Show that if μ Ε Ω* (Μ)
is parallel {S7 χμ = 0 for all X G X(M)), then it is harmonic.
(9) Let Η := {(г^г;) Ε Μ2 : ν > 0} be the upper half-plane endowed with
the metric
g := - (d-u ® dw + dv ® rfv).
г;
Show that if has constant curvature К = — 1. Find which curves are
geodesies.
(10) (Killing fields) On a semi-Riemannian manifold (M, p), a vector field
X is called a Killing field if Cxg = 0. Show that the local flows
of a Killing field are isometries. Show that X is a Killing field if and
only if X (V, W) = (CXV, W) + (V, CXW) for all V, W e X(M). Show
that X is a Killing field if and only if (VVX, W) = - {VWX, V) for all
V,WeX(M).
(11) Show that if 7 is a geodesic in Μ and X is a Killing field (see the previous
problem), then Χ ο η is a Jacobi field along 7 and that (X 07,7) is
constant.
(12) Show that an M-linear combination of Killing fields is a Killing field and
that if X and Υ are Killing fields, then £[χ?γ] = [£χ, Су]. Deduce that
the space of Killing fields is a real Lie algebra. What are the Killing
fields of M3?
(13) Let (M, g) and (iV, h) be semi-Riemannian manifolds. Let / be a positive
smooth function on M. The warped product metric on Μ χ TV is defined
by
(g x/ h) := pris + (/ о рг2) РГ25,
where prx : Μ χ TV —>· Μ and pr2 : Μ χ Ν -^ Ν are the first and second
factor projections.
(a) Show that this is indeed a metric.
(b) Show that for each ρ e M, the map рг2|рхдт is a homothety.
(c) Show that each Mxgis normal to each ρ χ N.
(d) Let RM be the curvature on (M, g) and RMxN be the curvature
tensor on (Μ χ N,g Xf h). If X,У, and Ζ are lifts of X,У, Ζ e
X(M) as described in Problem 30 of Chapter 2, then what is the
relationship between R%X~NZ and R%YZ1
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Appendix A
The Language of
Category Theory
Category theory provides a powerful means of organizing our thinking in
mathematics. Some readers may be put off by the abstract nature of
category theory. To such readers, I can only say that it is not really difficult
to catch on to the spirit of category theory and the payoff in terms of
organizing mathematical thinking is considerable. I encourage these readers to
give it a chance. In any case, it is not strictly necessary for the reader to be
completely at home with category theory before going further into the book.
In particular, physics and engineering students may not be used to this kind
of abstraction and should simply try to gradually become accustomed to the
language. Feel free to defer reading this appendix on category theory until
it seems necessary.
Roughly speaking, category theory is an attempt at clarifying structural
similarities that tie together different parts of mathematics. A category has
"objects" and "morphisms". The prototypical category is just the category
Set which has for its objects ordinary sets and for its morphisms maps
between sets. The most important category for differential geometry is what
is sometimes called the "smooth category" consisting of smooth manifolds
and smooth maps. (The definition of these terms is given in the text proper,
but roughly speaking, smooth means differentiable.)
Now on to the formal definition of a category.
Definition A.l. A category С is a collection of objects Ob(£) = {X, У,
Z,...} and for every pair of objects X, У, a set Hom^X, Y) called the set of
morphisms from X to Y. The family of all morphisms in a category £ will be
637
638
A. The Language of Category Theory
denoted Mor(C). In addition, a category is required to have a composition
law which is defined as a map о : Hom^X, У) χ Нот<г(У, Ζ) —ϊ Hom^X, Z)
such that for every three objects X, У, Ζ G Ob(£) the following axioms hold:
(1) Hom(r(X, У) and Hom(r(Z, W) are disjoint unless X = Ζ and Υ =
W, in which case Uom€(X,Y) = Hom€(Z,W).
(2) The composition law is associative: / о (д о h) = (/ о д) о h.
(3) Each set of morphisms of the form Hom(r(A', X) must contain a
necessarily unique element idx, the identity element, such that
/ о idx = / for any / G Hom^X, Y) (and any У), and idx of = f
for any / e НоШ(р(У, X).
Notation A.2. A morphism is sometimes written using an arrow. For
example, if / G Hom^(X, У) we would indicate this by writing / : X —> Υ
or by X Л У.
The notion of category is typified by the case where the objects are sets
and the morphisms are maps between the sets. In fact, subject to putting
extra structure on the sets and the maps, this will be almost the only type
of category we shall need to talk about. On the other hand there are plenty
of interesting categories of this type. Examples include the following.
(1) Grp: The objects are groups and the morphisms are group homo-
morphisms.
(2) Rng : The objects are rings and the morphisms are ring homo-
morphisms.
(3) LinF : The objects are vector spaces over the field F and the
morphisms are linear maps. This category is referred to as the linear
category or the vector space category (over the field F).
(4) Top: The objects are topological spaces and the morphisms are
continuous maps.
(5) Manr: The category of Cr differentiable manifolds and Cr maps:
One of the main categories discussed in this book. This is also called
the smooth or differentiable category, especially when r = oo.
Notation A.3. If for some morphisms fi : Xi ->Yi , (i = 1,2), gx : X\ —>
X2 and gy : Y\ —> У2 we have gy о f\ = /2 о дх, then we express this by
saying that the following diagram "commutes":
9X\ \9Y
Υ Υ
Χ2—Γ+Υ2
h
A. The Language of Category Theory
639
Similarly, if h о f = p, we say that the diagram
commutes. More generally, tracing out a path of arrows in a diagram
corresponds to composition of morphisms, and to say that such a diagram
commutes is to say that the compositions arising from two paths of arrows
that begin and end at the same objects are equal.
Definition A.4. Suppose that / : X —> Υ is a morphism from some
category €. If / has the property that for any two (parallel) morphisms <ji,
#2 · Ζ —> X we always have that fogi = /o#2 implies g\ = #2, i-e. if / is "left
cancellable", then we call / a monomorphism. Similarly, if / : X —> Υ is
"right cancellable", we call / an epimorphism. A morphism that is both
a monomorphism and an epimorphism is called an isomorphism. If the
category needs to be specified, then we talk about a £-monomorphism,
C-epimorphism and so on).
In some cases we will use other terminology. For example, an
isomorphism in the smooth category is called a diffeomorphism. In the linear
category, we speak of linear maps and linear isomorphisms. Morphisms
which comprise Hom<r(.X",.X") are also called endomorphisms and so we also
write End<r(X) := Hom^X, X). The set of all isomorphisms in Hom^X, X)
is sometimes denoted by Aut<r(.X"), and these morphisms are called
automorphisms.
We single out the following: In many categories such as the above, we
can form a new category that uses the notion of pointed space and pointed
map. For example, we have the "pointed topological category". A pointed
topological space is a topological space X together with a distinguished
point p. Thus a typical object in the pointed topological category would be
written as (X,p). A morphism / : (X,p) —> (W, q) is a continuous map such
that f(p) = q.
A functor J7 is a pair of maps, both denoted by the same letter J7, that
map objects and morphisms from one category to those of another,
7":Ob(Ci)->Ob(<!:2),
•F:Mor(Ci) -»Mor(C2),
so that composition and identity morphisms are respected. This means that
for a morphism / : X —> У, the morphism
T(f) : T(X) -* НУ)
640
A. The Language of Category Theory
is a morphism in the second category and we must have
(1) J-(id€l)=idea.
(2) If / : X -> Υ and д : Υ -+ Ζ, then J"(/) : J"(*) -»· 7"(У),
Л5) : НУ) "> Л^) and
Л<?°/)=Л<?)°П/)·
Example А.5. Let Lin^ be the category whose objects are real vector
spaces and whose morphisms are real linear maps. Similarly, let Line be
the category of complex vector spaces with complex linear maps. To each
real vector space V, we can associate the complex vector space С ®ш V,
called the complexification of V, and to each linear map of real vector spaces
I: V —> W we associate the complex extension £c : С®к V—> С®к W. Here,
С ®м V is easily thought of as the vector space V where now complex scalars
are allowed. Elements of С®к V are generated by elements of the form c®v,
where с G C, ν G V and we have i(c ® v) = ic ® v, where г = у/^Л. The
map ic '- C®r V—> C®r W is defined by the requirement lc(c®v) = c®£v.
Now the assignments
VhC®kV
define a functor from Lin^ to Line- In practice, complexification amounts
to simply allowing complex scalars. For instance, we might just write cv
instead of c® v.
Actually, what we have defined here is a covariant functor. A con-
travariant functor is defined similarly except that the order of
composition is reversed so that instead of (2) above we would have Т(д о f) =
F(f) of(g). An example of a contravariant functor is the dual vector space
functor, which is a functor from the category of vector spaces Lin^ to itself
that sends each space V to its dual V* and each linear map to its dual (or
transpose). Under this functor a morphism V —> W is sent to the morphism
V* ^— W*.
Notice the arrow reversal.
One of the most important functors for our purposes is the tangent
functor defined in Chapter 2. Roughly speaking this functor replaces
different iable maps and spaces by their linear parts.
Example A.6. Consider the category of real vector spaces and linear maps.
To every vector space V, we can associate the dual of the dual V**. This is
A. The Language of Category Theory
641
a covariant functor which is the composition of the dual functor with itself:
V W* V**
A*
A**
w
v*
w*
suppose we
have two functors,
Tx
Ά
Ъ
Тг
: Ob(Ci) ->
Ob(C2),
: Mor(Ci) ->· Mor(C2)
Ob(Ci) -»·
Ob(C2),
Mor(Ci) ->· Mor(C2)
and
A natural transformation Τ from T\ to Тч is given by assigning to each
object X of d, a morphism T(X) : F\{X) —> ^(X) such that for every
morphism / : X —> Υ of Ci, the following diagram commutes:
7i(y)
Т(У)
^"2 (/)
•^2(V)
A common first example is the natural transformation ι between the identity
functor I : LiniR —> Lin^ and the double dual functor ** : Lin^ —> Lin^:
V-
w
t(V)
l(W)
V*
r*
w*
The map V —> V** sends a vector to a linear function υ : V* —> Ш defined
by v(a) := a(v) (the hunted becomes the hunter, so to speak). If there is
an inverse natural transformation T~l in the obvious sense, then we say
that Τ is a natural isomorphism, and for any object X G <£i we say that
Τ\{Χ) is naturally isomorphic to ^{X)- The natural transformation just
defined is easily checked to have an inverse, so it is a natural isomorphism.
The point here is not just that V is isomorphic to V** in the category Lin^,
but that the isomorphism exhibited is natural. It works for all the spaces V
in a uniform way that involves no special choices. This is to be contrasted
with the fact that V is isomorphic to V*, where the construction of such an
isomorphism involves an arbitrary choice of a basis.
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Appendix В
Topology
B.l. The Shrinking Lemma
We first state and prove a simple special case of the shrinking lemma since
it makes clear the main idea at the root of the fancier versions.
Lemma B.l. Let X be a normal topological space and {Ui,U2} an open
cover of X. There exists an open set V with V С U\ such that {V, U2} is
still a cover of X.
Proof. Since Ui Uf72 = X, we have (X\Ui)n(X\U2) = 0. Using normality,
we find disjoint open sets О and V such that X\U\ С О and X\U2 С V.
Then it follows that X\0 С f7]_and X\V С U2 and so X = U2 U V. But
ODV = $soV С X\0. Thus V С Х\0 С J7i. D
Proposition B.2. Let X be a normal topological space and {/7i, {/2,..., Un}
a finite open cover of X. Then there exists an open cover {Vi, V2,..., Vn}
such that Vi С Ui for г = 1, 2,..., η.
Proof. Simple induction using Lemma B.l. D
The proof we give of the shrinking lemma uses transfinite induction. A
different proof may be found in the online supplement [Lee, Jeff]. It is a
fact that any set A can be well ordered, which means that we may impose a
partial order -< on the set so that each nonempty subset S С A has a least
element. Every well-ordered set is in an order preserving isomorphism with
an ordinal ω (which, by definition, is itself the set of ordinals strictly less
than ω). For purposes of transfinite induction, we may as well assume that
the given indexing set is such an ordinal. Let A be the ordinal which is the
indexing set. Each a e A has a unique successor, which is written as a + 1.
643
644
В. Topology
The successor of a + 1 is written as a + 2, and so on. If /3 G A has the form
β = {α, α + Ι,α + 2,...}, then we say that /3 is a limit ordinal and of course
α -< β for all α G /3. (This may seem confusing if one is not familiar with
ordinals.) Suppose we have a statement Ρ (a) for all a e A. Let 0 denote
the first element of A. The principle of transfinite induction on A says that
if P(0) is true and if the truth of Ρ (a) for all a -< β can be shown to imply
the truth of Ρ (β) for arbitrary β, then Ρ (a) is in fact true for all a G A. In
most cases, a transfinite induction proof has three steps:
(1) Zero case: Prove that P(0) is true.
(2) Successor case: Prove that for any successor ordinal a + 1, the
assumption that Ρ(θ) is true for all θ -< a +1 implies that Ρ (a +1)
is true.
(3) Limit case: Prove that for any limit ordinal ω, Ρ (ω) follows from
the assumption that Ρ (a) is true for all a -< ω.
Definition B.3. A cover {Ua}a^A of a topological space X is called point
finite if for every ρ e X the family A(p) = {a : ρ G Ua} is finite.
Clearly, a locally finite cover is point finite.
Theorem B.4. Let X be a normal topological space and {Ua}aeA Q> point
finite open cover of X. Then there exists an open cover {Va}a^A of X such
that Va С Ua for all a G A.
Proof. Assume that A is an ordinal. The goal is to construct a cover
{К*}аел °f X such that ^а С Ua for all a G A. We do transfinite induction
on A. Let Ρ (a) be the statement
(*) For all θ -< a there exists V$ with V$ С U$ such that {Ve}e^a U
{Ue}a^e is a cover of X.
The goal is to prove that Ρ (a) is true for all a -< A + 1 since this entails
that Ρ (A) is true.
Case 1: a has a successor а + 1. Assume that Ρ (α) holds. We show
that Ρ (a + 1) is true, which entails constructing Va. Let
^ = ^\{(и^)и(иа+и^)}·
Clearly, F is closed and F С Ua. By normality there is a set Va with
F С Va С Va С ί/α.
Then {V£}0^Q+iU{E/0}Q+i-^0 is a cover of X, which is the statement P(a+l).
Case 2: Limit ordinal case. Suppose that ω is a limit ordinal and
that Ρ(β) is true for all θ -< ω. We want to show that this implies that
{Ув}в^и U {Ue}u^e is a cover of X. Suppose this is not the case. Then
В.2. Locally Euclidean Spaces
645
there is an element χ e X which is not in the union of this family of open
sets. We know that there exists a finite collection of sets C/Q1,..., C/Qn, each
containing χ and such that no other Ua contains x. We have that cti -< ω for
each г = 1,..., η, and since ω is a limit ordinal, there exists an ordinal δ such
that oti -< δ -< ω for all i. Then the point χ is in the union {Ve}e^sU{Uo}s-<e,
and since we know that χ is not in any of the sets in {Ue}s<e-> we must have
χ e Vq for some θ -< δ so that
* e {\Je*M и (lU^) ,
which contradicts our assumption that {Ve}e^u U {Ue}u:<e is not a cover of
X. Thus Ρ (a) is true for all a -< A + 1. D
B.2. Locally Euclidean Spaces
If every point of a topological space X has an open neighborhood that is
homeomorphic to an open set in a Euclidean space, then we say that X is
locally Euclidean. A locally Euclidean space need not be Hausdorff. For
example, if we take the spaces R χ {0} and R χ {1} and give them the relative
topologies as subsets of R χ R, then they are both homeomorphic to R. Now
on the (disjoint) union (R χ {0}) U (R χ {1}) define an equivalence relation
by requiring (x,0) ~ (ж, 1) except when χ = 0. The quotient topological
space thus obtained in locally Euclidean, but not Hausdorff. Indeed, the
two points [(0,0)] and [(0,1)] are distinct but cannot be separated. It is as
if they both occupy the origin.
A refinement of a cover {ΙΙβ}β^Β οι ά topological space X is another
cover {Vi}i£i such that every set from the second cover is contained in at
least one set from the original cover. We say that a cover {V^e/ of a
topological space X is a locally finite cover if every point of X has a
neighborhood that intersects only a finite number of sets from the cover. A
topological space X is called paracompact if every open cover of X has a
refinement which is a locally finite open cover.
Proposition B.5. If X is α locally Euclidean Hausdorff space, then the are
following properties equivalent:
(1) X is paracompact
(2) X is metrizable.
(3) Each connected component of X is second countable.
(4) Each connected component of X is a-compact.
(5) Each connected component of X is separable.
For a proof see [Spv, volume I] and [Dug].
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Appendix С
Some Calculus
Theorems
For a review of multivariable calculus and the proofs of the theorems below,
see the online supplement [Lee, Jeff].
Theorem C.l (Inverse mapping theorem). Let U be an open subset ofRn
and let f : U —> Шп be a Cr mapping for 1 < r < oo. Suppose that xo G U
and that Df(xo) : Mn —> Rn is a linear isomorphism. Then there exists an
open set V С U with xo e V such that f(V) is open and f : V -» f(V) С Rn
is a Cr diffeomorphism. Furthermore the derivative of f~l at у is given by
Df-\ = (Df\,.Hv))-\
Theorem C.2 (Implicit mapping theorem). Let О С Шк χ Μ1 be open. Let
f : О —> Rm be a Cr mapping such that /(xo> J/o) = 0· If ^2/(^0? J/o) : ^ ->
Rm is an isomorphism, then there exist open sets U\ С Шк and U2 С Ш1
such that U\ χ U2 С О with xo G U\ and a Cr mapping g : U\ —> U2 with
g(xo) = Уо such that for all (ж, y) G U\ x U2 we have
f{x,y) = 0 if and only ify = g(x).
We may take U\ to be connected. The function g in the theorem satisfies
f(x,g(x)) = 0, which says that the graph of g is contained in (U\ χ U2) Π
/_1(0), but the conclusion of the theorem is stronger since it says that in
fact the graph of g is exactly equal to (U\ x U2) Π /_1(0).
Corollary C.3. IfU is an open neighborhood of0eRk and f : U С Rk -»
Шп is a smooth map with /(0) = 0 such that Df(0) has rank k, then there
is an open neighborhood V o/0GRn, an open neighborhood W of 0 G Μη,
647
648
С. Some Calculus Theorems
and a diffeomorphism g : V —> W such that g о f : / l(V) —> W is of the
form (a1,... ,ak) ь-> (a1,... ,α^,Ο,... ,0).
Corollary C.4. If U is an open neighborhood of 0 G Mn and f : U С
Шк χ Mn_/c —>· M^ is a smooth map with /(0) =0 and г/ ί/ie partial derivative
Di/(0,0) is a linear isomorphism, then there exist a diffeomorphism h : V С
Rn —> J7i, ti;/iere V is an open neighborhood of 0 G Mn and J7i is an open
neighborhood of 0 G M^ swc/i ί/ιαί ί/ie composite map f о h is of the form
(α\...,α")^(α\...,α*).
Theorem C.5 (The constant rank theorem). Let f : (Mn,p) -» (Mm,g) &e
α /oca/ map swc/i ί/ιαί D/ has constant rank r in an open set containing
p. Then there are local diffeomorphisms g\ : (Mn,p) —> (Mn,g) and g2 :
(Mm, q) -» (Mm, 0) sucft ί/ιαί #2 о / о pf1 ftas ί/ie /orm
(ж1,...,жп)^(х1,...,хг,0,...,0)
on a sufficiently small neighborhood o/O.
Theorem C.6 (Mean value). Let U be an open subset ofRn and let f :
U —> W71 be of class Cl. Suppose that for χ, ζ G U the line segment L given
by χ + t(z — x) (for 0 < t < 1) is contained in U. Then
\\f(z) - f(x)\\ < \\z - x\\ sup{||£>/(y)|| : у € L},
where \\Df(y)\\ := supw=1 {\\Df(y) ■ v\\}.
Appendix D
Modules and
Multilinearity
A module is an algebraic object that shows up quite a bit in differential
geometry and analysis (at least implicitly). A module is a generalization of
a vector space where the field F is replaced by a ring or an algebra over a field.
For the definition of ring and field consult any book on abstract algebra. The
definition of algebra is given below. The modules that occur in differential
geometry are almost always finitely generated projective modules over the
algebra of Cr functions, and these correspond to the spaces of Cr sections
of vector bundles. We give the abstract definitions but we ask the reader
to keep two cases in mind. The first is just the vector spaces which are the
fibers of vector bundles. In this case, the ring in the definition below is the
field F (the real numbers R or the complex numbers C), and the module is
just a vector space. The second case, already mentioned, is where the ring
is the algebra Cr(M) for a Cr manifold and the module is the set of Cr
sections of a vector bundle over M.
As we have indicated, a module is similar to a vector space with the
differences stemming from the use of elements of a ring R as the scalars
rather than the field of complex С or real numbers R. For an element ν of
a module V, one still has Ov = 0, and if the ring is a ring with unity 1, then
we usually require lv = v. Of course, every vector space is also a module
since the latter is a generalization of the notion of vector space. We also
have maps between modules, the module homomorphisms (see Definition
D.5 below), which make the class of modules and module homomorphisms
into a category.
649
650
D. Modules and Multilinearity
Definition D.l. Let R be a ring. A left R-module (or a left module over
R) is an abelian group (V, +) together with an operation R χ V —> V written
as (а, г;) ь-> αν and such that
1) (a + b)v = αν + bv for all a, b G R and all dgV;
2) a(v\ + v%) = clv\ + av2 for all α G R and all V2,v\ G V;
3 (ab)v = a(bv) for all a, b G R and all ν G V.
A right R-module is defined similarly with the multiplication on the
right so that
1) v(a + b) = va + vb for all a, b G R and all υ G V;
2) (i>i + г>2)а = νια + v%a for all α G R and all г>2, vi G V
3) v(ab) = (i>a)b for all а, Ь G R and all υ G V.
If R has an identity and lv = υ (or г;1 = г;) for all υ G V, then we say
that V is a unitary R-module.
If R has an identity, then by "R-module" we shall always mean a unitary
R-module unless otherwise indicated. Also, if the ring is commutative (the
usual case for us), then we may write αν = να and consider any right module
as a left module and vice versa. Even if the ring is not commutative, we will
usually stick to left modules in this appendix and so we drop the reference
to "left" and refer to such as R-modules. We do also use right modules in
the text. For example, we consider right modules over the quaternions.
Remark D.2. We shall often refer to the elements of R as scalars.
Example D.3. An abelian group (Д +) is a Z-module, and a Z-module is
none other than an abelian group. Here we take the product of η G Ζ with
χ G A to be nx := χ -\ h χ if η > 0 and nx := —(x -\ \-x) if η < 0 (in
either case we are adding \n\ terms).
Example D.4. The set of all m χ η matrices with entries that are elements
of a commutative ring R is an R-module with scalar multiplication.
Definition D.5. Let Vi and V2 be modules over a ring R. A map L : Vi —>
V2 is called a module homomorphism or a linear map if
L{av\ + bv2) = aL{y\) + bL{v2)-
By analogy with the case of vector spaces we often characterize a module
homomorphism L by saying that L is linear over R.
Example D.6. The set of all module homomorphisms of a module V over
a commutative ring R to another R-module Μ is also a (left) R-module in
its own right and is denoted HomR(V, M) or Lr(V,M) (we mainly use the
D. Modules and Multilinearity
651
latter). The scalar multiplication and addition in Lr(V, M) are defined by
(/ + 9)(v) := f(v) + g(v) for f,g€ LR(V, M) and all υ € V;
(α/)(ν) := af(v) for a £ R.
Note that ((ab)f)(v) := (ab)f(v) = a(bf(v)) = a((bf)(v)) = (a{bf))(v).
Also, in order to show that a/ is linear we argue as follows: (af) (cv) =
af(cv) = acf(v) = caf(v) = c(af)(v). This argument fails if R is not
commutative! Indeed, if R is not commutative, then Lr(V,M) is not a
module but rather only an abelian group.
Example D.7. Let V be a vector space and £ : V —> V a linear operator.
Using I, we may consider V as a module over the ring of polynomials R[t]
by defining the "scalar" multiplication by the rule
p{t)v := p(£)v
for ρ G R[i], υ G V. Here, if p(t) = ^antn^ then p(£) is the linear map
Since the ring is usually fixed, we often omit mentioning the ring. In
particular, we often abbreviate £r(V, W) to L(V, W). Similar omissions will
be made without further mention.
Remark D.8. If the modules are infinite-dimensional topological vector
spaces such as Banach space, then we must distinguish between the bounded
linear maps and simply linear maps. If Ε and F are infinite-dimensional
Banach spaces, then L(E; F) would normally denote bounded linear maps.
A submodule is defined in the obvious way as a subset S С V that is
closed under the operations inherited from V so that S itself is a module.
The intersection of all submodules containing a subset А С V is called the
submodule generated by A and is denoted (A). In this case, A is called
a generating set. If (A) = V for a finite set A, then we say that V is
finitely generated.
Let S be a submodule of V and consider the quotient abelian group V/5
consisting of cosets, that is, sets of the form [v] := υ + S = {v + χ : χ G S}
with addition given by [v] + [w] = [v + w]. We define scalar multiplication
by elements of the ring R by a[v] := [av] for a G R. In this way, V/5 is a
module called a quotient module.
Many of the operations that exist for vector spaces have analogues in
the module category. For example, if V and W are R-modules, then the set
V χ W can be made into an R-module by defining
(v\,wi) + (ί>2,^2) ·= {v\ + V2,wi + W2) for (v\,wi) and (v2,W2) in V x W
652
D. Modules and Multilinearity
and
a(v,w) := (av,aw) for α G R and (v,w) G V x W.
This module is sometimes written as V0 W, especially when taken together
with the injections V-)VxW and W-)VxW given by ν \-+ (i>, 0) and
w h-> (0, w) respectively. Also, for any module homomorphism L : Vi —> V2
we have the usual notions of kernel and image:
KerL = {г; G Vi : L(v) = 0} С Vb
Im(L) = L(V\) = {w G V2 : w = Lv for some υ G Vi} С V^.
These are submodules of Vi and V2 respectively.
On the other hand, modules are generally not as simple to study as vector
spaces. For example, there are several notions of dimension. The following
notions for a vector space all lead to the same notion of dimension. For a
completely general module, these are all potentially different notions:
(1) The length η of the longest chain of submodules
0 = Vn С ... С Vi С V.
(2) The cardinality of the largest linearly independent set (see below).
(3) The cardinality of a basis (see below).
For simplicity, in our study of dimension, let us now assume that R is
commutative.
Definition D.9. A set of elements {ei,..., e/J of a module are said to be
linearly dependent if there exist ring elements n,..., r^ G R not all zero,
such that r\e\ + · · · + r^e^ — 0. Otherwise, they are said to be linearly
independent. We also speak of the set {ei,..., e/J as being a linearly
independent set.
So far so good, but it is important to realize that just because ei,..., e^
are linearly dependent does not mean that we may write each of these ei as
a linear combination of the others. It may even be that some single element
ν forms a linearly dependent set since there may be a nonzero r such that
rv = 0 (such а г; is said to be a torsion element).
If a linearly independent set {ei,..., e/J is maximal in size, then we say
that the module has rank k. Another strange possibility is that a maximal
linearly independent set may not be a generating set for the module and
hence may not be a basis in the sense to be defined below. The point is
that although for an arbitrary w G V we must have that {ei,..., e/J U {ги}
is linearly dependent and hence there must be a nontrivial expression rw +
т\е\ + · · · + r/св/с = 0, it does not follow that we may solve for w since r
may not be an invertible element of the ring. In other words, it may not be
a unit.
Ό. Modules and Multilinearity
653
Definition D.10. If β is a generating set for a module V such that every
element of V has a unique expression as a finite R-linear combination of
elements of B, then we say that Б is a basis for V.
Definition D.ll. If an R-module has a basis, then it is referred to as a free
module. If this basis is finite we indicate this by referring to the module
as a finitely generated free module.
It turns out that just as for vector spaces the cardinality of a basis for a
finitely generated free module V is the same as that of every other basis for
V. If a module over a (commutative) ring R has a basis, then the number of
elements in the basis is called the dimension and must in this case be the
same as the rank (the size of a maximal linearly independent set). Thus a
finitely generated free module is also called a finite-dimensional free module.
Exercise D.12. Show that every finitely generated R-module is the homo-
morphic image of a finitely generated free module.
If R is a field, then every module is free and is a vector space by definition.
In this case, the current definitions of dimension and basis coincide with the
usual ones.
The ring R is itself a free R-module with standard basis given by {1}.
Also, Rn := R Θ · · · 0 R is a finitely generated free module with standard
basis {ei,..., en}, where, as usual, ег· := (0,..., 1,..., 0); the only nonzero
entry is in the г-th position. Up to isomorphism, these account for all finitely
generated free modules: If a module V is free with basis ei,..., en, then we
have an isomorphism Rn = V given by
(rb...,rn) ь-> nei Η h
Definition D.13. Let V;, г = 1,..., /с, and W be modules over a ring R. A
map μ : Vi χ · · · χ V/- —> W is called multilinear (Zc-multilinear) if for each
г, 1 < г < /с, and each fixed (i>i,..., щ,..., Vk) £ Vi χ ■ · · χ V; χ ■ · · x Vfc
we have that the map
г-th
obtained by fixing all but the i-th variable, is a module homomorphism. In
other words, we require that μ be R-linear in each slot separately. The set
of all multilinear maps Vi x · · · χ V/- —> W is denoted Lr(Vi, ..., V/-; W).
If Vi = · · · = Vk = V, then we abbreviate this to Z^(V; W).
If R is commutative, then the space of multilinear maps Lr(Vi ,... ,V/c; W)
is itself an R-module in a fairly obvious way: If a,b G R and μι,μ2 £
Lr(Vi, ..., Vfc; W), then αμ\ + 6μ2 is defined pointwise in the usual way.
654
D. Modules and Multilinearity
Note: For the remainder of this chapter, all modules will be taken to
be over a fixed commutative ring R.
Let us agree to use the following abbreviation: V^ = V χ · · · χ V (Zc-fold
Cartesian product).
Definition D.14. The dual of an R-module V is the module V* := LR(V, R)
of all R-linear functionals on V.
Any element w G V can be thought of as an element of V** := Lr(V*, R)
according to w(a) := a(w) where a G V*. This provides a map V c-> V**,
and if this map is an isomorphism, then we say that V is reflexive.
If V is reflexive, then we are free to identify V with V**.
Exercise D.15. Show that if V is a finitely generated free module, then V
is reflexive.
For completeness, we include the definition of a projective module but
what is important for us is that the finitely generated projective modules
over C°°(M) correspond to spaces of sections of smooth vector bundles.
These modules are not necessarily free but are reflexive and have many
other good properties such as being "locally free".
Definition D.16. A module V is projective if, whenever V is a quotient
of a module W, there exists a module U such that the direct sum V Θ U is
isomorphic to W.
Given two modules V and W over some commutative ring R, consider
the class CvxW consisting of all bilinear maps V χ W —> X where X varies
over all R-modules, but V and W are fixed. We take members of CvxW
as the objects of a category (see Appendix A). A morphism from, say
μι : V χ W —> Xi to μ2 : V χ W —> X2 is defined to be a homomorphism
^ : Xi —» X2 such that μ2 = to μι- There exists a vector space Ty,w together
with a bilinear map ® : V χ W —> Ty,w that has the following universal
property: For every bilinear map μ : V χ W —> X, there is a unique linear
map μ : Ty,w —^ X such that μ = μ ο ®. If a pair (Ty,w,®) with this
property exists, then it is unique up to isomorphism in CvxW· We refer to
such a universal object as a tensor product of V and W. We will indicate
the construction of a specific tensor product that we denote by V ® W with
the corresponding map ® : V χ W —> V ® W. The idea is simple: We let
V ® W be the set of all linear combinations of symbols of the form υ ® w for
ν eV and wGW, subject to the relations
(v\ + V2) ® w = v\ ® w + V2 ® гу,
ν ® (w\ + W2) = ν ® w\ + υ ® г^2,
r (v ® w) = rv ® w = υ ® rw, for r G F.
D. Modules and Multilinearity
655
The map ® is then simply ® : (v,w) -+ ν ®w. Let us generalize this idea
to tensor products of several vector spaces at a time, and also, let us be a
bit more pedantic about the construction. We seek a universal object for
the category of /c-multilinear maps of the form μ : Vi χ · · · χ V/c —> W with
Vi,...,Va; fixed.
Definition D.17. A module Τ = Tvb...,vfc together with a multilinear map
® : Vi x · · · x V/c —> Τ is called universal for /c-multilinear maps on
Vi x · · · x V/c if for every multilinear map μ : Vi χ · · · x Vk —> W there is a
unique linear map μ : Τ —> W such that the following diagram commutes:
Vi χ · · · χ V* -^ W
τ
Τ
i.e. we must have μ = μ о ®. If such a universal object exists, it will be
called a tensor product of Vi,..., V/-, and the module itself Τ = TVb...,vfc
is also referred to as a tensor product of the modules Vi,..., Vk·
The tensor product is again unique up to isomorphism:
Proposition D.18. If (Ti,®i) and (Т2,®г) are both universal for
к-multilinear maps on Vi χ · · · χ V^ then there is a unique isomorphism
Φ : Τι -» T2 such that Φ ο ®χ = ®2;
Vi χ ■ · ■ χ νΛ
02
Proof. By the assumption of universality, there are maps ®i and ®2 such
that Φ ο ®! = (g)2 and Φ ο ®2 = ®ι· We thus have Φ ο Φ ο ®χ = ®ь and
by the uniqueness part of the definition of universality of ®i we must have
Φ ο Φ = id or Φ = Φ"1. D
The usual specific realization of the tensor product of modules Vi,..., Vk
is, roughly, the set of all linear combinations of symbols of the form v\ ®
• · · ® Vk subject to the obvious multilinear relations:
v\ ® · · · ® avi ® · · · ® Vk = a(v\ ® · · · ® V{ ® · · · ® Vk)
and
Vi®'"®{vi+Vfj)®"'®Vk
= V\ ® · · · ® Vi ® · · · ® Vk + V\ ® · · · ® v\ ® · · · ® Vk-
656
D. Modules and Multilinearity
This space is denoted by Vi <g> · · · <g> V/~ or by ®i=1 Vi and called the tensor
product of Vi,..., Vk. Also, we will use V0/c or ®*V to denote V® · · -®V
(/c-fold tensor product of V). The associated map ® : Vi χ · · · χ Vk —>
Vi ® · · · ® V/c is simply
® : (г>1,...,г>А;) н-> г>1 ® · · · ® v*.
A more pedantic description is as follows. We take Vi ® · · · ® Vk := F(V\ x
• · -xV/J/i/o, where F(ViX· · -xVk) is the free module on the set V\x- · -xV/-
and Uq is the submodule generated by the set of all elements of the form
(v\,..., avi,..., vk) - a(vi,..., vi,..., vk)
and
(vi,..., (vi + v'i) ,...,vk)
- (vi,...,Vi,...,Vk) - (vi,...,V·,...,^),
where i>i, v[ G Vi, and α G R. Each element (v\,... ,vk) of the set Vi χ · · · χ Vk
is naturally identified with a generator of the free module F(Vi χ · · · χ Vk)
and we have the obvious injection Vi x · · · χ Vk c-> F(Vi χ · · · χ Vk).
Its equivalence class is denoted v\ ® · · · ® vk and the map ® is then the
composition Vi χ · · · χ V^ ч· F(V\ χ · · · χ Vk) -» F(Vi χ · · · χ Vk)/U0.
Proposition D.19. ® : Vi χ · · · χ Vk —> Vi ® · · · ® Vk is universal for
multilinear maps on V\ χ · · · χ Vk.
Exercise D.20. Prove the above proposition.
Proposition D.21. If f : Vi —> Wi and g : V2 —> W2 are module ho-
momorphisms, then there is a unique homomorphism f ® ρ : Vi ® V2 —>
Wi ® W2, ί/ie tensor product, which has the characterizing properties that
f ®g is linear and that (/ ® g) (v\®V2) = (fv\)®(gv2) for all v\ G Vi,i>2 G
V2. Similarly, if fi : V» -> W», we may obtain ®г/г : ®J=i Vi -» ®*=1 W».
Proof. Exercise. D
Definition D.22. Elements of ®i=1 Vi that may be written as v\ ® · · · ®ι>/~
for some i>i are called simple or decomposable.
Remark D.23. It is clear from our specific realization of ®i=1 Vi that
elements in the image of ® : Vi χ · · · χ Vk —> ®f=1 Vj span ®f=1 V». I.e.,
decomposable elements span the space.
Exercise D.24. Not all elements are decomposable but the decomposable
elements generate Vi ® · · · ® Vk.
It may be that the Vi are modules over more than one ring. For example,
any complex vector space is a module over both Ш and С Also, the module of
D. Modules and Multilinearity
657
smooth vector fields Xm(U) is a module over C°°(U) and a module (actually
a vector space) over R. Thus it is sometimes important to indicate the ring
involved, and so we write the tensor product of two R-modules V and W
as V ®r W. For instance, there is a big difference between Xm(U) ®c<x>(U)
Xm(U) and Xm(U) ®r %m(U).
Lemma D.25. There are the following natural isomorphisms:
(1) (V ® W) ® U 9* V ® (W ® U) ^ V ® W ® U, and under these
isomorphisms, (v ®w) ®u <—> υ ® (w ®u) <—> υ ®w ®u.
(2) V ® W = W ® V, and under this isomorphism ν ® w <—> w ® v.
Proof. We prove (1) and leave (2) as an exercise.
Elements of the form (г; ® w) ® и generate (V ® W) ® U, so any map
that sends (v ® w) ® и to ν ® (w ® u) for all г>, г^;, и must be unique. Now
we have compositions
(V χ W) χ U Θ-^υ (V ® W) χ U A (V ® W) ® U
and
V χ (W χ U) id^40 V χ (W ® U) A V ® (W ® U).
It is a simple matter to check that these composite maps have the same
universal property as the map VxWxU—>V®W®U. The result now
follows from the existence and essential uniqueness (Propositions D.19 and
D.18). D
We shall use the first isomorphism and the obvious generalizations to
identify Vi ® · · · ® V/c with all legal parenthetical constructions such as
(((Vi ® V2) ® · · · ® Vj) ® · · ·) ® V/c and so forth. In short, we may construct
Vi ® · · · ® V/c by tensoring spaces two at a time. In particular, we assume
the isomorphisms (as identifications)
(Vi ® · · · ® Vfc) ® (Wi ® · · · ® Wk) = Vi ® · · · ® Vk ® Wi ® · · · ® Wk,
where (v\ ® · · -®Vk)®{wi ® · · ·®ι^) maps to v\ ® · · · ®vk ®w\ ® · · -®wk.
Proposition D.26. // V is an R-module, then we have natural
isomorphisms
V®R^V^ R®V
given on decomposable elements asv®r\-^rv\-^r®v. (Recall that we are
assuming that R is commutative.)
The proof is left to the reader. The following proposition gives a basic
and often used isomorphism.
658
D. Modules and Multilinearity
Proposition D.27. For R-modules W, V,U, we have
LR(W®V,U)^L(W,V;U).
More generally,
LR(Wi ® · · · ® Wfc, U) * L(W1;..., Wfc; U).
Proof. This is more or less just a restatement of the universal property of
W®V. One should check that this association is indeed an isomorphism. D
Exercise D.28. Show that if W is free with basis (/i,..., /n), then W* is
also free and has a dual basis (/*,..., /n), that is, fl(fj) = δ1·.
Theorem D.29. // Vi,..., Υ к are free R-modules and if (ej,..., eJnj) is a
basis for Vj, then the set of all decomposable elements of the form e\Y ® · · · ®
e%k is a basis for Vi ® · · · ® У к ·
Proof. We prove this for the case of к = 2. The general case is similar. We
wish to show that if (ei,..., eni) is a basis for Vi and (/i,..., /n2) is a basis
for V2, then {ei ® fj} is a basis for Vi ® V2. Define ф^ : Vi χ V2 —> R by
0//с(бг, fj) = $iujl, where 1 is the identity in R and
I 0 otherwise.
Extend this definition bilinearly. These maps are linearly independent in
£(Vi, V2; R) since if Σιΐς aik<filk — 0 in R, then for any г, j we have
0 = ^ а1кф1к(еи fj) =Σ а^Щ1
Ik Ik
= aij.
Thusdim(Vi®V2) = dim((Vi ® V2)*) = dimL(Vb V2; R) > nin2. On the
other hand, {e; ® fj} spans the set of all decomposable elements and hence
the whole space Vi ® V2, so that dim(Vi ® V2) < ηχη2 and it follows that
{ei® fj} is a basis. D
Proposition D.30. There is a unique R-module map ι : L(Vi, Wi) ® · · · ®
^(V/e, Wk) -» L(Vi ® · · · ® V/e, Wi ® · · · ® Wk) such that if /1 ® · · · ® fk is
a (decomposable) element o/L(Vi, Wi) ® · · · ® L(Vk, Wk) then
t{fi ® · · · ® fk)(vi ® · · · ® Vk) = fi(vi) ® · · · ® fk(vk).
If the modules are all finitely generated and free, then this is an isomorphism.
D. Modules and Multilinearity
659
Proof. If such a map exists, it must be unique since the decomposable
elements span L(Vi, Wi)® · · -<8>L(Vk, W^). To show the existence, we define
a multilinear map
ϋ : L(Vb Wi) x · · · x L(Vk, Wk) χ Vi χ · · · χ V* -* Wi ® · · · ® Wk
by the recipe
(/ΐ,...,Λ,^ι,...,^) \-> fl(vi) ® · · · ® fk{Vk)>
By the universal property there must be a linear map
ϋ : VJ ® · · · ® V£ ® Vi ® · · · ® V/, -» Wi ® · · · ® W*
such that $ о ® = τ?, where ® is the universal map. Now define
*(/ι®···®Λ)(<>ι®···®ν*)
:= u(fi ® · · · ® fk ® vi ® · · · ® νΛ).
The fact that *, is an isomorphism in case the Vi are all free follows easily
from Exercise D.28 and Theorem D.29. D
Since R ® R = R, we obtain
Corollary D.31. There is α unique R-module map ι : V\ ® · · · ® V£ —>
(Vi ® · · · ® Vfc)* such that if oti ® · · · ® ak is a (decomposable) element of
VJ® •••®V]J, then
t(ai ® · · · ® ak){v\ ® · · · ® Vk) = ai(v\) · · · ak(vk).
If the modules are all finitely generated and free, then this is an isomorphism.
Corollary D.32. There is a unique module map lq : W ® V* —> L(V,W)
such that if'ν ® β is a (decomposable) element o/W®V*, then
lq(w ® β)(ν) = β(ν)νϋ.
If V and W are finitely generated free modules, then this is an isomorphism.
Proof. If we associate to every w G W the map wm&p G L(R, W) given by
wmap(r) := rw, then we obtain an isomorphism W = L(R, W). Use this and
then compose
W®V*^L(R,W)®L(V,R)
^L(R®V,W®R)^L(V,W),
V*®W^L(V,R)®L(R,W)
-» L(V ® R, R ® W) ^ L(V, W). D
By combining Corollary D.31 with Proposition D.27 and taking U = R
we obtain the following assertion.
660
D. Modules and Multilinearity
Corollary D.33. There is α unique R-module map ι : V* ® · · · ® V£ —>
L(Vi,..., Vfc; R) such that if a\ ® · · · ® α/~ is a (decomposable) element of
VJ ®··-®ν£, йеп
*,(c*i ® •••®а/с)(г;ь...,г;/с) = αι(υι) · · · ak(vk).
If the modules are all finitely generated and free, then this is an isomorphism.
Theorem D.34. // ψ\ : Vi χ W{ —> Ui are bilinear maps for г = 1,..., к,
then there is a unique bilinear map
φ : (gtjV* X <g)?=1Wi -+ <g)f=1Ui
swc/i ί/ιαί /or i>i G Vi and W{ G Wi,
(/?(г>1 ® ··· ®^,^ι ® ··· ®г^) = y?i(vi,wi) ® ··· ® (pk(vk,™k)>
Proof. We sketch the proof in the к = 2 case. If φ exists, it is unique since
elements of the form v\ ® г>2 span Vi ® V2 and similarly for Wi ® W2. Now
by the universal property of tensor products, associated to ψι for г = 1,2,
we have unique linear maps fi : Vi ® Wz —> Ui with /i ο ® = ψ{. Then we
obtain the linear map /1 ® /2 : (Vi ® Wi) ® (V2 ® W2) -» Ui ® U2. On the
other hand we have the natural isomorphism S : (Vi ® V2) ® (Wi ® W2) —>
(Vi ® Wi)® (V2 ® W2) induced by the obvious switching of factors of simple
elements. Now define φ : (Vi ® V2) χ (Wi ® W2) -» Ui ® U2 by φ(χ, у) =
(/ι ® /2) (S(x ® j/)) for χ G Vi ® V2 and у G Wi ® W2. Then we have
у>(г>1 ® г>2, гУ1 ® w2) = (/ι ® /2) (νχ ® w\ ® г>2 ® w2)
= /l (^1 ® ™l) ® /2 (^2 ® ™2)
= φι (νι,νυι) ® (/?2(г>2,ги2). □
Corollary D.35. Lei Vi and Wi be R-modules for г = 1,..., к. If ψ% \
Vi χ Wi —> R are bilinear maps, г = 1,..., к, then there is a unique bilinear
map
Ψ ■ ®tiVi x ®?=iWi -+ R
swc/i that for Vi G V2- and W{ G Wi,
<^(^i ® -"®vk,wi ® •••®^/c) = ^ι(ϊ;ι,^ι)···^(^,^).
Proof. Imitate the proof of the previous theorem or just use the previous
theorem together with the natural isomorphism R ® · · · ® R = R. D
D.l. R-Algebras
Definition D.36. Let R be a commutative ring. An (associative) R-algebra
21 is a unitary R-module that is also a ring with identity lgt, where the ring
addition and the module addition coincide and where r(aia2) = (rai)a2 =
ai(ra2) for all αϊ, α2 G 21 and all r G R.
D.l. R-Algebras
661
As defined above, an algebra is associative. However, one can also define
nonassociative algebras, and a Lie algebra is an example of such.
Definition D.37. Let 21 and 25 be R-algebras. A module homomorphism
h : 21 —> 25 that is also a ring homomorphism is called an R-algebra
homomorphism. Epimorphism, monomorphism, and isomorphism are defined
in the obvious way.
If a submodule 3 of an algebra 21 is also a two-sided ideal with respect
to the ring structure on 21, then 21/3 is also an algebra.
Example D.38. Let U be an open subset of a Cr manifold M. The set
of all smooth functions Cr(U) is an M-algebra (R is the real numbers) with
unity being the function constantly equal to 1.
Example D.39. The set of all complex η χ η matrices is an algebra over
С with the product being matrix multiplication.
Example D.40. The set of all complex nxn matrices with real polynomial
entries is an algebra over the ring of polynomials Щх].
Definition D.41. The set of all endomorphisms of an R-module W is an
R-algebra denoted Endp(W) and called the endomorphism algebra of W.
Here, the sum and scalar multiplication are defined as usual and the product
is composition. Note that for r G R
r(fog) = (rf)og = fo(rg),
where /,# <G EndR(W).
Definition D.42. A set A together with a binary operation * : A x A —> A
is called a monoid if the operation is associative and there exists an element
e (the identity) such that α * e = e * α = α for all α G A.
With the operation of addition, Ν, Ζ and Z2 are all commutative mon-
doids.
Definition D.43. Let (A, *) be a monoid and R a ring. An A-graded
R-algebra is an R-algebra with a direct sum decomposition 21 = ΣΐβΑ ^
such that 2li2lj С 2l;*j. An N-graded algebra is sometimes simply referred
to as a graded algebra. A superalgebra is a Z2-graded algebra.
Definition D.44. Let 21 = Σίβζ^ΐ an(^ ® = Σζ<ξΖ^ be Z-graded
algebras. An R-algebra homomorphism h : 21 —> 25 is called a Z-graded
homomorphism if /i(2l;) С 25г· for each г G Z.
We now construct the tensor algebra on a fixed R-module W. This
algebra is important because using it we may construct by quotients many
important algebras. Consider the following situation: 21 is an R-algebra, W
662
D. Modules and Multilinearity
an R-module, and φ : W —> 21 is a module homomorphism. If h : 21 —> 53 is
an algebra homomorphism, then of course h ο φ : W —» 53 is an R-module
homomorphism.
Definition D.45. Let W be an R-module. An R-algebra il together with a
map φ : W —> il is called universal with respect to W if for any R-module
homomorphism φ : W —> 53 there is a unique algebra homomorphism h :
il —> 53 such that h ο φ = φ.
Again if such a universal object exists, it is unique up to isomorphism.
We now exhibit the construction of this type of universal algebra. First we
define 0°W := R and (g)1 W := W. Then we define ®*W := W0/c =
W ® · · · ® W. The next step is to form the direct sum ® W := ΣΖο ®*w·
In order to make this a Z-graded algebra, we define ®г W := 0 for г < 0
and then define a product on 0W := ^ iezffiW as follows: We know
that for г, j > 0 there is an isomorphism Wz<8> ® W®J —> W®^) and so a
bilinear map W*® χ W®' -» W®^) such that
(w\ ® · · · ® ^) x (wj ® · · · ® tuj) н-> г^1 ® · · · ® г^ ® wj ® · · · ® ^·.
Similarly, we define <g)°W χ W®* = RxW®* -» W®* by scalar multiplication.
Also, W®z χ W®J —>· 0 if either г or j is negative. Now we may use the
symbol ® to denote these multiplications without contradiction and put
them together to form a product on (g)W := Х)^е^®г W. It is now clear
that
(g)lWx(g)%^(g)^W,
where we make the needed trivial definitions for the negative powers:
(g)<W = 0, г<0.
Definition D.46. The algebra 0 W is a graded algebra called the R-tensor
algebra.
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Index
adapted chart, 46
adjoint map, 220
adjoint representation, 220, 221
admissible chart, 13
algebra bundle, 287
alternating, 345
associated bundle, 300
asymptotic curve, 159
asymptotic vector, 158
atlas, 11
submanifold, 46
automorphism, 201
base space, 258
basis criterion, 6
bilinear, 3
boundary, 9
manifold with, 48, 49
bundle atlas, 260, 263
bundle chart, 260
bundle morphism, 258
bundle-valued forms, 370
canonical parametrization, 180
Cartan's formula, 374
causal character, 547
chart, 11
centered, 11
Christoffel symbols, 168
closed form, 366
closed Lie subgroup, 192
coboundary, 444
cocycle, 263, 444
conditions, 263
Codazzi-Mainardi equation, 173
codimension, 46
coframe, 121
complete vector field, 97
complex, 444
orthogonal group, 196
conjugation, 202
connected, 8
connection forms, 506
conservative, 118
locally, 119
consolidated tensor product, 311
consolidation maps, 309
contraction, 317
coordinate frame, 87
cotangent bundle, 85
cotangent space, 65
covariant derivative, 501, 503
covector field, 110
cover, 6
covering, 33
map, 33
space, 33
critical point, 74
critical value, 74
curvature function, 146
curvature vector, 146
cut-off function, 28
Darboux frame, 160
de Rham cohomology, 367
deck transformation, 34
decomposable, 656
deformation retraction, 453
degree, 463
derivation, 61, 89
of germs, 64
determinant, 353
668
Index
diffeomorphism, 4, 25
differentiable, 3
manifold, 14
structure, 12
differential, 81, 111
Lie, 211
differential complex, 444
differential form, 359
discrete group action, 42
distant parallelism, 56
distinguished Frenet frame, 148
distribution (tangent), 468
divergence, 398, 623
effective action, 41
embedding, 128
endomorphism algebra, 661
equivariant rank theorem, 230
exact 1-form, 112
exact form, 366
exponential map, 213
exterior derivative, 363, 366
exterior product, 347
wedge product, 359
faithful representation, 247
fiber, 138, 258
bundle, 257
flat point, 162
flatting, 334
flow, 95
box, 96
complete, 95
local, 97
maximal, 100
foliation, 482
chart, 482
form, 1-form, 110
frame, 277
bundle, 292
field, 120, 277
free action, 41
free module, 653
Frenet frame, 146
functor, 640
fundamental group, 36
G-structure, 264
gauge transformation, 297
Gauss curvature, 157, 557
Gauss curvature equation, 172
Gauss formula, 166
Gauss' theorem, 399
Gauss-Bonnet, 186
general linear group, 193
geodesic, 567
geodesic curvature, 160
vector, 161
geodesically complete, 569
geodesically convex, 634
germ, 28
global, 30
good cover, 455
graded algebra, 661
graded commutative, 359
Grassmann manifold, 21
group action, 40
half-space, 9
chart, 49
harmonic form, 419
Hermitian metric, 280
Hessian, 76, 123
homogeneous component, 359
homogeneous coordinates, 19
homogeneous space, 241
homologous, 402
homomorphism presheaf, 289
homothety, 634
homotopy, 32
Hopf bundles, 295
Hopf map, 238, 239
hypersurface, 143, 153
identity component, 191
immersed submanifold, 130
immersion, 127
implicit napping theorem, 647
index, 332, 337
of a critical point, 77
raising and lowering, 335
induced orientaiton, 382
integral curve, 95
interior, 9
interior product, 374
intrinsic, 169
inversion, 190
isometry, 338
isotropy group, 228
isotropy representation, 248
Jacobi identity, 92
Jacobian, 3
Koszul connection, 501
Lagrange identity, 164
left invariant, 204
left translation, 191
length, 333
lens space, 52
Levi-Civita derivative, 550
Index
669
Lie algebra, 92
automorphism, 206
homomorphism, 206
Lie bracket, 91
Lie derivative, 89, 330
of a vector field, 103
on functions, 89
Lie differential, 211
Lie group, 189
homomorphism, 201
Lie subgroup, 191
closed, 192
lift, 37, 126
lightcone, 548
lightlike curve, 548
lightlike subspace, 588
line bundle, 281
line integral, 117
linear frame bundle, 292
linear Lie group, 193
local flow, 97
local section, 259
local trivialization, 260
locally connected, 8
locally finite, 6
long exact, 445
Lorentz manifold, 337
Mobius band, 261
manifold topology, 13
manifold with corners, 54
Maurer-Cartan equation, 387
Maurer-Cartan form, 224
maximal flow, 99
maximal integral curve, 98
Mayer-Vietoris, 447
mean curvature, 157
measure zero, 75
meridians, 160
metric, 279
connection, 543
minimal hypersurface, 179
Minkowski space, 337
module, 649
morphism, 638
moving frame, 120
multilinear, 653
musical isomorphism, 334
n-manifold, 16
neighborhood, 2
nice chart, 383
nonnull curve, 548
nonnull vector, 547
normal coordinates, 572
normal curvature, 158
normal field, 153
normal section, 159
nowhere vanishing, 277
null vector, 547
one-parameter subgroup, 202
open manifold, 50
open submanifold, 16
orbit, 41
map, 232
orientable manifold, 377
orientation cover, 380
orientation for a vector space, 353
orientation of a vector bundle, 375
oriented manifold, 377
orthogonal group, 195
orthonormal frame field, 165, 279
outward pointing, 381
overlap maps, 11
paracompact, 6, 645
parallel, 171
translation, 516
parallelizable, 278
parallels, 160
partial tangent map, 72
partition of unity, 30
path component, 8
path connected, 8
Pauli matrices, 203
piecewise smooth, 117
plaque, 482
point derivation, 61
point finite cover, 644
presheaf, 289
principal bundle, 293
atlas, 293
morphism, 297
principal curve, 159
principal frame field, 165
principal normal, 146
principal part, 56
principal vector, 158
product group, 191
product manifold, 20
projective plane, 18
projective space, 18
proper action, 231
proper map, 33
proper time, 550
property W, 133
pull-back, 92, 114, 322, 323
bundle, 268
vector bundle, 276
push-forward, 92, 115, 323
R-algebra, 660
radial geodesic, 571
670
Index
radially parallel, 517
rank, 78
of a linear map, 127
real projective space, 18
refinement, 6
reflexive, 654
reflexive module, 309
regular point, 74
regular submanifold, 46
regular value, 74
related vector fields, 93
Riemannian manifold, 337
Riemannian metric, 279
Sard's theorem, 76
scalar product, 193, 331
space, 193
second fundamental form, 156
section, 87, 259
along a map, 269
sectional curvature, 557
self-avoiding, 42
semi-Euclidean motion., 339
semi-Riemannian, 337
semiorthogonal, 195
shape operator, 155
sharping, 334
sheaf, 289
short exact, 444
shuffle, 347
sign, 599
simple tensor, 656
simply connected, 37
single-slice chart, 46
singular homology, 402
singular point, 74
smooth functor, 283
smooth manifold, 15
smooth map, 22
smooth structure, 12
smoothly universal, 129
spacelike curve, 548
spacelike subspace, 588
spacelike vector, 547
sphere theorem, 622
spin-j, 250
spray, 544
stabilizer, 228
standard action, 250
standard transition maps, 274
stereographic projection, 17
Stiefel manifold, 51, 243
Stokes' theorem, 396
straightening, 102
structural equations, 386
structure constants, 205
subgroup (Lie), 191
submanifold, 46
property, 46
submersion, 138
submodule, 651
summation convention, 5
support, 28, 101, 391
surface of revolution, 160
symplectic group, 196
tangent bundle, 81
tangent functor, 71, 84
tangent map, 67, 68, 81
tangent space, 58, 61, 65
tangent vector, 58, 60, 61
tautological bundle, 281
tensor (algebraic), 308
bundle, 319
derivation, 327
field, 320
map, 308
tensor product, 251, 319, 654, 655
bundle, 282
of tensor fields, 319
of tensor maps, 311
of tensors, 311
theorema egregium, 176
tidal operator, 558
time dependent vector field, 110
timelike curve, 548
timelike subspace, 588
timelike vector, 547
TM-valued tensor field, 320
top form, 378
topological manifold, 7
torsion (of curve), 150
total space, 258
totally geodesic, 582
totally umbilic, 162
transition maps, 261, 265
standard, 274
transitive, 228
transitive action, 41
transversality, 80
trivialization, 84, 260
typical fiber, 258
umbilic, 162
unitary group, 196
universal, 655
cover, 39
property, bilinear, 251, 654
VB-chart, 270
vector bundle, 270
morphism, 271
vector field, 87
along, 89
Index
671
vector subbundle, 271
velocity, 69
volume form, 378
weak embedding, 129
weakly embedded, 132
wedge product, 347
Whitney sum bundle, 276
zero section, 276
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Titles in This Series
108 Enrique Outerelo and Jesus M. Ruiz, Mapping degree theory, 2009
107 Jeffrey M. Lee, Manifolds and differential geometry, 2009
106 Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, 2009
105 Giovanni Leoni, A first course in Sobolev spaces, 2009
104 Paolo Aluffi, Algebra: Chapter 0, 2009
103 Branko Grunbaum, Configurations of points and lines, 2009
102 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009
101 Ward Cheney and Will Light, A course in approximation theory, 2009
100 I. Martin Isaacs, Algebra: A graduate course, 2009
99 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to
Schrodinger operators, 2009
98 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable
structure, 2008
97 David С Ullrich, Complex made simple, 2008
96 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008
95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008
94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category
O, 2008
93 Peter W. Michor, Topics in differential geometry, 2008
92 I. Martin Isaacs, Finite group theory, 2008
91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008
90 Larry J. Gerstein, Basic quadratic forms, 2008
89 Anthony Bonato, A course on the web graph, 2008
88 Nathanial P. Brown and Narutaka Ozawa, C*-algebras and finite-dimensional
approximations, 2008
87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra
Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology,
2007
86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations,
2007
85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007
84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007
83 Wolfgang Ebeling, Functions of several complex variables and their singularities
(translated by Philip G. Spain), 2007
82 Serge Alinhac and Patrick Gerard, Pseudo-differential operators and the Nash-Moser
theorem (translated by Stephen S. Wilson), 2007
81 V. V. Prasolov, Elements of homology theory, 2007
80 Davar Khoshnevisan, Probability, 2007
79 William Stein, Modular forms, a computational approach (with an appendix by Paul E.
Gunnells), 2007
78 Harry Dym, Linear algebra in action, 2007
77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton's Ricci flow, 2006
76 Michael E. Taylor, Measure theory and integration, 2006
75 Peter D. Miller, Applied asymptotic analysis, 2006
74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006
73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006
72 R. J. Williams, Introduction the the mathematics of finance, 2006
71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006
70 Sean Dineen, Probability theory in finance, 2005
69 Sebastian Montiel and Antonio Ros, Curves and surfaces, 2005
TITLES IN THIS SERIES
68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems,
2005
67 T.Y. Lam, Introduction to quadratic forms over fields, 2004
66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An
introduction, 2004
65 S. Ram a nan, Global calculus, 2004
64 A. A. Kirillov, Lectures on the orbit method, 2004
63 Steven Dale Cutkosky, Resolution of singularities, 2004
62 T. W. Korner, A companion to analysis: A second first and first second course in
analysis, 2004
61 Thomas A. Ivey and J. M. Landsberg, Cart an for beginners: Differential geometry via
moving frames and exterior differential systems, 2003
60 Alberto Candel and Lawrence Conlon, Foliations II, 2003
59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic,
2003
58 Cedric Villani, Topics in optimal transportation, 2003
57 Robert Plato, Concise numerical mathematics, 2003
56 Ε. Β. Vinberg, A course in algebra, 2003
55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003
54 Alexander Barvinok, A course in convexity, 2002
53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002
52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis,
geometry and physics, 2002
51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002
50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002
49 John R. Harper, Secondary cohomology operations, 2002
48 Y. Eliashberg and N. Mishachev, Introduction to the /i-principle, 2002
47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation,
2002
46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and
Lie groups, 2002
45 Inder K. Rana, An introduction to measure and integration, second edition, 2002
44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002
43 N. V. Krylov, Introduction to the theory of random processes, 2002
42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002
41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002
40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable,
third edition, 2006
39 Larry C. Grove, Classical groups and geometric algebra, 2002
38 Elton P. Hsu, Stochastic analysis on manifolds, 2002
37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular
group, 2001
36 Martin Schechter, Principles of functional analysis, second edition, 2002
35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001
34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001
33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001
32 Robert G. Bartle, A modern theory of integration, 2001
For a complete list of titles in this series, visit the
AMS Bookstore at www.ams.org/bookstore/.
I
Differential geometry began as the study of curves and
surfaces using the methods of calculus. In time, the notions
of curve and surface were generalized along with
associated notions such as length, volume, and curvature. At the
same time the topic has become closely allied with
developments in topology. The basic object is a smooth manifold,
to which some extra structure has been attached, such as a
Riemannian metric, a symplectic form, a distinguished group
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This book is a graduate-level introduction to the tools and structures of modern
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cohomology, the Frobenius theorem and basic Lie group theory. The book also
contains material on the general theory of connections on vector bundles and an
in-depth chapter on semi-Riemannian geometry that covers basic material about
Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is
the inclusion of an early chapter on the differential geometry of hypersurfaces in
Euclidean space.There is also a section that derives the exterior calculus version of
Maxwell's equations.
The first chapters of the book are suitable for a one-semester course on
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