A.Mishchenko and A.Fomenko
A Course
of Differential
Geometry
4fGOY
 '"
;;: -
"
'1'1\ .
Mir Publishers Moscow

A.Mishchenko and A.Fomenko
A Course
of Differential
Geometry
4fGOY
 '"
;;: -
"
'1'1\ .
Mir Publishers Moscow

'I'ranalated from Ruslan
by Anatoly Talsohev
First published t 988
Bevispd from the 1980 Russian edllion
Ra ClHMu4cICOM 118WICe
Prlnla in the ail/oil 01 Soulel Soclal!II Republic'
ISBN 5-03-000220-0
@ t13l1aTenbCTBo MOCI<OBcKoro
YHIIDepcIIT8Ta, 1 Y80
@ English tranlation, Mir
pulilishers, t 988

Contents
Prelaco to the English Edition . . . . . . . . . . . . . . , 8
Preface to tho Russian Edition . . . . . . . . . . . . . , . 9
Chapter I, INTRODUCTION TO DIFFERENTIAL GEOMETRY 12
1. 1. Curvilinear Coordinate Systems. Simple Examples . , ,. 12
1.1.1. Background. . , , . . .: . , . . . . . , , , , . , . .. 12
1.1.2. Cutesian and Curvilinear Coordinates , , . . . . . .. is
1.1.3. Simple Examples of Curvilinear Coordinate Systems 21
1.2. The Length of Curve in a Curvilinear Coordinate System . .. 25-
1.2.1. The Length of Curve in la Euclfdean Coordinate System 2$
1.2.2. The Length of CUrn in a Curvilinear Coordinate System 28
1.2.3. The Concept of tbe Riemannian Metric in a Euclidean
. Domain ..,...."..... 32
1.2.4. Indefinite Metrics .......... 35
1.3. Geometry on a Sphere and on a Plane . . . 3S
1.4. Pseudoephere IInd Lobachevskian Geometry 46
Chepter 2. GENERAJ., TOPOLOGY . . . . . . . . , 67
2.1. Definition and Basic Propeti6B of Metric and Topological
SpaC68 ..."., , 61
2.1.1. Metric Spaces . . . . . . . 61
2.1.2, Topological Spaces . . . . . 70
:U.3, Continuous Mappings . . . , 72
2.2. Connectedness. Separation AX;ioms 76
2,2.1. Connectedness . . . . . . , , 71
2.2.2. Seperation AxiolllB , , , . . . 79
2.3. Compact Spaces . . , . . , , . . 82
2.3,1. Definition "...,..". 82
2.3.2. The Properties of Compact Spaces . . 83
2.3.3. Metric Compact Spaces , '. . . . . . , 84
2.3.4. Operations Over Co:npact Spaces . . , 86
2.4. Functional Se r ltl'abilit y . PartlUon of Unity 87
2.4.1. Functiona Separablllty ....... 81
2.4.2. Partition 01 Unity . . , . . . . , , . 90
Chapter 3. SMOOTH MANIFOLDS (GENERAL THEORY) 92-
Introduction ......... 92
3.1. The Concept of a Mani!old . 93
3. i. i. Fundamental Definitions 93

6
Cont..1S
3.1.2. Functions of Coordinate Transformalion. Definition of a
Smooth Manifold . . . . . . . . .
3.1 3. Smooth Manifolds Diffeomorphism
3.2. Definition of Manifolds by EquationS .
3.3. Tangent Vectors. Tangent Space .
3.3.1. Simple Examples .... . .
3.3 2 General Defmition of a Tangent Vector
33.3. Tangent Spacc Tp,(M) . . . . .
3.3 4. Sheaf of Tangent Curves . . . . . .
3.3.5. Directional Derivative of a Function
3.3.G. Tan!:;enl Bundles . . .
3.4. Submalllfoids . ,. .... .
3.4.1 Differential of a Smooth Manifold . . . . . .
3.4 2. Diffe«mtial and Local Properties of Mappings
3.4.3. Sard's Theorem . . .. .........
34.4. Cmbedding of Manifolds in a Euclidean Space
Chapter 4. SMOOTH MANII'OLDS (EXAMPLES) . . . . .
4.1. The Theory of CUI'Ves On a Plane and in a Three-Dimensional
Space .......................
4.1.1. The Theory of Curves on a Plane. Io'renet Io'ormulas
11.1.2. Tha Theory of Spatial Curves. Frenel Formulas . .
4.2. Surfaces. First and Second Fundamental Forms .
4.2.1. The First Fundamental t'orm .,.....
4.2.2. The Second FundamenlaL Form .....
<4 2.3. An Elementary Theory of Smooth Curves on a Hyper-
surface '. . . . '. . . . . ., ....
4 2.4. The Gaussian and Mean Curvatures of a Two-Dimensional
Sm'face . . . , . . . . . . . . . . . .
4.3. Transformalion Groups . . . . . . . . . . . .
4.3.1. Simple Examples of , Transformation Groups
4.3.2 Matrix Transformation Groups . . . . .
4.; Dynamical Systems ...........
4.5. Classification of Two-Dimensional Surfaces
4.5.1. Manifolds with Boundary . . . .. .
4 5 2. Orientable Manifolds . . . . . . . . .
4.6. Riemannian Surfaces of Algebraic Functions
Chapler 5. TENSOR ANALYSIS AND RIEMANNIAN GEOMETRY
5 i The General Concepl of a Tensor Field On a Manifold
.5 2. Simple Tensor Io'ields ..............
5.2.1. Examples ..... .... . . . .
:5 2.2. Algebraic Operations over Tensors . . . . . . .
5.2.3 Skew-Symmetric Tensors . .. . . .
:s 3. Connection and Covariant Differentia.tion . . .. .
53:1. The Definition and Properties of Affine Connection
5.3 2. Riernauni<lll Connoctions .. ........
:; 4. Parallel Displacement Geodesics . . . . .
5.4.1. Preliminaries ...........
5.4.2. The EquaLion of Parallel DispLacement
5.4 3 GeodesIcs .. . . .. .....
:5 5. Curvature Tensol .........
55.1. Preliminaries . ..... .......
.5 5 2 Coordinate Definition of the Curvature Tensor
98
104
108
13
it.!
117
tiS
120
122
127
129
130
135
138
141
147
147
147
154
160
160
163
169
178
200
200
215
229
246
247
249
270
294
294
300
300
305
309
322
322
332
336
336
339
341
.157
357
360

Conlent.
5.5.3. Invariant Definition of the Curvature Tensor . . . . .
5.5.4. Algebraic Properties of the Riemann Curvature Tensor
5.5.5. Certain Applications of the Riemann Curvature Tenser
Chapter 6. HOMOLOG\' THEORY. . . , . . . . . . . . . . . . .
6.1. Calculus of Exterior Differential Forms. Cohomology Groups
6.1 1. Differentiation of Exterior Differential Forms . . . . .
6.1.2. Cohomology Groups of a Smooth Manifold (tho de Rham
Cohomology Groups) . . . _ . . , . . . . . . . . . .
6 \.3. Homotopio Properties of Cohomology Groups . . .
6.2. Integration of Exterior "forms . . . , , . . . . . .
6.2.1. Tho Integral of a Dif!ereni.lal Form over a Manifold
6 2.2. Stokes' Theorem . . . . . . . . . . . .
6 3. The Degree of Mappiog and Its Applications
6.3.1. Example ...............
6.3.2. The Degree of Mapping . . . . . . . .
6.3.3. The Fundamental Theorem of Algebra
6.34. Integration of Forms . , . . . . . .
6.3.5. Gaussian Mappiog of a Hypersurface
:8ElR:IML. RI.AIA. P.RLS_ N. tN
7.1. t'unctional. Extremal Functions. Euler's Equations .
7.2. ExtremalilY of Geodesics . . . . . . . . . . . . , .
7.3. l>linimal Surfaces . . . . . . . . . . . . . . . .
7.4, Calculus of Variations and Symplectic Geometr)'
Name Index ...,.,.....""."...,
Subject Index , . , . , , , . . . , , . . . , , . , . . ,
361
3(;.!
301\
3i1
372
372
378
381
387
388
393
399
399
400
402
403
404
401
407
4t6
424
430
451
452

Preface to the English Edition
The English edit.ion has been thoroughly revised in line
with comments and suggestions .made by our readers, and the mis-
takes and misprint that were detected have been corrected. This is
essentially a textbook for a modern course on differential geometry
and topology, which is much wider than the traditional courses on
classical differential geometry, an it covers many branches of mathe-
matic'! a knowledge of which hasjnow become essential for a modern
mathematical education. We hope that a .l:eader who has mastered
this material will be able to do independent research both in geometry
and in other related fields. To gain a deeper understanding of the
material of this book, we reconimend the reader should solve the
questions in A,S. Mishchenko, Yu.P. Solovyev, and A.T, Fornenko,
Problems in Differential Geomety and Topology (Mir Publishers,
Moscow, 1985) which was specially compiled to accompany this
course.

Preface to the Russian Edition
Differential geometry and topology constitute an impor-
tant branch of mathematics. The rapid progress in theoretical phys-
ics and mechanics during the 20th century has shown t.at geomet-
ric concepts lie at the heart of certain fundamental theories. Suffic&
it to point out such fields as relatiity theory, the mechanics of con-
tinuous media, and electrodynamics, where the mathematical appa-
ratus rests upon geometric methods.
The present course deals with the fundamentals of differential
geometry and topology whose present state is the culmination of
contributions from generations of mathematicians, It is rather dif-
ficult to survey the history of geometry because we now treat most
geometry »roblems quite differently from the way mathematicians
did in the past.
Almost aU the methods of geometry (and geometry is not the only
such field) have been reworked. E'rom the roodern point of view,
geometry and topology are based on several fundamental ideas,
which may be ou tlined as follows\
Set-theoretic (or general) topology is the branch of geometry that
has "absorbed" the once general methods for studying convergence
and continuity. One important idea in geometry-the use of curvi-
linear coordinate systems-eventu'ally led to tensor analysis and
the theory of invariants. Whereas mathematical analysis and the
theory of differential equations deal primarily with the "local" prop-
erties of a function (only infanitely adjacent points are considered),
geometry studies the "global" properties of functions (that is, their
properties are analysed at Iiltitely spaced points). This intuitive idea
of the study of the "global" properties of geometric objects has given
rise to the fundamental concept of manifold as a geMralization of
the concept of domain in EuclidC1an space.
Later, the .emarriage" of these idas in various particular applica-
tions became equally important in geometry: the geometric interpre-
tation of integration, cohomology theory as a formal geometric
representation of an intuitive transition from the "local" to a "glob-
al" study of a space, geometric interpretation of a linear approxima-

10 Pr/n('c tll 11,;0 llll.'.siall f..dilwrt
tion of non-linear functions and mappings, the principle of general
posiLioll ill geometry.
All these geometric ideas were invented to facilitate the anal'sis
of particular problems, and they have furthered both the solution
of these problE:'l11s nnd the dl'velopment of geometric concepts anel
methods. rt is sometiml's rather rlifficult to c1isting\lish between the
ideas that arc IIOW conc.erlled with differential geometry oIl1d topology
.and those that/have rleveloped into separaLl' directions of analytic
geomctry and lillear algebra, or evcn belong to some other branches
of mathomatics.
.\nl'ient geometry cuhninated in Euclid's /:,'lelll<!llt. /Iud had little
influence 011 the rlevelopment of differl.'ntial geometry. Only one
problem, which for mallY centuies seemed to be the most impor-
taut, found a simple ancl lIatllra;1 solution within the framl.'work of
diffE:'rential geometry. We mcan l the proof of Euclid's Mtll axiom.
The I1xiom was so obscure IIncllsO intimately rcolated to the other
.axioms that for many centuries mathematicians vainly att<.>mpted to
prov<.> it by proceeding from tile others. In all probability, Leo
Gersonide (1288-1344) WitS the fjrst mathematician in Europe who
tried to prove the parallel axiom. The proof of the fifth axiom be-
came the foc\IS of keen atLention by f;nnous matlwmaticians such
as Ch. C[avins (1574), P. Cataldi', (1603), G.A. Borelli (1rJ58), J Vi-
tali (!G80), 1. Wallis (Hi63), GG. Saccheri (1733), J.H. Lambert
(1766), A.M, Legendre (1800), F.iK. Schweikart (1818), F. Taurinus
(1825), and even C.F. Gal1ss. Althbugh their attempts were unsuccess-
fnl, they were of tremendous significance, for they laid the founda-
tions for a lIew, non-Euclidean geometry. This geometry was based
on a rejection of the fifth axiom sind was invented by N.!. Lobachev-
sky (1792-1856), the great Russian mathemalic.ian. In 1826 he deliv-
ered his first communication oni non-Enclidean geometry, and thi.!!
opened up a now era in the development of geometry. In 1832, a
similar paper of 1. Dolyai was published. The same ideas were put
forward by C.F.. Gauss, who did not, however, publish his work.
Whereas Lobachevsky arrived .at his non-Euclidean geometry by
the axiomatic method, further ne\" ideas in geometry were developed
concllrrantly with other fundamental principles of defining geometry
objects via a coordinate system! Coordinates were employed even
in anc!<.>nt geography and Rstron;omy. In Ptolemy's Geo{!raphy, we
come across latitude and 10ngiUude as numerical coordinates. III
Europe, N. Oresme (1323-1382) came very close to the idea of coor-
dinates in the graphic representation of a function, but he never
mentioned the term "coordinates'. The discoverers of the coordinate
method and analytic geometry 'ere R. Descartes (1596-1650) and
P Fermat (1601-1665). The consistent use of ci'rvilinear coordinates
.and Lobachevsky's ideas led to the rapid devolopment of diff(!ren-
tial geometry. C.F.B. Riemann (1826-1866) took a novel step when

Preface /0 the RussIan £<1,111011
it
he pioneered the study of arbitrary, so-called Riemannian spaces,
whirh were found to have wide aplications in mechanics, relativity
theory, and other Helds. Riemann's studies stimulated vigorous
growth of vast new branches of geo'metry: vector and tensor calculus
and Riemannian geometry.
At the same time, the foundations of topology were laid as part of
geometry deating with continuity properties, Topology, of cOllrse,
.owes its origins to the infinitesimal calculus, although tile first
purely topological papers were wri:tten by L. Euler (1707-1783) and
C. Jordan (1838-1922). At the beginning of the 20th century, the
fundamental principles of topology were elaborated by M. Frechet
(1878-HJ73), F. Hausdorff (1868-1942), H.L. Lebesgue (1875-1941),
and L.E.J. Brouwer (1881-1966). J .H. Poin(',are (1854-1912) contrih-
uted a lot to progress in topology' and its applications. Nowadays,
topology is a rapidly growing br4nch of mathematics.
L. Euler and G. Monge (1746-1818) initiated an independent study
within differential geometry: surfce theory. The union of the con-
('opts of non-linear coordinates, veCtor and tensor calculus, and sur-
face theory, as well as the fruitfuless of geometric ideas in the nat-
ural sdences led to the most fundamental notion in geometry, the
mllnifold. Poincare was apparentIy[the first to assign a clear meaning
Lo the nolion. The concept of manifold has now taken such firm
root. in mathematical thinking tl l ' at mathematicians are no longer
deliberately developing manifold theory.
Finally, we should mention one more thing that has played an
important role in the evolution arid understanding of geometry and
the key problems of modern theQretical physics. It is the use of
group methods, These are basedi on the invariance of geometric
-objects with respect to a space symmetry group. Group methods
emerged in geometry from the study of projective geometry and they
are associated with the names of Poncelet, Staudt, Mobius, PHicker,
and Cayley. In 1872, F. Klein (1849-1925) published a famous paper,
lat()r called "The Erlangen Program", in which he summarized ad-
vances in geometry and rigorously. formulated the group plinciples
for dcveloping geometry. Group f.ethods have strongly influenced
{Iilforential geometry, topology, 3r;Id the various applications of ge-
ometry in the other branches of mathematics and the natural sciences.
It should be borne in mind however that the group technique is not
the only geometric method.
!\Jany mathematicians have cotributed to differential geometry
and topology. We believe it is worth noting that Russian and Soviet
scholars have made a strong impct on this progress.

Cha pter 1
Introduction to Differential
Geometry
1.1. CURVILINEAR COORDINATE SYSTEMS.
SIMPLE EXAMPLES
t.t.t. BACKGROUND
Let us consider an n-dimensional Euclidean space which
is usually denoted by Rn. We assume that this space is provided
with Cartesian coordinates Xl, ' . ., x" which permit a unique deter-
mination of the position of any point in R" by associating with it a
set of real numbers, the coordinates relative to a fixed orthogonal
basis formed by mutually orthogonal unit vectors e 1 , e2' . . ., en.
The idea of describing a point in an Euclidean space by a set of real
numbers (which may also be considered as the coordinates of the
radius vector emerging from the origin tlrthe point) underlies ana-
lytic geometry which solves various geometric problems by purely
algebraic methods, This important approach was first introduced
(explicitly) into mathematics by !Descartes in whose honour we now
say "Cartesian coordinates", Algebraization of geometry has played
a key role in the development not only of geometry as such but also
of maihematics as a whole. We shU not concentrate on the problems
which are. easily and elegantly solved by algebraic-analytic methods
(viz., classification of second-ordr surfaces in a three-dimensional
space) and refer the reader to num'erous courses of algebra and analyt-
ic geometry. Let us only recall. tbat Cartesian coordinates in R n
are closely related to the concept of the Euclidean scalar product,
a bilinear form w,hich associates with each pair of vectors S, 1] E R"
a real number usually denoted by <s, 1]}; this operation is symmetric
and linear in each argument, andlthe form itself is positive definite.
In a Cartesian coordinate system we have
(s, TJ}=GI'II+... +;n'l)", where s=(!;', .... €on), 1]= ('II, .. .,1'}").
Simple examples however sho;W that Cartesian coordinates are
not always the most convenient ones to solve analytically many par-
ticular problems. We shall demontrate this by writing the equations
of curves on a plane in Cartesian coordinates x, y. Of course, for
rather simple curves, viz., a circle or ellipse, the analytic expres-
sions in Cartesian coordinates are 'Simple. Indeed, the equation of a
circle of radius R with centre at point A is (x - A 1)9 + (y - A 2)! =
RZ, where A = (AJ, AZ).

1,1. Curl1ll!ne/Jr Coordlnale SY"m.. Simple Ezampltl 13
The equation ot an ellipse is also simple: (x - Al)'Ja' + (y _
AZ}Zlb Z = R', where a and b are the principal semi.axes (Fig. 1,{),
However, in various applications and concrete mechanical and
physical problems we often deal With smooth curves (say, trajecto-
ries of the motion of a particle. in a force field) whose equations
y
y
o
Figure 1,1
I
)(
Figure 1.2
in Cartesian coordinates are rather cumbersome. For example, the
'-( tan-I.!!)
equation V X" + y' - e '" = 0 defines a spiral in Cartesian
coordinates (Fig, 1,2), Although this equation is rather simple, it
could be ritten in a simpler form 'in the so-called polar coordinates
(r, fj) related to the Cartesian coorainates x, y by X = r cos 1jJ, y =
r sin fj) (Fig. 1.3), In polar coordi'nates the equation of a spiral be-
y
y
Figure 1.3
Ie
o
Ie
Figure 1.4
comes r = eI-', thereby clearly demonstra ting the character of the mo-
tion along this trajectory. Below we shall return to a polar coordinate
system, and here we just note that the introduction of such coordi-
nates (called curvilinear coordinates) is not a caprice of mathemati-
cians trying to devise n( W entities but a practical necessi ty, some-
times quite useful in particular caloulatione, In this connection, we
shall briefly discuss a problem where polar coordinates appear to be
qui te useful.

14 J. J IIlr"dltrtian 10 n ilJrrmlla! Geometry
Let u. consider the motion. of a partide on a plane in a
("(.'ntral force field. Suppose the centre is at the point 0 and (r, q:)
are polar coordinates on thp plane. Let r be the radius vector of the
moving partide (this veClor orig,inates from 0), r be its length, and t
be time (the motion parameter); then the coordinates rand q> arc
functions of time. Consider at a point r (t) with polar coordinates
r = I r I, q> two orthogonal unit vectors: a vector e r along the radius
vector of the particle (note tha the relation r = r, e r holds in this
case) nnd a vector eq, orthogonal, to e r and so directed that the polar
angle rp increases (Fig. 1.4), Differentiation of a radius vector r (t)
with respect to time will be denoted by a dot. As is known from
mechanics, the motion of a particle (its mass is taken equal to 1,
for simplicity) in a central force:field on a plane is described by thl!'
following differential equation: ' = / (r) ero where f is a smooth func-
tion of a single argument r. Incidentally, here is a useful exercise
for the reader: write this differentia) equation in Cartesian coordi-
nates on a plane,
The motion of a partide an be dt'scribed by two func-
tions: r = r (t) and <p = rp (t), that is. in II polar coordinate sys-
tem. I t is a simple matter to verify that_when a material particle
moves in a central force field the quantity "(y is conserved. This is
one of Kepler's laws which he discovered while studying the motion
of the planets of the Solar syste (at that time Kepler already em-
ployed the tables of the coordina,tes of the r.lanets on the celestial
sphere as a function of time). THis conserved quantity can be given
clear geometrical meaning. Kepler introd\lced a convenient notion
of areal velocity v as the time rate of change of the area s (t) swept
Ollt by the radius vector r (t), Le; v = as (t)/dt. In terms of the areal
velocity Kepler's law can be fornpJated as follows: the radius v{'ctor
weeps out equal areas in equal tmes; in other words, the areal veloc-
ity is constant, ds (t).'dt = const, We can also prove (tho proof is
omitted here) that this law is ope of the formulations of the prin-
ciple of conservation of nngular1momentum. The reader can easily
see that Ihis law is much simpler to derive in polar coordinates rather
than in Cartesian coordinates (though calculations may, of course,
be performed in the latter as well).
Solution of parti('u]ar problems in mechanics and physics has
called for the invention of other curvilinear coordinate systems:
cylindrical, spherical, and so on.: Close examination of all the afore-
mentioned ways of associating  point in spate with a set of real
numbers (the coordinates of this; point) shows that this association
relies upon a general idea adn1itting a reasonable formalization
which comprises all the "curvilinear" coordinates mentioned (invert-
ed commas are used for the wor4 curvilinear, since we have not yet
defined the concept strictly, put consider only graphic example,.)

1.1. Curvilillar Coortlinal Systems. $lm,.l. £.rom,.lrs 1:>
1.1.2. CARTESIAN AND CURVILINEAR COORDINATES
Let us consider an arbitrary domain in a Eudidean spare-
R". We recall that, just as in mathematical analysis, by a domain we
mean an arbitrary set C in a Euclidean space whose every point P
is contained in C together with al ball of sllfficientl' smaIl radius
with centre at P. Consider also a second copy of the Euc1idean space,
which is denoted by R. To deline1the coordinates of the point P in,
the domain C is to associate with tlhis point a set of numbers, called
coordinates. Obviously, an arbitrar\}' association (i.e. the one without
additional requirements) will not lead to a good result in that such
a correspondence may be devoid of, sense (it is desirable that mathe-
matical concepts should be of some luse, for instance in computations,
just as was the case with Cartesian; coordinates). Here is an example-
of senseless association: the same et of numbers, say (0. 0, 0, . . .
. , ., 0), is associated with each point P in C. Thus, we arrive at
the first requirement to the association: it is desirable that distinct
sets of numbers (coordinates) shoul correspond to different points of
the domain. The example just mentioned does not satisfy this requi-
rement (all the points of the domain C have the same "coordinates",
zeros) .
Thus, our aim is to associate with each point P oC a domain C a
set of n real numbers. Apparently" this opration gives rise to a set
of n functions Xl (P), . . ., .x" (P) defined in the domain C; here-
xl, . . ., x" are coordinates in the Euclidean space R7.
These functions are usually req\\ired to be continuous and even
smooth (at least for almost al1 the ' points of the domain C), that is,
a small change in. the position of  should lead to a small change in
its coordinates, and a smooth deformation of P should generate a
smooth variation of its coordinates.
So, let us consider two copies of a Euclidean space: R" with Carte-
sian coordinates yl, . . ., yn and R with Carttasian coordinate
xl, . . ., x n ; let C be a domain in R".
Remark. The Euclidean space R could be considered as an "arith-
metic space" by identifying its points with leal sequences of length n.
Definition t. A continuous coordl7!ate system in a domain C of Eu-
clidean space R" is said to be a l system of functions Xl (yl, . . .
. . ., yn), . . ., x" (yl, . . " yn) which map the domain C conti'l-
uously and bijectively onto a certain domain A of Ri. In other
words, the system of functions Xl (P), . . " 3!' (P) defines a map-
ping, sometimes cal1ed a homeomorphism of C onto A (the concept
of a homeomorphism will be considered in detail below).
Defmition i is a formal expression of our desire that as a point P
moves continuously in C its coordinates should also change contin-
uously The f\mctions Xl (P), . . ., x" (P) are caJled the coordinates
of point P re!ative to the coordinate mapping f: C -+ A.

16 1. Introduction 10 D10trtn!lal Geometry
For instance, the coordinate mapping f: C -A may be chosen in
the form of an identity mapping defined by the linear functions,
r = yl, , . '. x n = y".
Sometimes we shall write a point P with coordinates Xl (P), , . .
, , " x" (P) in tbe form P (Xl, . . " x n ) assuming that the coordinate
mapping f: C _ A has already Deen defined and fixed.
Among all continuous coordinate mappings of special intexest are
those that define a smooth mapping of a domain C onto A. i.e, when
all functions Xl (yl, ' , " yn), . , ., x n (y1, ' . " yn) are continuous
functions of their arguments lIt . . " yn, But the smoothness of
the coordinate mapping / without the assumption of the smoothness
of the inverse mapping ,-I does not lead to a meaningful coordinate
system, Therefore, we now turn to defining coordinate systems in
which I and /-1 are both smooth', To this end, we shall need a new
concept, the Jacobi matrix of a !Jmooth mapping,
Let f: C _A be a smooth mapping defined by a set of functions
.xl (yl, .." yn), . . " x" (111, ' 'I', 'yn).
Definition 2. The Jacobt matr;iz of a mapping / is a functional
matrix
df = ( :z ) = (  .' :.., : ) ,
u 8zn 8z'"
 ... agsr
composed of partial derivatives of',the coordinates %1 (P), . , ., :x" (P).
The determinant of this matrix is denoted by J (f) and called the
JacobIan of the mapping I.
Remark. We hope that the notation dl for the .Jacobi matrix wiIJ
not be confused with the differential of a smooth function I, for
this differential (when interpreted appropriately) coincides with the
.Jacobi matrix in this particula1' Case. This topic is discussed below.
Let us note once. more that a Jacobi matrix is a variable matrix, i,e.
it depends on a point P in a domain Ci similarly, the 1 acobian J (j)
is a smooth function on C.
Definition 3. A regular coordinate 8ystem in a domain C of Eu-
clidean space R n is a system of smooth functions Xl (yt, , . ..
yn), ' .., x n (yl, ' , ., yn) which map hijectively the domain C
onto a domain A in R'l and arEl such that the Jacobian J (f) (P)
is not zero at all points of C.
Let us note that the condi(ion that the Jacobian does not
vanish at all points of C means that the inverse mapping 1_1 is not
only continuous, but also smoot. This follows from the implicit-
function theorem. Thus.. a regular coordinate system is defined by
two smooth mutually inverse mppings establishing a homeomoI'-

1.1. Curullfnear CoordtnfJ/ Sill/ems, SI/Ilpu EZfJ/IIpl.. 17
phism between the domains C and A, Definition 3 makes formal our
desire that when a point P changes smoothly in C its coordinates
should also change smoothly; moreover, smooth variation of a "coor.
dinate point" B in A should alsolresult in smooth variation of the
point P induced by the mapping 1_1, The definitions presented above
clearly show that the very concep of a "smooth and regular coordi-
nate system" automatically implies that at least two copies of a
standard Euclidean space should be considered. Certain domains of
these copies are identified by a dontinuous and bijective mapping
with an additional requirement of smoothness (in both directions).
These definitions can be interpreted from another point of view.
We could assume that a Cartesian coordinate system is initially
introduced in a domain C of Eu,:lidean space R n (via an identity
mapping of C onto A under natural identification of both copies,
R n and R1). Then, the introduction in C of another coordinate sys-
tem defined by a regular mapping I (Le, a smooth, one-to.one map-
ping with a non-zero Jacobian) may be considered as a coordinate
transformation: we simply pass from the initial Cartesian coordinate
system to a new one in the same domain C,
Definition 4. A regular coordinl\te system in a domain C is some-
times called a curvilinear coordinate system in C,
Consider two arbitrary curvilinear coordinate systems in a domain
C: x 1 (P), ,. ., x n (P) and Zl (P),.. , ., zn (P). This means that two
regular mappings f: C......A c: R? (xl, . . " x n ) and g: C......B c:
R (%1, ' . " %n) are defined whioh map smoothly and bijecti vely
the domains C, A and C. B. respectively. In other words, each ?oint
P in C Is associated with two sets of curvilinear coordlnstes {x (P)}
and (Zl (P)}, 1  i  n. Since tis correspondence is bijective, we
may consider a correspondence which relates the coordinates {xl (P)}
of point P to the coordinates {Zl (P)}. this operation defining the
mapping 11>"".: A -+B, Le, 11>",..:' Xl (P) _Zl (P), 1  i  n. The
mapping 11>",. is called coordtnate:transformation in the domain C.
Under this transformation the initial curvi1inear coordinates {xl (P»
of point P change to new curvilinear coordinates (Zl (PH.
Lemma 1. The tran$formation 11>"". is a bijective and smooth mapping
01 the domain A onto B with a non-zero Jacobian.
Prool. That 11>"". is one-to-one directly follows from Definition 3.
The sYIloothness of 1\>"". foHows from the fact that the composition
of two smooth YIlappings is also a smooth mapping. It remains to
verify that the Jacobian J (11),,,.) qf 11>",. is non-zero a t each point of
the domain B,
Indeed, the mapping 1/1"". splits into the composition of t.....o
mappings: IjJ ,.= g.f"I: A......B (Fig, 1.5), The J!lcobi matrix of If"".
splits into the product of Jacobi matrices of the mappings I-I
nrl g, Indeed, d1j>",. z = ( a d: ) . Consider the derivative ; since
Z lJrJ

18 1. 1 nlroducllM 10 D iff.rent/al Geome/ry
2;i=Zi(yl, ,..,yn)=zi(yl(XI, ...,;Z;n), ..., Y"(XI, "..x"), where
the functions {y" (Xl, ., ., ;z;"), 1';;;;n} define the smooth mapping
j-I: A -+ C. we obtain from the Iormull\ for differentiation of a
"
. . a:1 '" a:/ ayk . h h J b .
composite functIon --:- = £J  iJ k - ,WhlC means that t e aeo I
. iJzJ II a;z:J
k=\ ,
ma.trix d1j)",.% splits into the product of two matrices dg and d/-I.
We have used here the formula which expresses the elements of the
product of two matrices in terms of the elements of each matrix,
c
,-, 
--;---R:I A
.
g
Figure I. 5
It remains to establish a relation between the J acohi matrices d/
and d (j-I), Since the compositJo j-l 0 f is an identity mapping of
the domain C into itself (see thel definition of a regular coordinate
system), we find from the formula just proved that d (j_l 0 f) =
df-l 0 dj = E, where E is an identity matrix of order n, i.e. finally
d (I-I) = (dl)-I. Thus, we have proved that the identity d1j)",. =
(dg) '(dt)-1 holds true for the matrx d'/1",., i,e, J (1j)",.) = J (g)/J (I),
and since the Jacobians J (g) andf J (I) are both non-zero, J (1j1", r)
is also non-zero. The lemma is p:roved. '
If the mapping I: C _A defin curvilinear coordinates in C, the
mapping I-I; A -+ C defines curvilinear coordinates in A (through
Cartesian coordinates in C). We shall often use this simple remark,
when passing from' the mapping 'I :to /_1.
Let a set of smooth functions {!t' (P)}, 1  i  n, be given on a
domain C. A question arises: does t.his set define a regular coordinate
system in C"i
Lemma 2, Let the set oj smoothltunctlons {xi (P»), 1  i .;;;; .n, be
such that the Jacobian of this system of junction$ J (I "'" {x' (P),
1  i  n}) is non-zero in the domain C. Then, for each point P in C
there exists an open neighbourhoodi such that "{Xi (P)} defines in this
neighbourhood a regular coordinate system (such a coordinate system
may be called a local coordinate sY$tem).

1.1. Curvilinear Coordinate, Sv,ttm,. Simpl. E",ampl.. 19
Proo/. The lemma does not prepuppose that the set of functions
{Xi (P)} defines (at least locally) a one-to-one mapping of the domain
t onto a domain A of Euclidean space R1. Using the implicit-func-
tion theorem (and the existence theorem for inver8e mapping), we
see that a non-zero Jacobian implies the existence (at least in an
open neighbourhood) of an inverse mapping which is also smooth.
Thus, the proof of the lemma foll6ws from the definition of a regu-
Jar coordinate system.
Let us note that the set of functions satisfying Lemma 2 may not
define a regular coordinate system in the whole domain C, i e. the
smooth mapping /_1 of the domain A onto C may not exist. Here is
y
,2
x
Figure 1.6
a simple example. Let a two-dImensional Cartesian plane with punc-
tured origin 0 be chosen as the domain C and let the mapping f
(defined by two functions Xl (P) ! and x' (P» represent the smooth
mapping 1 (yt, yt) = (Xl (y), x' (y», where Xl lyI, y2) = (yl)t _ (y')Z
and :r. 2 (yI, y2) = 2yly', i.e. if we put z = yl + iy2, w = Xl + Ix.
(i is the imaginary unit), then w = Z2, This mapping transforms a
complex number z into this number squared (for convenience, one
may assume that the two copies of the Euclidean plane, R2 (y) and
Rt (x), are identified with each other). The same mapping can easily
be written in the polar coordinates (r, q» to give 1 (r, q» = (r2, 2cp).
Geometric interpretation of the mapping 1 is shown in Fig. 1.6.
Let us find the Jacobian J (I) (we .9hall calculate it, for example, in
the initial Cartesian coordinate system yl, y2 on R2 (y». The Jacobi
matrix dl is of the form
_ ( 2yl 2 yt ) T
dl- 2 2 2 1 ' i.e. J(f)=4 «(yl)2+(y2)2) >0.
- y y
We see that the Jacobian is positive at aU points of C (since the ori-
gin is punctured). Hence, according to Lemma 2, our mapping estab-
lishes a local (regular) coordinate 'system in an open neighbourhood
of each point in C, At the same time, the mapping f does not have
the inverse mapping I_I because :1 is not bijective. Indeed, every
point w = Xl + Cx 2 E RZ (x), other than the origin, always has exacl-

2IJ 1. 1nlroducIlon to DI04r4ntlal G40IMlrg
Iy two inverse images under the mapping f: namely, the points
(r, q» and (r, Ip + 11) which are, f course, distinct points of the do-
main C, Thus, if a set of functions is chosen as a regular coordinate
system in a Euclidean domain C, we should not only verify that the
Jacobian of this system is non-zero (at every point of C), but also
that the mapping defined by this set is bijective, Note also that in
the above example the 1 acobian I of the system of functions van-
ishes as the point P tends to zero; A question which often arises in
geometry is: will a smooth mapping of a Euclidean space into itself
Figure 1.7
be bijective, provided 0 < e :s:;;; ,f (j) :s:;;; N < 00 (e and N are con-
stants)? This topic is beyond the scope of the book.
Every curvilinear coordinate system in C specifies families of the
so-called coordinate lines defined as follows; the ith coordinate line
is given by the equations
Xl (P) = c 1 ' XZ (P) = C 2 . .,. I Zl-I (P) = CI_ lf x,' (P) = t,
Zl+\ (P) = C'+I' ..., zn (P) = Cn.
where all c, are constants and t is a continuous parameter, As t
varies, the point P runs a smooth trajectory in th domain C, Thus, n
smooth trajectories emerge from each point P in C, and these tra-
jectories are called coordinate li*es of a given coordinate system
(at point P), For another point P !we obtain another famiJy of Coor-
dinate lines which are smoothly 'deformed as point P varies. For
example, if we consider a Cartesian coordinate system, the coordinlite
lines are straight lines through P parallel to the coordinate axes.

1.1. Curvillnoar Caordlnal SY8t.m.. Simple Ezampl.. 21
In graphic representation of a curvilinear coordinate system it is
often useflll to draw coordinate lines; in particular, transformation
to a curvilinear coordinate system' is especially clear if the coordi-
nate network is depicted (see Fig. 1.7).
1.1.3. SIMPLE EXAMPLES
OF CURVILINEAR COORoDINATE SYSTEMS
We start this section DY noting that a polar coordinate
system (r, cp) on a Euclidean plane is not a regular coordinate sys-
tem defined on the whole plane RI: Indeed, let us consider the func-
tions of coordinate transformation from a polar system to a Carte-
of'
Xl
R(r.:) _
2T C
o
Figure 1.8
sian one: Xl = r cos!f! and X2 = r sin cpo Direct calculation of the
transformation Jacobian J ('i') yields
d ( COS cp sin!f! ) T
'i'= , J(1J;)=r,
- r sin cp r cos cp
Thus, since the Jacobian is zero at the origin and the Cartesian
coordinate system Xl, Xl is, obviously, regular on the plane
R' (x', .x 2 ), the polar coordinate system is not regular (in the sense
of Definition 3). Furthermore, this is not the only disadvantage of
our polar coordinate system. This ystem is not a bijective mapping
of the whole Euclidean plane onto itself, since the points (r, cp)
and (r, cp + 2n) are transformed into the same point.
A domain C in which a polar coordinate system is regular deserves
special attention. Let us analyse this example in greater detail.
Consider a Euclidean plane RI (r, cp), where yl = r, y2 = !p, and
take an infinite strip defined by 0 < q> < 2n, 0 < r < + 00

22 1. I/lirodu'llon 10 D lU.renlial G.om,'ry
as a domain C. Then, the domain A in the plane R' (x', x')
should be chosen as the entire plane except the ray Xl  0, x a = O.
The mapping j: C ......A is given' by the relations Xl = r cos q> and
x a = r sin cp, Figure 1.8 shows how the coordinate lines arc trans-
'"
R3(X', x2, xJ)
x J
z
x
.I' x'
Figure 1.9
formed under this mapping. A rectangular network of Cartesian
coordinates goes over to a polar network. Obviously, the mapping j
is bijective and regular.
We now turn to a three-dimensional Euclidean space and consider
so-called cylindrical coordinates'. Coordinate transformation is d-
scribed by the relations::x l = r cos '1', x a = r sin rp, and x 3 = z.
Consider R3 (yl, ya, y3), where yl = r, ya = q>, y3 = Z, and take for
C the domain (0 < r, 0 < q> < 2n, -00 < Z < +(0). These rela-
tions define a smooth mapping j: C __ A C R3 (x', x a , x 3 ), where
the domain A is obtained from .Jt3 (x 1 , x\ x 3 ) by the elimination of
the half.plane (Fig. 1.9). The J a'cobi matrix is of the form
( cos q> sin !p O ) T
-rsinrp rCOSIj) 0
o 0 1
and the transformation Jacobian is equal to r. Thus, in the domain
A a cylindrical coordinate system is regular (the Jacobian is zero
only at points of the z axis); th half-plane (q> = 0, r  0) is elimi-
nated to make the mapping bijective.
Let us now turn to an n.dimenional Euclidean space and introduce
in it a spherical coordinate sytem. Although the transformation
formulas, the Jacob! matrix, aIJd the Jacobian are written for the
n-dimensional case, the structure of the domains C and A is ana1ysed

1.1. Curvilin.ar Coordinai. SIJ$I.ms. Simple EZIJ.mpl.. 23
(for clarity) only for the three-dimensional case. The transformation
formulas are as follows;
In: C (r, e l , e., ..., 0n_j) - A (Xl, .x 1 , .... .x n ).
x' = r cos 9.
x 2 = r si II 0\ cos 6 l
x 3 = r sin 9. sin 9a cos 9 3
I .. I . 9 . 9 ' . n e
x -II = r S11 1 sn ... sn "n-2 cs n-I
.x = r sm 6 1 Sin e l . . . sm 9n_. sin en_I
Remark. It is clear [rom thesci formulas that they all have the
same structure, but the parameters e" starting with the number
i = n. are equal to zero. The Jacobi matrix is of the form
cosO t
sin 9 1 cos 6 J
dfn=
sill 0 1 . . . sin 6n_2 CDS 6n_1
sin 9, ., . sin 9n_2 sin 9n_1
-r sin 6,
reDs 0 1 cos 6 2
o
-rsin6 j sine l ...
r cos 6. . . . sin On_2 cos 9n_1
r cos 6 1 . . . sin 9n_2 sin 9"_1
o
J n = r sin 9) . . . sin 6"-2 cos a n _) Cos 6 n _J n-l
+ r sin e) ., sin 6n-2 sin enol sin On-II n-l
= r sin 6 1 . . . sin 6 n - 2 / n -\.
1 2 = r, 13 = +r2 sin 9). /4 = r 3 sin191 sin 9 2 ,
13 = r' sin 3 9 1 sin 2 6 2 sin 9 3 and so on.
Spherical coordinates in a threetdimensional Euclidean space are
usually denoted by (r, a, q»j in this notation the transformation
formulas become Xl = r sin 9 cos <p. x 2 = r sin 9 sin <p, x 3 =

24 1. Introduction to DtQerenttal Geometry
r CDS e. 0 < e < 11, 0 < cp < 211, r;;;;" 0, In these coordinates J =
r 2 sin 6. Consider the structure of domains C and A (see Fig. 1.10).
The Jacobian of this transformation vanishes only at points of the
)(3
1(2
Figure 1,10
:z:3 axis, the remaining points belonging to the half-plane (.x 2 = 0,
Zl;;;;" 0) are eIJminated to make it possible to introduce II bijective
coordinate system. For a fixed r, the coordinate lines of the para-
figure I. II
meters 6 an<J cp ar shown in Fig, 1.11. These angular parameters are
sometimes called latitude and longitude (they provide 8 coordinate
network on a globe), The Jacobi matrix in the three-dimensional
case is
( sin 6 cos cp
dljJ= sin6slncp
cos 9
r cos;!) cos cp
rcos6sincp
- r sin 6
r sin 9 sin (j) )
r si n  cos cp ,
Problems
1, Prove that the system of functions u = .x + sin y,
II = Y - {. sin :c on a plane is a regular coordinate system.

1.2. Tho Length of Curve in Curvllinoar Coordtnal.. 25
2, Demonstrate that
defined on a circle 8 1 ,
3, Write the Lllpilice
coordinate systenl.
a globa'i coordinate systenl cannot b
atu a'u
equation .6.u = a;o + iJy' = 0 in a pollir
L2. THE LENGTH O}' CURVE IN A CURVILINEAR
COORDINATE SYSTEM
1,2.1. THE LENGTH OF CURVE IN A EUCLIDEAN
COORDINATE SYSTEM
Consider a Euclidean space R n and define in it a Eucli-
dean scalar product (t, TJ) = tlt} , t, 1) ERn. Then, with each vec-
Lor S E R n we can associate a real number called the absolute value-
or Jength and defined as I S I = V (6, t> . This formula gives the
length of a vector emerging from point 0 to a point 6 ERn, To find
the distance between any two points 6, t} E R", we have to calculate
the length of the vector t - 'I). ,It is known from analytic geom-
etry that the angle (j) between two Vectors t, 11 E R" can also be
expressed in terms of the scalar product as cos cp = I i',  I ' We
thus see that sitch important metric concepts as the length of a
vector and the angle between t"1'0 vectors are closely related to
the Euclidean scalar product, While defining other important geo-
metric conc.epts, we shall often proceed from the scalar product of
vectors.
We know how to calculate the length of a straight segment; it
would, however, he desirable to now how to calculate the length
of a smooth curve, Let us deline the length of a smooth curve 1 (t),
assuming that it is given in parametric form, i,e. by a set of n smooth
functions Xl (t), . '.., X" (t) in a i Euclidean space, where the pa-
rameter (time) runs either the entire rea) axis or the segment [a, b)
(the latter case is of major impornce to us), We shall also assume
that Xl, , , " x" are Cartesian coordinates in R",
Definition 1, The length of a curve 'I' (I) from point 1 (a) to point
y (11) (or from the parameter value t = a to t = b) is the number
b
1(1): = S y (V (t), .;. (t)} at; where y (t) is the vector with the coordi-
e
nates ( dZ;t(t) , ' . ., az;e (I) ) called sometimes the velocity vector of
the curve y (t) at point t or the tangent vector to the curve y (t).

21) 1. 1ntrodu<tlon to DiOor"fl/tal Geometry
Thus, the leng'\} of a curve is the integral of the length of the ve-
locity vector (r vI the velocity vector magnitude). The explicit
expression for "_ 0 length of a curve is
b /"
l (y) =  V ;I ( dzt(t) ) 2 dt.
Lemma 1. Given a smooth curve "I (t) and two fixed points "I (a) and
y (b) corresponding to the parameter values t = a and t = b. Let
t = t ('t') be an arbitrary smooth ransformation of the parameter t
into a new one 't' such that  > O. Then, the length of the curve
I ("I (t» remains unchanged, i.e. the equality l ("I (t» = l ("I ('t'»
holds, where a = t (ex), b = t ().
Proof. Straightforward calculation yields
b _ 1\
r J" r /. . ( d. ) 2 dt
l(y(t»= ,) Y (Yt.'I',)dt= j J. (i'Ii't) dI "d-rd'T:
a a
 -/d. ClI  I
= ., Y (y" "It) did:< d't =r j 1 ("I.. y) d't',
a 
which is what was required. Here we have introduced the nOlation
, _ { dzl (t) dX"(I) }
'1', - """"dt"'.,.,"""'di"""" '
Let us assume that we have two smooth curves '1'1 (t) and '1'2 ('t')
intersecting at a point P of a Euclidean space, that is, there exist
parameter values t = a and 't' =ib such that P = "11 (a) = '1'2 (b).
Let us define the angle between the two curves at the intersection
point.
Definition 2, The angle between; two smooth trajectories "11 (t) and
1'2 ('t') intersectiqg at point P = '1'1 (a) = '1'2 (b) is defined as cos cp =
7 1 (<1), .(b» , provided the veloity vectors Yl (a) and Y2 (b) are
1'1'1 (a}l. Iv.(b)!
both non-zero at p,
Remark, Strictly speaking, this formula defines not one, but two
angles equal together to n. If we ssume however that the curves are
numbered, we arrive at the concept of an oriented angle which is
uniquely defined by the above formula. Consider the condition that
the two velocity vectors be nonzero at the intersection point. At the
points where the velocity vector of a smooth curve vanishes the
curve may have a cusp and abruptly change its direction. Therefore,
tit ere is an uncertainty in definirg the angle. at the point of inter-
section with such a curve. in regard to the choice of one of the two

1.2. The Long/I< 0' Curve /11 Cllrvilinear Coord/nalt$ 27
smooth segments of the CUfVe separated by the cusp (Fig. 1,12). It
should also be noted that the exitence of a cusp on a smooth curve
(fOf those parameter values for which the velocity vector vanishes)
by no mellns contradicts the condition that the curve is smooth (see
?
;'Y:
Figure 1.12
the definition of a smooth curve). Figure 1.13 gives an example of a
smooth curve with a cusp; here the "cusp angle" at the singular point
is equal to 11,/2. It is a simple malter to invent a smooth curve with
the cusp angle at the singular poittt equal to 11 (see Fig. 1.14). (Exer-
t
x(1) Y(I) Y
7(1) = (X(I). Y(I))
x
FIgure 1.13
cise. Write a parametric equation' of the curve of Fig. 1.14, Ques-
tion: can this curve be defined by analytic functions x (t) and y (t)?)
Let us now discuss to what extent the above definition of the
length of a curve is in conformity with the customary ideas about
>-
Figure 1.14
length based on the concept of the Euclidean length of a segment
and on the concept of a thin inextensible string with the aid of which
we can measure the length of a more complicated curve. We know
that the circumference of a circle is sometimes defined as the limit
of the perimeter of a polygon inscribed in (or circumscribed about)

28 J. I nlroductloll 10 D IDtrentto.l Geometry
the circle when the number of the polygon sides tends to infinily.
We start with a segment, of course, .
(1) Let Lhe curve 'I' (I) be defied by the linear functions x'.(t) =
alt, where a l = const, 1  1  n, a  t  b. Then, caJcuJatlon of
the length of this smoot h curve f rom t = a to t = b yields
bin /"n
1 (y)g =  Y 2i (a i )2dt= (b-a) Y  (a!)2.
o 11 (=J
Since the segment originates at the point {a.fa} and terminates !It
tho point {a l b}. the ordinary (Euclidean) length of the gmellt is
(b-a)" f f (a'F, \hich coincides with the value of the integral
V 1=1
1 (y)g,
(2) Let the curVe 'I' (t) be defined on the plane R 2 (x, y) by the
parametric equations :r; = R cos t Y = R sin t, Straightforward
evaluation of the integral gives l (y) = 2nR (where a = 0, b = 211),
which, apparently, coincides with the familiar expression for the
length of a circle of radius R.
1.2.2. THE LENGTH OF CURVE IN A CURVILINEAR
COORDINATE SYSTEM
Let us now consider an arbitrary curvilinear coordinate
system in a Euclidean domain Cand let 'I' (t) be an arbitrary smooth
curve in this domain, What is the expression for the length of
i' (t) In this curvilinear coordinate system? To answer this question,
we analyse the behavior of the cqmponents of the velocity vector of
y (t) under coordinate transformation. Denoting the curvilinear coor-
dinates by ZI, ' . "z", i,e. xl =iXI(z), we obtain from the law of
differentiation of'a composite function
(kl (I) = (kl (: (I» ==:  l..i !.=!.. 
dl dt a.k dl' 1in,
(k)
that is,
b I
1 (y)=  V
b /"
=V
 ( d",1  (I» r dt
fal
n .
"" (  ;}:r;1 (:(1»)  ) 2 dt
L.J" ;}zh dt
!al (k)

1.2. Th. L.nlIth of CII'V' III Curv/ltnear Coordllllll.. 29
b y n
= r 2; ""..2!!..:   dzJJ dt
J L.J a. m a.r> de T
a {m11 m. p
r V '" d im dsJJ
= j LJ ({ml' (Z) tit di- de,
4 m. p
n
where the functions ({mp (z) are of the form ({mp (z) = h :: ::; ,
...1
Obviously, these func.tions are symmetric. in m and P, i.e. ({Trlr> =
g [ m' Hence, the set of functions 8mp can be represented in the form
o a symmetric matrix denoted in the sequel by G = (imp), In the
formula just derived the coef.liciertts of G are expressed as the sums
of the products of elements of the Jacobi matrices. Indeed, d1j!"", ""
( : ), so that G is the product of two matrices G = AT_A, where
A "" d1l'.,,,,. Clearly, the matrix G depends on the curvilinear coordi-
nate system z and may, in general, vary under coordinate transfor-
mation. What is the law of v!lriation of G (z)? Let us make another
coordinate transformation {z'}...... {y}, that is, consider a regular
coordinate transformation zl = :1 (yl, ' . 'I yn), 1::;;;; i ::;;;; n. (The
set {} is again considered as curvilinear coordinates in C,)
In this case the coefficients Imp (z) are transformed as
,. n
( ) "az l (y) azl (y) '" "" az! az m 8.,1 a."
g I Y = L.J 8giI Q;i"""" = LJ L.J B;rn avr DiP 8i1
ir:::lt 1lm.1J
n
'" a. TrI ( ",aX' azl ) a." '" a. m 8.r>
= L.J ¥ L.J '8iiii" aiP 8;1 = L.J ¥ ({mp (z) 8;1'
m. p fc::1 m, p
i.e. G (y) = (d'l'II' r)T G (Z)(d1jJII,')'
Remark, To simplify the notation, in expresiolls of the form
 a1b l . where summation extends over repeatod indices. we shall
(H . j I
omit the symbol .t and simplr write a b , For example, the formula
"
I (y) = "" az l (y) oz' (y) is written as g (y) = 8z' (y) , a:zl (y)
l L.J 8y lJ I I 8ylr. 8 I .
1=1 Y Y
Sometimes. we shall denote new coordinates by Zl', uing primed
indices for II new coordinate system, viz., Kj'J'.
The functions ({mp (z) have a clear geometric meaning. Consider an
arbitrary point P in a domain C and coordinate lines of the curviJin-

30 1. 1 ntroduWon to DID..nllal Gomtlry
ear coordinate system ZI, ' , .j zn through P. Each of these lines
can be defined by the following' parametric equations: :1= CI1 . , .,
Z'-1 = C/- l . Zl = t, Zi+1 = C/+'I' . . ., Z" = Cn' where c" (1::;;:;
Cl. (op i) ::;;:; n) are constants suc that P has the coordinates (in the
system z). {z" = C. 1 ::;;:; Cl. ::;;:; n}. Denote the mth coordinate line
by Vm (t), 1  m ::;;:; n. Then, in the coordinate system x the mth
coordinate line of the system z is written as {Xl (c l , . . .. cm-I>
zm = t, c m + lI ., '. cn)}. 1 ::;;:; i  n.
The velocity vector of this smooth curve at point P has the
d . { iJz 1 } 1 . S .  8z 1 iJzI
coor males em = 8%m ' ::;;:;t::;;:;n, mee gmp (z) = L1 azmBiP ' We
i
can write gmp (z) = (em, e p ), i.e. the functions gmp are scalar prod-
ucts of the vectors tangent to the corresponding coordinate Jines
Figure 1.15
(Fig. 1.15) (we recall that the scalar product is Euclidean, i.e. it is a
characteristic of the surrounding space).
Thus we see that under coordinate transformation the matrix
G (z) is transformed as the matrix of a quadratic form. In particular,
if the initial coordinates are Cartesian, G is an identity matrix, so
that in any other (curvilinear) coordinate system i can be written
as G (z) = A TEA. where A = d'l'.,;r. provided {x'} are Cartesian
coordinates (then G (x) = E).
Before proceeding further, we shall consider formulas for the
length of a curve in various curvilinear coordinate systems and pre-
sent explicit expressions for the matrix G in these coordinate systems,
(1) Polar coordinates 00 a two-dimensional plane R2 (r, cp). In
Cartesian coordinates (Xl, x 2 ) the matrix G (x) is of the form
( 1 0 ) , { 0, i *; .,
o 1 ' I.e. giJ=61J= 1, i=j The Jacobi matrix (calculaterl

1.2. The Length of CurlJt! In Curvilinear Coordinates
31
( caS Ip sin Ip ) T
above) is d1j>=. , whence
- r sm !p r cos q>
G (r, Ip) = d'l'T (d1\»
( cos q> sin!P ) ( COS <p - r sin !P ) ( 1 0 )
= - r sin Ip r sin q> sin cp r cOs Ip = 0 r 2 .
Hence. in a polar coordinate syJtem the length of a curve defined
as y (t) = (r (t), Ip (t» is given by.
b
I (y) =  y ( ; I) 2 + r2 (  ) 2 dt.
.
(2) Cylindrical coordinates i11 a three-dimensional Euclidean space-
R3 (r, <p, z). The matrix G (z) in Cartesian coordinates (Xl, .:r, z3) is
G (or) = E. The Jacobi matrix for this case has been calculated
above
( cos Ip sin q> O ) T
d1j>= -rsinlp rcoslp 0 ,
o 0 1
whence
G (r , <p, z) = (d1\»T (dip) = (  2 g ) ,
o 0 {
Thus, in a cylindrical coordinate system the length of a curve-
y (t) = (r (t), Ip (t), z (t» is of the form
b
I (y) = } y (  r + r 2 (  ) 2 + ( : r dt.
G
As an example, we calculate the length of a helix on a right cirC\l
lar cylinder given parametrically as: r (t) = R = const, Ip (t) = wt, .
.; = qt, Vie have then
b
l(y)= ,\ VR 2 w 2 +q Z dt.= VR2+q2(b-a).
Q
(3) Spherical coordinates in a three-dimensional Euclidean space
R3 (r, e, q», The matrix G (x) in Crtesian coordinates (Xl, x 2 , z3) is
G (x) = E. The Jacobi matrix (calculated above) is of the (oru)
( sin e cos <p r cos e COB fP - r sin e sin <P )
sin e sin <p .. cos a sin fP r sin e cos cp ,
cose -rsine 0

32 J. Inlroducllon to D'lfferenlial GeoIM1rv
whence
G (r, e t CP) -= (d1J1)T (dq,) = (  2  ) ,
o 0 r 2 sin 2 6
Hence, in a spherical coordinate system the length of a curve
given by i' (t) = (r (t), e (t), cp (t» is expressed as

1 (y): =  V ( :; ) z + r 2 (  ) 2 + r2 sin 2 a ( i ) 2 elt.
a
It is sometimes convenient to deal with an elementary arc dl in-
stead of the whole arc. In thel examples considered ahove these ele-
mentary arcs squared (in the respective coordinates) are of the
form: (dl)1 = \dr)1 + r ll (dcp)2 (polar coordinates on a plane), (dl)' =
(dr)' + r' (d(jJ) + (dz)1 (cylidrical coordinates in RS); (dl)' =
(dr)' + r (de)' + r l sin ll e (dcp)1 (spherical coordinates in R3).
1.2.3, THE CONCEPT OF THE RIEMANNIAN METRIC
IN A. EUCLIDEAN DOMAIN
In the preceding section we "have associated with each
curvilinear coordinate system z in C a smooth matrix function
G (z) which is transformed under coordinate transformation as a
quadratic form (at each point). The role of this set of matrix func-
tions is quite clear: once it is given, the length of a curve can be
<:alculated in a curvilinear coordinate system. Among all the prop-
erties of this set of matrix functions of major importance is the
property of being transformed (at each point) as a quadratic form,
Consider now various sets satisfying this property.
Definition 2. A Riemannian metric is said to he given in a Euclid-
ean domain C if in any regular coordinate system zl, ' , '. z"
there is defined a set of smooth functions g",p (zl, , , " z")
such that: (1) g""p (z) = iI'''' (z) (G (z) is symmetric), (2) G (z) =
(g",p) is non-singular and positive definite, (3) under coordinate tram'
formation z _ y the matrix G (z) is transformed 8S G (y) =
d,pTG (z) (d1jl) (recall that on]y regular coordinate transformations are
considered). liere d1jl stands for the Jacobi matrix of the coordinate
transformation d1j>",
Definition 3. Let the Rie'ilnnian metric G (z) = g,) be given in
C and let a smooth curve y (t) = {Zl. (t)} be given in the coordinate
system (Zl). The length of the curve from point y (a) to point y (0) is
the number

l(y) = 
.
( ) (  ) ( 1i,)!I» ) dt
g 11 z dt dl .

1.2. The Length 0/ Curve In Curvlltnear Coord/"al" 33
If two smooth curves 1'1 (t) and 'Va (t) satisfying '1'1 (0) = 'I' (0) = P,
";1 (0) =1= 0, Va (0) =1= 0 intersect at a point PEe, the angle between
these curves (for a given Riemannian metric) is the number q> such
that
( ) d.: (/) d.t (I)
iIJ'
COS qJ =
... ( ( ) <I.: (t) d.f 1t) ... ( ( ) d.: (t) d.{ (t)
Y m. -----aI Y ill. 
The definition of the Riemannian metric presented above can be
formulated in more invariant terms without explicit coordinate
writing of the metric. Namely, once the Riemannian metric has
been given, we can define (at each point of a domain) a bilinear form
(,), on the set of all vedors tangent to smooth trajectories through
this point. Indeed, if '1'1 (0) = P = '1'. (0), where PEe, then we
may assume (s, 1)}g = gl!;T}j for the vectors S = 11 (0) and 1) =
12 (0), where S = (1;1, . .., "), TJ = (T}l, . . ., TJ"). Note that the
coordinates of the vectors sand TJ are calculated in the system
(Zl, . . ., z").
Lemma 2. The mapping S, TJ -+ (5, 1)}g defines a non-degenerate,
p08ittve definite bilinear form smoothly dependent on a point.
Proof. That the mapping is symmetric and bilinear follows from
Definition 2. We now verify that the mapping defines a bilinear form,
We perform a regular coordinate transformation {Zl} -+ {Zl'} and
have
'l'dt) = {z: (t), ..., z (t)}, '1'2 (t) = {z (t). ..., % (t)} ,
;' _ d-r _ ;;-;' d.1 _ ;;.;' I j' _ 8. 1 ' i _ a. 1 8. l
S -T-B;lT-B;lS' 11 -o;I1J, g;'j'-7 8.i' g'l
whence
" I' , " 8.1 0.1 ;;.;'  8.;- p
(I; ,11 }g' = gi';'S 1) = ---"-........,,. gjJ- ;; . .., _ 8  1J
8_' 8.' 'A .r
( 8.' ;;.1' ) ( ;;_1 ;;_;' ) loR P _ i)il6i SA n
= a. I ' "'8Zh "JiP gill> 11 - A pgl! I}
= giJSI1JI = (6, 11>"
i.e. (S, '1>/1 is
iI,1 ih l ' i
---,,- - ;; __ = 6 A
a.' V'
ln\n!ol is pro\'ed.
Thus, we arrive at the folJowing definition: the Riemannian metric
rea]ly a bilinear form. We have used the relation
which follows from ;the identity (d1j1) (d1jJt ' =E. The

34 1. Introdu.ction to Differential Geometry
is said to be defined in a Euclidean domain C if at each point of the
domain thero is given a hiHnear form (scalar product) on vectors
tangent to smooth curves through this point, the form being non-
dpgencrate, positive definite, sYlpmetric, and smoothly dependent on
the point. '
Lemma 2 shows that this de/inition is equivalent to Definition 2.
The lemma implies, in particular, that the length of a smooth curve
does not depend on a curvilinear coordinate system (provided the
Hiemannian metric is given in a certain coordinate system and is
transformed under coordinate substitution in accordance with the
law considered above).
Do Riemannian metrics exist? An example has already been giv-
en. Indeed, if the matrix G (x) = (6 tl > is defined in a domain C
in the Cartesian coordinates (Xl,:, . " xn), then in any other curviHn-
ear coordinate system z obtaine'd from z by a regular transformation
we can set, by defmition, G (z) = d1jlTG (x) (d1jl) = (d1jl)T (d1jl), where
dlj) ill the Jacobi matrix of this transformation. Apparently, since any
regular coordinate system z can be obtained from a Cartesian coordi-
nate system by a regular transformation (see the above definition
of a regular coordinate system), we have defined the matrix function
G (z) in an arbitrary regular coordinate system, This consideration
shows that conditions (1)-(3), which the Riemannian metric must
satisfy, hold true. Indeed, properties (1) and (2) can be verified
direcLJy by the definition of G (z), property (3) follows from the
fact that tbe Jacobi matrix of the composition of two coordinate
transformations is equal to the product of tht! Jacobi matrices of
each tranformation: d1jl". 1, = d'lpz.. z. .d1jlz.. Zo" Thus, we have defined
a Riemannian meLric in a Euclidean domain, This metric is Euclid-
ean, and the squared differential of the length of a smooth C\lrve in
a Cartesian coordinate system is written as (dl)2 =  (dx i )'. How-
i=1
ever, it should not be thought that, given an arbitrary Riemannian
metric, an nppropriate coordinate transformation may be found
n
in C which will reduce the metric to the form  (dx i )1.
i-I
Definition 4. A Riemannian metric G defined in a domain C is
called Euclidean if in C there exists a coordinate system y (generally
curvilinear) such that G (y) is an identity matrix.
Given a Euclidean metric (relative to a coordinate system), we can
describe all other coordinate systems in which this metric is also
Euclidean (there are many such systems). This description will be
given in the sequel, for at tbis moment we do not have at our dispos-
al an appropriate apparatus. Here we only note that al1 such systems
can be obtained from one system by rotations,. translations, and
reflections in a Euclidean spac.

1.2. The Le1Iglh of Curve In Curvilinear Coordlnat$ 35
The existence of a "non-Euclidean" metric, i.e. a metric which
n
cannot be represented as  (dx i )2 in any coordinate system, can-
.=1
not be demonstrated for the time being. At this moment we are
unable to find such a metric about which we could say with certainty
that it is non-Euclidean. It is intuitively clear that to find such a
metric, one has to find its invariants preserved under a regular coor-
dinate transformation. Once these invariants are found for two metrics
and shown to be distinct, we can prove the non-equivalence of the two
metrics, Such invariants do exist and we shall define them soon.
Only after that shall we be able io prove rigorously the existence of
a non-Euclidean metric defined in a Euclidean domain.
As was noted earlier, the Euclidean metric written in an arbitrary
coordinate system z "loses" its simple "Euclidean form" and is given
by the matrix G (z) in which it is rather difficult to "recognize" this
metric, especially if we do not know the invariants discriminating
different metrics (that is, the metrics which are not transformed
into one another under a suitable coordinate transformation). We
now show how the Euclidean metric is written in simple curvilinear
coordinate systems coru;idered above. It is often more convenient to
deal with he squared elementary arc of a smooth curve (dl)2 =
g/J (z) dzidz J rather than with the matrix G (z) of the Riemannian
metric,
(1) Polar coordinates on a plane: dl 2 = dr 2 + r 2 dq>2.
(2) Cylindrical coordinates in three-dimensional space: dl 2 =
dr 2 + r 2 drp2 + dz 2 .
(3) Spherical coordinates in three-dimensional space: dl 2 = dr 2 +
,2 (dB2 + sin 2 B drp2).
1.2.1.. INDEFINITE METRICS
Until now we have i considered only positive definite
metrics caUed Riemannian. In ,various applications, however, we
often come across the so-called indefinite metrics.
Definition 5. An indefinite metric is said to be given in a Euclidean
domain C if in any regular coordinate system Zl, ' , ., z" there is
defined a set of smooth function {gmp (ZI, . . '. zn)} satisfying all
the conditions imposed on the Riemannian metric (see Definition 2)
except the condition of positive 'definiteness, i.e. the corresponding
quadratic form is indefinite.
As a simple example of an indefinite metrk, we shall consider the
so-called pseudo-Eudidean metric of index s in a pseudo-Eudidean
space R. To construcl such a metric, it is sufficient to provide an
ordinary Euclidean space R" with Cartesian coordinates x 1 , ' . ., x n
and define at each poinl P E R n the following bilinear form (with

3 6 1. / nlr.duclion I. Differential GeometrJl
constanl, i.e. point-independent coefficients): <S, TJ >. =
..
illi +
iet
"
 ,i11j. Then, for any smoobh curve 'I' (t) = {Xi (I)}, 1 :s;;;; i :s;;;; n,
j::!!!!.+l
the length of an arc is expressed as
b /
l(y)=  Y
_  (r +  C:;;i ) 2 dt.
;:01 1=.+1
For n "" 4, pseudo-Euclidean space of index 1 is sometimes called
Minkowski's space (in the special theory of relativity); we shaH also
tEl, = 0
IE'. < 0
. E' R
Figure 1.16
consider spaces R and R:. Not.e that pseudo-Euclidean space of
index 0 coincides with an ordinary Euclidean space.
Remark. The study of spaces R::_, is equivalent, in a sense, to
the study of R, since aU lengths in R_. can be multiplied by i
(the imaginary unit); the form (,).._. changes then to <.>.. Therefore,
we shaH assume,. for simplicity, that the inequality s:S;;;; (n12J holds
true (square hrackets denote the integral part).
As in an ordinary Euclidean space, the length of a vector S in R
is given by I 6 I. = V (5,5>" but in R, unlike in R, the length of
a vector may be zero or imaginary. Indeed, since the form (.>. is
r,ot positive definite, the set of all vectors S E R" emerging, say,
from the origin can he decomposed into three disjoint subsets:
(6, 51. < 0 (time-like vectors), <5: s>. = 0 (light or isotropic vectors),
<5, 5>. > 0 (space-like vectors). Owing to this circumstance, there
may exist ver.tors with zero, real, and purely imaginary length
Indeed, the length of a time-like vector is purely imaginary, that of
a Jight vector is zero, and a space-like vector has a real length. The
positions of these three types of vectors in pace are also different
Consider the veclor5 emerging from the origin. The definition implies

J .2. The Length f Curue In Curvilinnear Coordinates 37
that isotropic vectors form the cone: - 1: (XI)2 + t (.ri)2 = 0
fl:::ll j;;o"'+1
with the vertex at the origin, time-like vectors are located "inside"
the cone, i.e. in its hollow bounded by the coordinate plane Xl, . . .
. . ., x\ and space-like vectors are located "outside" the light cone
(Fig. 1.16). This fact in itself shows that the indefinite metric defmes
a much richer geometry (in the metric sense) than the Euclidean
metric.
Remark. In Minknowski's space R1 (in the special theory of rela-
tivity) the isotropic cone consists entirely of the so-called "light
vectors" S (i.e. (S, S)l = 0) and is termed the light cone, since a
light ray emerging from the origin propagates along one of the gen-
erators of this cone (provided the parameterct is chosen as the coordi-
nate x', the constant c is the velocity of light).
As in a Euclidean space, we can now define the length of any smooth
curve in the pseudo-Euclidean space R of index s by setting I. (1') =
 i-
  (y, y). dt, the only difference from the Euclidean case being
G
that certain curves may have zero, purely imaginary or complex
length. Indeed, since the integral along a curve is additive, it can
he decomposed into a sum of several terms (for simplicity, their
number is assumed to be finite), each term being such that at each
point of a given segment of the curve the scalar product (y, V). does
not change t1l(' sign (segments of zero length may also occur). Thus,
the length of a curve is, in general, a complex number.
The space R introduced for conveniently expressing certain quan-
tities in the special theory of relativity has played an important role.
Let us refer R to the coordinates::r;O = ct, Xl = x, x 2 = y, x 3 = z,
i.e. dl 2 = _c 2 dt 2 + dx 2 + dy2 + Idz 2 . Here t is time and c is the
velocity of light. Consider in R1 I\n orthonormal 4-frame e l , e 2 , es,
e, with respect to the Euclidean metric in R'. Here the space R: is
superposed, for the sake of clarity', with the Euclidean four.dimen-
sional space. Consider also th'e so-called "world line" of a
particle y (), this line being a smooth trajectory in R1. If x, y, z
are spatial coordinates, the motion I)f the particle along the trajectory
"i () can be interpreted as the evolution in space and time of a
particle moving in a three-dimensional Euclidean space. Let y
be, as before, a vector tangent to the trajectory l' () at point .
Since, according to the special th'eory of relativity, no signal can
pr opagate at veloc ity exceeding th'e speed of light c, we have cdt >
11 dx 2 + dy + dz 2 , where cdt, dx, i dy, dz are the coordinates of an
infmltesimal translation along the trajectory l' () in the direction
of the tangent vector y (). In other words, the length of any path

38 I. 1 nlmduclion 10 DIDrtnttal G<ometry
cannot exceed the distance travelled by light for a given time. Hence,
we find that the relation _a 2 dt 2 + dx 2 + dy2 + dz 2 < 0 always
110Ids true along the world line Y (T) of any particle, i.e.
(.y (T), Y (T»I < O. This means that any vector tangent to a world
line is a time-like vector and the world line of a particle always
has imaginary length. In particular. a worJd line lies entirely
inside the light (isotropic) cone with the axis t. This condition
Figure t. t 7. Th world line depicted
in t he bottom cannot exist
must be satisfied at any point pf a world line (see Fig. 1.17). Wf:
recall that the isotropic cone can: be defined at any point of a pseudo-
Euclidean space. In !:ections to ollo"'; we shall discuss the geometry
of Minkowski's space at greater length.
Problems
1. Demonstrate that the length of a curve is the limit of
the sum of the lengths of segmel1ts of a broken line connecting fInite-
ly many points on the curvo 'When the maximal segment length
tends to zero.
2. Prove that in EucJidean space a straight line is the shortest
among aU other lines connecting two points.
1.3. GEOMETRY ON A SPHERE AND ON A PLANE
We start with a two-dimensional Euclidean plane refer-
red to Cartesian coordinates x, y and endowed with Euclidean metric
dl 2 = dx 2 + dy2 (note that the symbol ds 2 is also used sometimes

1.9. Geonu:lry on a Sphere a..d on a Plane
39
for an infinitesimal elementary arc of a smooth curve; we shall use
both symbols). This Riemannian metric (apparently, it is positive
definite) induces the scalar product I (;, 1]) = £J1]1 + £'Tj'. The con-
cept of scalar product naturally leads to the concept of circle as the
set of points which are the ends of :Vectors of length R. If we intro-
duce polar coordinates on a plane, then circles with centre at point 0
will be coordinate lines of the formi' (t) = const. In a polar coordi-
nate system the infinitesimal elementary arc of a circle is rdlp.
Consider now a standard embed&ing of a two-dimensional sphere
in a three-dimensional space referred to Cartesian coordinates x, y, z.
The sphere is defined as a set of points which are the ends of vectors
of length R emerging from point O. Before studying the geometry
of a two-dimensional sphere in greater detail, let us discuss the fol-
lowing general topic. Suppose a smodth curve l' (t) lies on a sphere S'
and we need to calculate the length of the entire curve (or of its
part). Since we deal with an Eucliaean space, we may consider the
ambient three-dimensional Euclidean metric ds' = dx' + dg 2 +
dz', write the smooth curve paramericallY y (t) = (3: (t), g (t), z (t»,
and calculate the quantity in question by the methods described
above. The same procedure can be used if we need to measure the
angle between two curves 1'1 (t) and!1'. (t) intersecting on a sphere S'
(both curves are assumed to lie entiely on 8'). To this end, we have
to fmd the Cartesian coordinates or: the vectors 11 and Y2 (in a three-
dimensional Euclidean space) and alculate this angle, using again
the ambient Euclidean metric.
It should be noted, however, that in such calculations we use the
properties of the Euclidean metric ionly at points in the vicinity of
S', In other words, we could consider the Euclidean metric only for
points of a two-dimensional sphere: and express the metric in terms
of coordinates on this sphere. Since the sphere 8 2 is given in R3 by
a single equation, the position of a. ]Joint on the sphere is determined
by two parameters (i.e. less by one Ithan in three-dimensional space).
This is especially clear with spherical coordinates (r, a, q» in R3.
In this case a two-dimensional sphere of radius R is defined by a
single equation r = R = const. Let us now obtain an explicit ex-
pression for the scalar product of two vectors tangent to curves lying
entirely on S' (these vectors aro therefore tangent to the sphere).
Let 1'1 (t) = (R, 6 1 (t), C'j>1 (t», Y. (t) = (R, 6. (t), q>2 (t»). We have
then
101 (t) Q (0. a l . I)' 1» (t) = (0, 6 2 , .), i.e. (1'1' 'Yt}
= RZ (Sle. + sin z 6 (l) 12)'
where (6 (t), If> (tn are the coordinates of the point of intersection of
the curves '11 (t) and 1'2 (t). This iplies that the scalar product just

40 1. I nlrQdutllon 10 Differenllal Geometry
calculated coincides with the scalar product of two vectors (8 L . l)
and (e 2 , 2) relative to a new bilinear form: R (aiel + sin" e (t)12)'
This form induces the quadratic form R2 (d6 2 + sin 2 a dcp2) which
is obtained from the corresp\lnding quadratic form in Euclidean
space dr 2 + r 2 (d6 a + sin 2 e dqJ2) by substituting the functions of
a, '1': r = R = const, e = a, w = 'I' for the variables r, e. cp. The
Riemannian metric R2 (de 2 + sin 2 a d!jJ) thus obtained on S is said
to be an iruluced ambient Euclidean metric of a three-dimensional
spaco.
Since the position of a point on S2 can be specified by two para-
meters 9 and cp (latitude and longitude), the radius vector of a point
on a sphere can be reprented in the form: x = x (6, cp) =
R cos e CDS cp, Y = Y (9, fI')1 = R cos e sin cp, z = z (9, fI') =
R sin a. Substitution of these three functions (of two parameters) in-
to the expression for the squared elementary arc in a three-dimension-
al space dx 2 + d y 2 + dz 2 ields (dx (a, cp»2 + (dy (a. cp»2 +
(dz (6, q>))2 = R (de' + cos 2 a dcp2).
This concrete example will b'e generalh;ed below and will serve as
a particular case of "induced Riemannian metrics". To this end, in
the following chapters we shall introduce .II rigorous concept of a
"surface" to which the ambint Riemannian metric is restricted.
A two-dimensional sphere stan4ardly embedded in a three-dimension-
al space is precisely a "surface". We shall give some other examples
of "surfaces", which are mainly defined as vector functions of sever-
al variables. For example, if the vector function (radius vector)
x = z (u, u), y = y (u, v), z = Z (u, JI) with the parameters u, u
varying in a domain on a plane i given in a three-dimensional Euclid-
ean space, this radius vector sweeps out (as u and u change) a set
which we shall call (provided the vectors ( : ' : . : ) and ( ;; ,
, ) are linearly independent at each point (u, u» a "two-dimen-
sional surface" in a. a three-dimensional space. (The general defini-
tion will be given below.) Once such a surface (or a piece of it) is
defined, there arises the follo\ving quadratic form induced by the
EucHdean Riemannian metric: (dx (u, U»2 + (dy (u, V»)2 +
(dz (u, u»".
Let us consider a two-dimensional sphere 8 2 standardly embedded
in R3 and endowed with an induced Riemannian metric. We have
already obtained an explicit expression for this metric: R2 (da 2 +
sin 2 e dlp2), where a, 'I' are spherical coordinales. Some other curvi-
linear coordinates can also be introduced on a sphere 8 2 Here are
several examples.
We first consider the stereographic projection of a sphere S2 onto
a plane n 2 . To do this, we place; the centre of the sphere of radius R
at the origin 0 and consider the coordinate plane R2 (x, y) through

1.3. Geometry on a Sphere and on a Plane
41
0; the north and south poles on S2 are denoted by Nand S, respec-
tively. Let P be an arbitra.ry poil).t on the sphere, distant from N.
Connect the north pole N with P and extend the segment N P to the
point Q where it intersects the plane R2 (x, y). Associating point Q
with P, we obtain a mapping <jJo: 'S2 __ R2 which is called the ste-
reographic projection of a sphere onto a plane. The construction im-
plies that the mapping <jJo is defined! atall points of the sphere except
at the north pole N. We may assume, conditionally of course, that
the north pole "maps" infinitely distant points of a two-dimensional
plane (see Fig. 1.18). Introducing coordinates both on the sphere
and on the plane, we can write the mapping <jJo in analytic form.
Figure 1.18
N
Q
s
Figure 1.19
Consider, for example, spherical <\Qordinates r, e, cp in R3. These
coordinates induce coordinates on !the sphere S2 and on the plane
R2 (x, y), Indeed, the coordinates (6, <p) arise on the sphere and po-
lar coordinates (r, Ip) arise on RZ' i Since the mapping <Po preserves
the coordinate cp, it is sufficient, fo:r determining 1/'0' to find an ex-
plicit relation between the radius r and angle e. Consider the section
of 8 2 by a plane through points P, 0, and N (Fig. 1,19). Since the:
angle ONT is equal to rr./2 - 6/2, fwe obtain from the right-angled
triangle ONQ: r = OQ = R tan (1/2 - 812) = R cot{, Thus, the
final equations for the coordinate .transformation are: <jJ = <Pi r =
R cot (6/2).
The Jacobi matrix of the coordinate transformation is
_ I , i.e. J=-,
2 sin'  2 sill' _ 2
2 I
and the transformation is regular at aU points except at the nortl-a
pole. Hence, we can introduce on the sphere SZ coordinates induced
by polar coordinates on a Euclidealt plane. The Riemannian metric
1
o

<12 1. Introduction (0 Dlffer.nllal Geometry
of sphere in these coordinates takes the form
ds z =R2(d6 2 +sin z 6d<p2), dr=de,
2. in' 2:
. 2 e R2 2 6 r 2
sm"2= R2+r' cos"2= R'+r"
d ..2- (d 2 + 2 d y. )
.- - (R'+r'/2 r r q .
Note that this Riemannian metric differs from tho Euclidean met-
ric on a plane in polar coordinates (dr 2 + r 2 dlp2) only by the var-
iable factor (U:!:.-:')" Such metrics are called conformal.
Definition 5. The Rie!{1annian metric ell (z) defined in a Euclidean
domain C in curvilinear coordinates ZI, . . ., z" is caiJed conformal
if it can be represented in the form gij (z) = A (z) ell (z), where
').. (z) is a smooth function on Card el; (z) are the components of the
Euclidean metric written in the coordinates ZI, . . " z". In other
words, the metric g II (z) is called conformal if there exists a coordi-
nate system x such that ell (x) = A (x) 1: (dx")2.
k=1
Thus, on a Euclidean plane (referred to-polar coordinates) we can
-consider two Riemannian metrics: dr 2 + r 2 dCpB (the Euclidean
metric) and (R::')' (drt + r B dcp') (the metric of a sphere). Both
-can be assumed to be defined on the sam domain, RZ (x, y). Let us
now turn back to the topic considered above, namely, the equivalence
.of metrics. We may ask: does a regular coordinate transformation
.exist on the plane R2 which sends the metric dr 2 + r 2 dlp2 into the
metric (R':;:')' (dr 2 + r Z d'p2)? Here we may only say. intuitively,
that these two metrics are not equivalent. Let us calculate the cir-
<:umference of the circle ;iZ + y' '= a 2 in the Euclidean and "spher-
ical" metrics. By circle we mean 'a smooth trajectory on a plane R Z
the length of which can be calculljted in different metrics introduced
on R2. We shall calculate this quantity as a funct,ion of the circle
radius. The Euclidean formula is quite familiar: l. = 2na, where a
is the radius (calculated in the Euclidean metric). To find the cir-
cumference in the "spherical" metric, we first derive a relation be-
tween the Euclidean radius a and the radius p in the spherical metric
(see Fig. 1.20)
r 2  R'r' dr= 2R tan-' ( ;t ) ,
o
2..
lc = 2 \' : = ;:::-: = 2nR sin (  ) .
'0

J 3. Geometry on a Sphere and an Q Pla1le
43
It is clear from the figure that p is ellual to the length of the meridian
connecting the north pole with a variable point of the circle. Our
formal calculations in terms of the Riemannian metrics on a plane
can also be performed with the aid of e]ementary geometric rela-
tions. Thus, if we could denne a circle of radius a with centre at a
certain point in terms of quantitieslindependent of a local coordinate
system, we would aMain a formula for the circumference which does
not depend on the local coordinate, system either. Such a definition
docs exist. Let us call the minimal length of a curve connecting
s
Figure 1.20
points P and Q the distance between these points (the minimum is
determined for all smooth curves connecting the points). Then the
circle of radius a with centre ati point P is the set of points
Q such that their distance from P ir equal to a. To apply this defini-
tion to a sphere, we have to provOjthat the distance between points
P and Q on the sphere S2 is equal to the length of the arc of the large
circle through these points. This proposition is easily proved if
any curve connecting P and Q is: approximated by a broken line
composed of finitely many arcs of:large circ]es.
In particular. for p .....0 (i.e. for a circle of a small radius in com-
parison wilh R) we obtain le  1tp, that is, the formula derived
above coincides with the Euclidean expression for the circumfer-
encc. Comparison of "the Euclidean: circumference of radius p, 2np"
I\'ith "the spherical circumference bf the same radius, 21tR sin plR"
shows that these two functions are jessentially diHerent: one is linear
and the other is periodic.
One can easily understand why a convex spherical surface cannot
be deformed into a domain on a Euc1idean plane in such a way that
the length of a curve on the sphhical surface is preserved if one
recalls that it is much more difficult to flatten a spherical segment
than a cylindrical segment.
The above calculation of the cirumrerence on a sphere could also
be carried out directly in spherical coordinates (6. (p) in which the
metric of a sphere is R2 (d8 2 + sin 2 e dq>2). This metric can be defined

44 1. Introduction to DiDertntlal Geometry
on a disk of radius 1t on a Euclidean plane referred to the coordi-
nates 0, !p. Obviously, in these coordinates the circumfere,!-ce of, a
circle of radius e is equal to 21tR sin a (Fig. 1.21); the pomt 0 ]5,
naturally, identified with the north pole and the circumference of
the boundary circle (of radius n) is zero, since ,dO = 0,. sin. (n)  0,
so that the entire circle is transformed into a smgle pomt Identified
with the south pole of the sphere.
Let us now turn to calculating the length of a smooth curve on a
sphere. By way of example, we shall consider the so-called loxo-
drome, the trajectory intersecting each meridian of a sphere at the
S
S
Figure 1.21
same angle a.. (Problem. Find th length of this curve between points
y (a) and 'I' (b).) This curve is well known in navigation theory, for
it ilJ very convenient for choosing flight routes between two fixed
points. The fact is that the angle cr; can easily be measured: it is
equal to the angle between the aircraft velocity vector and the com-
pass direction, Thus, to travel along a pre-determined loxodrome
connecting the starting and terminal points, we may utilize only
this parameter (the real situation is, of course, muoh more compli-
c.ated).
A convenient method of finding an explicit parametric expression
for a loxodrome is to project sterographical1y the sphere (and hence
the loxodrome) onto a plane referred to polar coordinates (r, Ip).
After this operation, the loxodrome is mapped into a plane trajecto-
ry. We have already calculated the metric of a sphere in these coor-
dinates: (R.4:.). (dr' + r 2 dip'). The stereographic projection trans-
forms meridians into rays emerging from the origin on the plane.
We assert that the image of a loxodrome is a curve OJ) the plane,
which intersects all these rays at the same aJ)gle cr; (this curve is
therefore uniquely determined by this angle, up to a rotation). This
assertion follows from a more general fact: a stereographic projection
preserves angles between intersecting curves. The transformation
satisfying this condition is called conformal. To be more exact, we

1,11. G.omelry On a Sphere and an a Plan.
<15
mean the following. Let us considel.' a spherical metric (see above)
induced on a sphere by the ambient Euclidean metriG, and let
two curves intersect at an angle ct (VI' 1".) (the angles are assumed
oriented) (see Fig. 1.22), The number ct (1t V,) has been defined in
Section 1.2. The mapping 'Po (stereographic projection) transforms
Figure 1.22
the curves 1'1 and Y. into curvesiql and 9.; the angle between the
latter, as calculated in the Euclid'ean metric dl' on a plane, is de-
noted by  (ql' Q2)'
Lemma 1, For any intersecting curves 1'1 and Y. the equality
ct (VI' Y.) =  (ii' 92) holds true.
Proof. It is sufficient to compare the explicit formulas for ct and
 and use the expression for the spherical metric (under the mapping
CPo) written in polar coordinates on fR2, We have £ls2(r, cp) = (R,<I:.). x
(dr' + r' dcp.); dl 2 (r, cp) = dr' + ,2 d<p', It is clear that ds' (r, q» =
}." (r) dl 2 (r, cp), whence
ct (I' Y2) = V "" (;I;.+r21')  (9" ql)'
"'2 (; + r') "" (H r2t)
The lemma is proved.
A more general statement is also valid.
Lemma 2. Let giJ (z) and qiJ (z) be two metrics defined in a Euclid-
ean domain C in curvilinear coordinates (Zl, . . ., z"). /1 the iden-
tity g/J = 'J..qij. where A = A (z) is (l smooth function, holds true at
any point (z) E C, the angles bet/veen intersecting curves calculated in
these metrics coincide.

1.6 J. I ntrodu<tion 10 DIUer<ntlal C.omelru
The proof of this lemma is completely analogous to the proof of
Lemma 1.
Let us now return to the loxodrome problem. According to Lem-
ma 1 it is sufficient to fmd its eq(lation on a Euclidean plane. The
('ondi'tion that the angle a is prese\"ed means that «(;, ), (1, 0) > =
cos ct = const, where (;, <p) =  is the tangent vector to the loxo-
drome and (1, ) is the velocity vector of the ray (q> = canst, r = t).
Hence, y. '. = cos Ct = const and therefore r 2 sin 2 a =
r2 + r 2 q>2
. " I ) ' . t cot" where
r2cp2 cos 2 ct, r/r = Ij> cot ct, (n r = Ij> co ct, r = C'8 "
C = const. Putting !p = t, we have ; = C 'cot Ct .e<lCot«, IJ' = 1-
Calculating the length of the are, we obtain l (,\,):. = c' .ecpcot« -I- c W ,
where c", c' = canst. (Exercise, Find the constants c' and e W .)
Problems
1. Demonst.rate that tIfe sum of the angles of a triangle
formed by arcs of large circles on a spherB-S2 exceeds 2n,
2, Express the sum of the angles of a -triangle on 8 2 in terms of
its area (the triangle is formed by arcs of large circles).
3, Demcmstrate that the similarity transformation on a sphere
8 2 is possible if only the similarityj ratio is equal to unity.
4. Prove that for any Riemannian metric there can be found a
coordinate system such that the nlatrix of the Riemannian metric
at a given point is an identity mAtrix,
L4. PSEUDOSPHERE
AND LOBACHEVSKIAN GEOMETRY
Let us consider a pseudo-Euclidean space RZ of index
s. In a Euclidean space R n a sphere S"-t (hypersphere) can be defined
as a set of points located at a distante p from the origin. In a pseudo-
Euclidean space we can also consider a set of points at a distance p
from the origin (here the number p need not necessarily be real, it
may also be imaginary or zero). Th:is set of points is called a pseu-
dosphere of index s and is denoted by 8-', CJearly, 8-1 = 8n-l.
Be10w we shall distinguish between pseudospheres of real, imagi-
nary, or zero radius, A pseudosphete of 2ero radius is described by
, 'I
the second-order equation  (Xi)Z + 2J (Xi)2 = 0, where xl, . . ,
it ;-c::tI+1
. . ., x" are Cartesian coordinates in R n 011 which the pseudo-Euclid-
ean space R' is modeI1ed. Obviously, n pseudosphere of zero radius
coincides with an isotropic (zero) cone.

1.4. Puudo.phor alld Lubachev.ki(Jn Go1Mlr!l
47
Here are some examples. Let 11 = 2 and s = 1. The isotropic
cone consists of two straight lines x' = ::t:X2 (we consider a two-di-
mensional plane R2 referred to Cartesian coordinates Xl, x 2 , and
model pseudo-Euclidean geometry of index one on this plane). This
fone subdivides R2 into two domains: the domain defined by the
Figure f .23
Figure 1,24
inequality I x 2 I > I Xl I in which (S. S), > 0. and the domain
I Z2 I < ) Xl I in which (6. S),,< 0 (Fig. 1.23). Pseudospheres of
real radius are hyperbolas _(X L )2 + (X 2 J2 = a: 2 . where a: is a real
number. and pseudospheres of I imaginary radius are hyperbolas
_(XI)2 + (X 2 )2 = _a 2 (Fig. 1.24).
Let n = 3 and s = 1. The isotropic cone (a pseudosphere or zero.
radius) is an ordinary cone of th second order given by the equation
Figure 1.25
X'
(_(XI)2 + (X 2 )2 + (X')2 = 0 (Xl is the cone ax.is), This cone also
subdivides the entire space into two domains (using the customary
terms, we may say "subdivides into the interior and ex.terior do-
wains") (Fig. 1.25). Pseudosphere of real radius are one-sheet hyperbo-
loids _(XI)2 + (X2)2 + (,%3)2 = +a. 2 , and those of imaginary radius
are two-sheet hyperboloids _(X')2 + (X 2 )2 + (x")2 = _cr. 2 (p2 =

48 I. Introduction to Differential Geometry
_a. 2 ) (see Fig. 1.26). Let us now discuss the metric properties of
.space R:. which is modelled in R; if the Cartesian coot'dinates in R3
are denot.ed by x, y, and z, we have <, S)I = _x 2 + y2 + z!.
Consider a pseudosphere of imaginary radius. It is a two-sheet
hyperboloid given by the equation _a. 2 = -r + y2 + Z2, Since
this hyperboloid is embedded in , we may say that "the geometry
of the space RI induces a certain geometry on a pseudosphere of
Figure 1.26
imaginary radius", This idea canlalso be expressed in terms of in-
definite metric defined in R: "th metric of R: induces a metric on
a pseudosphere", The phrase "geo,metry induced on a hyperboloid"
Gan be given a reasonable meaning even without introducing the
concept of indefinite metric. Consider a pseudosphere -a.! = -x! +
11 + Z2 (for the sake of simplicity,: we shall be dealing only with one
sheet. say, the sheet defined by the inequality x > 0); by "points" of
the geometry induced on this pse1,1dosphere we shall mean ordinary
points of the hyperboloid and by "!!traight lines" of the induced ge-
ometry, all possible lines of intersection of the hyperboloid and the
planes a;r; + by + cz = 0 through the origin (Fig. 1.27). It turns
out that the geometry introduced in this way 011 a pseudosphere can
be studied by the methods of analytic geometry (i.e. without resort-
ing to the concepts of indefinite metric), using a far-reaching analo-
gy with geometry on an ordinary sphere. To this end, it is conve-
nienL to perform a transformation, similar to the stereographic pro-
jection of a sphere onto a plane. The origin, point 0, is taken as the
centre of the pseudosphere S = {-et 2 = _x 2 + y2 + z2}, the
point (-Ct, 0, 0) as the north pole, and the point (Ct. 0, 0) as the
south pole. The plane YOZ through the centre of the pseudosphere is
chosen as the projection plane (note that the restriction of the scalar
product <s. 1])1 to the plane YOZ is of the form 2'Y)2 + S3YJ3, i.e.
the pseudo-Euclidean scalar product <s, 1])1 induces the Euclidean
scalar product on YOZ). Conider now a variable point P on the
right-hand sheet of the hyperboloid and connect it with the north
-+
pole N. The segment PN meets the plane YOZ at a point which is
denoted by I (P) and called the im1age of P under the stereographic

1.4. Pseudo.phere and Lobachev.klan G.omtlrl/
49
projection t: S: - Ri, Exactly in the same way we can define the
stereographic projection of the left-hand sheet of the hyperboloid
onto the same plane YOZ, using the north pole N as a projection
centre. Figure 1.28 shows the section of pseudosphere by a plane
through the axis OX. The image of the right-hand sheet of the hy-
perboloid does not cover the entire plane YOZ. but only the inte-
rior of the disk yS + Z2 < ex 2 of radius C(, The image of the left-hand
! o.
Z by..C1-.....
...... V
.."t _- ......
.jJlL:.= ;'1
y
Figure 1.21
Figure 1.28
sheet of the hyperboloid covers the exterior of the circle yZ + ZZ =
ex 2 . Unlike the case of an ordinary sphere Si, the image of a pseudo-
sphere Sj under stereographic projection covers only part of the
plane YUZ, since the circle yi + Zi = a Z does not belong to the
projection image, The north pole N is mapped (as in the case of 8.
sphere) into an infinitely distant pc/int of the plane YOZ. Let point
P have Cartesian coordinates (x, yl, z) (x > 0) and let (u l , UZ) be
Cartesian coordinates of the point I (P) on the plane YOZ, where the
mapping t is a stereographic projection. The following lemma estab-
lishes an explicit relation between: these coordinates.
Lemma 1. Let P= (x, y, z), t (P) = (u l , u 2 ). Then,
I u 12+a 2 _ 2a: 2 U I _ 2a: 2 u l
x=a;' a 2 _lul" ' Y-a;I_!u\2 z-al_luI2'
where I u 1 2 = (ul)Z+ (UZ)Z; u = (U I ,U 2 ).
PrOGI From Fig. 1.28 it follows' that
1/ ,,+a ....:.. _ ,,+a . I z+a: z = u2!.:::.::..
ur=' u' - a I.e. y=u . 
Since _a;2= _x2+y2+Z2, we hae
_a;2= -x2+ (ul)Zi+ (u2)Z) (,,.,%)2 I
O "'+Ct 2
whence (becauso (x- ex) > ) .:r- ex = ' I u I , i.e.
luI 2 -;-a"
x= (;t, a'-'I u I' .
The lemma is proved

50 J. I "troduction to Dlffer,"t/al a.om.try
Lemma 2, The coordinates (u\ u 2 ) (varying in the open disk (1/.')" +
(U 2 )2 < eG") define a regular coordinate system on the right-hand sheet of
II hyperboloid, that is, the stereog,raphic projection defines II regular
coordinate transformation I (3:, y\ z) _ (u l , u 2 ).
Prool, Thl! coordinatl!s (x, y, ) of point P on the right-hand sheet
of the pseudosphere are related I by -cz" = _x 2 + y' + Z2, so that
the position of point P is uniquely determined by two numbers, sa",
11 and z, i.e. the coordinates of 'the orthogonal projection of Ponto
the plane YOZ, and we may assume that the ri ght-hand shee t of the
pseudosphere is given by the e!luation x = Va 2 + y2 + Z2. Thus,
the stereographic projection I can be meant as the coordinate trans-
formation (y, z) _ (u l , u 2 ), It remains to find the Jacobi matrix of
this transformation and verify that its Jacobian is not zero. Straight-
forward calculation yields
ay 2a. 2 to;O+(u'}2- (U')') Oy 2«'2u'u'
80= (a'-Iu,")' , tJU: =(a'-luj')"
. 2et'2u'u' O. 2a.' (to + (u")' _ (u l )")
8U' = (<'-I U \2)" tlu 2 = (a'-I u '2)'
'.:t..... I II I' 4a'z
J (f)=4C% (a......:j u 11)' = (0;''''''1 u \.). >0,
Thus, thl! transformation Jacobian is positive at all points of the
disk (u l )' - (u')' < CZ2. This completes the proof of the lemma,
Before proceeding further, let us consider once more a sphere S'.
What geometry will arise on a sphere if by "points" we mean ordin-
ar' points of the sphere and by "straight Jines" cross-sections of
the sphere by planes through the origin 0 (i.e. equators)? Let us
dwell on an elementary level and elucidate the geometric axioms
which this set of "points" and "straight Jines" satisfies. Clear! y, one
and only one "straight line" passes through a pair of non-antipo-
dal "points", but infinitely many "straight lines" may pass through
antipodl1l "points". Moreover, it; is not possible to draw a "straight
line" through a "point" not lying on a given "line", without inter-
secting the original "straight line". In other words in this geometry
on a sphere "parallel (i.e. non-intersecting) straight lines" do not
exist.
Some improvements are possible if this geometry could be approx-
imated, as close as possible, to the Euclidean geometry. For eX8m
pie, if in the new geometry by j'points" we mean pairs of antipodal
points (P, -P) on S', the first "«;Irawback" is eliminated, that is, one
and onlj' one "straight line" passes through any two "points" (provid-
ed these "points" do not coincide). This property is analogous to the
corresponding propert of the Euclidean geometry. It is a simple
matter to observe that in the new geometry all classical Euclidean
axioms are satisfied, except the so-called "fifth postulate": namely,

1.,1. P,eudo,phere and Lobachev,klan Geometry
51
given a "point" outside a "straight !ine". no "straight line" parallel
to the original one can be drawn through this point, that is, any two
"straight lines" either intersect I'! one "point" or coincide. Indeed.
any two (non-coinciding) equators on a sphere define one and only
one point (P, -P). The geometry th'(1s constructed (sometimes called
elliptic geometry) is as rich as the Euclidean geometry, though in the
former many customary properties of the Euclidean plane are re-
placed by other properties which are, perhaps. more "exotic" from the
classical point of view formed during the development of science on
the basis of simple physical ideas: about: the surrounding world.
Everyday human experience seems to be more prone to the "Euclid-
ean" concepts.
The above operation of identifying points P and -P, where P
runs a sphere, is equivalent to the factorization of a sphere with
respect to reflection symmetry at point O. Since any pair (P, -P)
defines one and only one straight lihe in a three-dimensional space
(in which the sphere is standardly embedded). we clln associate
with each "straight line" of elliptic geometry (equator) the orthogo-
nal straight line through point O. [l'hus. elliptic geometry can be
modelled on a two-dimensional real jprojective space encountered in
analytic geometry. Below we shall ifrequently deal with this type
of geometry. The ahove operation of associating a "straight line"
with a "point" becomes a duality on 'a projective space. This duality
permits one to deduce directly from any theorem of elliptic gom-
etry a new theorem by substituting "straight lines" for "points"
and vice versa. The statement thus obtained is not. generally.
equivalent to the original one.
Let us now return to pseudo-Euclidean geometry and the geometry
it induces on a pseudosphere of imaginary radius, The stereographic
projection I: +S _ {y + 1. 2 < ct. 2 ) 1= D2 (here "S: 8tands fQ the
right-hand sheet of hyperboloid) transforms points of a hyperboloid
into interior points of a two-dimensional disk D2 of radius ex, What
lire the curves on the circle D2 into hich the "straight lines" of our
geometry on a hyperboloid are tranformed? By "straight lines" we
mean the lines of intersection of the hyperboloid with planes
through the centre 0 ofthe pseudospl1ere (i.e. analoguesofequators on
a sphere).
Lemma 3. Each. line 01 the intersection of +S with tlu plane ax ..;-
by + cz = 0 is mapped by 1 into a circular arc intersecting the circle
y2 + 1.2 = ct. J at right angles (Fig. 1.29).
Proof. We recall that by the angle between intersecting smooth
rurves we mean the angle between their velocity vectors at the
intersection point. Lemma 1 implies that In order to fmd the curve
into which a "straight line" on "S i$ mapped, it is sufftcient to lIub-
stitute into the equation of the plane a.x + by + cz = 0 explicit
expressions for the variables x. y, 1. in terms of u t , u 2 . Suppose

52 1. Introductton to Dig.renttal Geometry
a 0;4 O. Then the equation
I u 1'+'" 2bc.'u' 2cc.'u'
aa.. 0:' _ I u I' + 0:' +--1 u I" + 0:"-1 u I" = 0
is reduced, after simple algebl'aic transformations, to
(U I + bar;. r + (U 1 + . c:_ r =f W+ c 2 -a 2 ),
Le. it defines Q circle of radius  Vb 2 +c 2 -a 2 =r centred at point
a
( - bar¥. , _ c: ) which intersects the circle y2+z2=a;2 at points A
and B at rIght angles (Fig. 1.30). That the angle at the intersection
Figure 1.29
Figure 1.30
points is equal to ",/2 fonows from the obvious relation a. 2 + r 2 =
(b':') c£' . Note that the imap' of a "straight line" on +S under
the mapping f is not the en tire IJircle (ul + b: ) 2 + ( u2 + c: ) 2 = r 2 ,
bllt only its part contained in ,the circle y2+ Z 2<a. 2 .
Thus, the geometry induced on a pseudosphere of imaginary radius
in R: coincides (after an appropriate coordinate transformation)
with the geometry in a circle of radius a. on a Euclidean plane R2,
provided by "points" of this geometry we mean ordinary points of
this circle (except for boundal1Y points), and by "straight lines"
circular arcs intersecting the circle at right angles (in particular, all
diameters of the circle are "straight lines", for these diameters can
be considered as arcs of infinitely large radius). This is the 50-called
Lobachevskian geometry, and its model in a circle of radius a; on a
Euclidean plane is caJled the Poincare model of Lobachevskian ge-
ometry. N.r. Lobachevski devel6ped his geometry in quite another
way, without using pseudo-Eue:lidean spaces, but proceeding from
snch a form of the "fifth axiom'! which a:;sumes the existence of in-
finitely many straight lines parallel to a given one.

1.4. P'eudosph., and LobruhelJ&klan Geometry
53
Using the Poincare model, we can easily verify that all Euclid's
axioms hold true, except for the fifth axiom, Figure 1.31 clearly
shows that infinitely many "straight lines" parallel to (i.e. not in-
tersecting) a given one can be drawn through a point outside the
original "straight line". From the point of view of the parallel a"(iom,
Lobachevskian geometry is just opposite to elliptic geometry. Note
also that if the parameter a: tends to infinity, in any finite domain on
the Poincare model Lobachevskian geometry will "tend" to Euclid-
ean geometry, since the arcs will be straightening, thereby turning
Figure 1.31
into Euclidean straight lines. The bf'undary of the Poincare model, the
circle yl + Zl = a;8, is called the absolute; infinitely distan points
of the Lobachevskian plane are located on' this boundary. While
studying the Lobachevskian plane, we sometimes assume, for sim-
plicity, a: = 1.
Remark. We could also consider geometry that arises on a pseu-
dosphere of real radius in R: (Le, a 'one-sheet hyperboloid). (Exercise.
Prove that this geometry coinides with Lobachevskian geom-
etry.) I
Let us now turn to calculating the Riemannian metric induced on
a pseudosphere of imaginary radius by the ambient indefinite
metric. We shall proceed by analogy with an ordinary sphere, in-
troducing in R: an analogue of spherical coordinates and writing
with their help the equation of a pseudosphere in a convenient form.
in the plane YOZ we shall introduce polar coordinates (r, q», where
'p is the angle with the axis y. M6reover, we shall introduce the pa-
rameter S', an analogue of the corresponding parameter in ordinary
spherical coordinates. Make the I following transformation: y =
Ct sinh S' cos q>, Z = a: sinh S' sin.p. x = a. cosh S'. In this "pseu-
dospherical" coordinate system the l equation of a pseudosphere takes
the form a. = const, This is a direct consequence of the equation of
a pseudosphere of imaginary radi:us.
We now calculate the Riemannian metric on a pseudosphere in
the coordinates u 1 , uS in the Pointare model. Substituting the for-
mulas for the stereographic projection into the expression for the

54
J. InlrDaUf/(OIl to DilJrtnt/al Geomtrll
squared elementary arc in R, we obtain (verify!)
- (dx (u l , UZ»)2+ (d.1I (u ' , U Z »2+ (dz (ul, uZ»Z= 4 «dU I )2+(du')2) a'
(a"- (U ' ) l _(U")Z)" .
Hence, in polltr coordinates (let a = 1) in the Poincare model
this metric js written as ds z =4 dr"+r"d<p" It can be seen from
. (1-r 2 )" '
this formula that the metric is conformal. Le. it differs from the
Euclidean metric by a variable factor A. (r) = 4 (1 - r") ", To write
this metric in pseudospherical coordinates, we go over from (r, 'p)
to new parameters (X. cp) by r = tanh (;.:/2). cp = cpo Straightforward
............
,ji mr!!/M"
S ::::i:ii::::::;::::::
::x " C;OShiJ'::
.: " $lntrQ' ::
. ::iJi!llllmllll/
Figure 1.32
calculation yields (verify!) dlfJ = 1/-"1,,2 + sinh" X dcp'. This form of the
metrk is analogous to that of the metric of a sphere in the coordin-
ates (a. cp). but with ordinary trigonometric functions replaced by
hyperbolic ones.
What is the geometric meaning of the parameter X? In the plane
XOZ the induced pseudo-Euclidean metric is of the form ds' =
-dx' + dz', The-section of pseudosphere by this plane is a hyperbola
written parametrically in pseudospherical coordinates as (we again
put Ct = 1): x = cosh e', z = sinh e' (Fig. 1.32). The length of a
hyperbolic segment from 0 to 9' in the pseudo-Euclidean metric is
e'
l= \ V -sinh 2 9'+cosh 2 9'd9'=9'. Thus, e' coincides with the
o
length of a "meridian" on a pseudosphere from the south pole S
to a variable point p, i.e. this parameter is completely analogous to
the ordinary parameter 9 on a sphere. In particular, we have eluci-
dated the meaning of pseudo-Euclidean coordinates (see Fig. 1.33).
It can be seen that pseudospherical coordinates are completely iden-
tical to spherical coordinates. II 0: ;;6 1, the Jength of a "meridian"
is equal to a9'.

1.4. Pseudospl.ere and Lobachevsklall Ceometrv
55
Consider now the stereographic projection (for the sake of simpli-
city. we again consider only thelplane XOZ because all calculations
hold true if the plane XOZ is rotated about the Axis OX), Since
z= u'''': . we obtain z= 1 2r, (because u 2 =r in the plllneXOZ).
a. -r -r
Furthermore,sincez=sinh9' sincp=sinh EI' (because q> =  ), we have
z
----------:
.. 1. x
Q- ......... :::.
COl)sr- ..:7
Figure 1.33
t'::-r' =sinh EI', whence r=tanh ( ' ), so that X=EI', Thus, we have
b . d h R . . t . 4 dr'+r2 dcp'. th P . . d 1
o talne t e lemanman me nc '(1_r 2 )2 In e omcare mo e .
Note that this metric is positive definite, though the ambient
metric is pseudo-Euclidean and, therefore, indefinite. Hence, cer-
tain surfaces (viz., a pseudospher of imaginary radius) can carry a
Figure 1.34
positive definite metric and be at the same time embedded in a
space with indefinite metric. That the metric induced on a pseudo-
sphere is positive definite, is clearly seen from geometric analysis,
Consider, for simplicity, the section of pseudosphere by the plane
XOZ, and let 6 be the velocity vactor of the hyperbola at point P.
Our task is to verify that the pseudo-Euclidean length of this veloc-
ity vector is real. This directly folJows from Fig, 1.34 which clearly

56 1. 1 nloductlon to Differentia! Geometry
shows that the vector s lies outside the light cone (with the vertex at
point P) and is therefore a space+like vector.
The Riemannian metric derived above is caUed the Lobachevskian
metric (as written in the Poincar model). This metric can thus be
considered as a new one defined on a circle, referred to ordinary po-
lar coordinates in a Euclidean plane. We have already presented two
other examples of the Riemannian metric given in a unit circle: the
Euclidean metric and the spherica.1metric. We have proved that these
metrics are not equivalent, Le us now demonstrate that the Lo-
bachevskian metric is equivalenti to neither of these two metrics.
We shall use the procedure that has already been employed above:
find the circumference of a circle on a Lobachevskian plane and ex-
press it in terms of the radius (calculated in the Lobachevskian met-
ric). We assume, for simplicity, that the cent!;e of the circle is at
point 0 and the Euclidean radius is a. Calculate the radius
in the Lobachevskian metric. By the definition of the length
a
f h i' 2dr 1+a. h X
o II curve we aye x= .' t=rr = In i-a ' I.e. a= tan 2'
o
2"
The circumference l = r ,24 d = 1 41t '\ = 2n sinh X' If 'X. is rather
J I-a -4
o
smaH, we may take approximatel;y l - 211X, that is, we obtain the
formula for the circumference in the Euclidean metric. Since, as in
the case of a two-dimensional sphere, we have expressed the circum-
ference (in the Lobachevskian met,ric) in terms of invariant (relative
to coordinate transformation) quantities, namely, the radius in the
Lobachevskian metric, the above1 formula for the circumference is
also invariant relative to coordihate transformation, so that the
Lobachevskian metric is equivalent to neither of the two metrics
mentioned previously. The table of Fig. f.35 compares the metrics
of a sphere and a pseudosphere.
There ara two more useful formSlof the above metrics, the so-called
"complex forms". Consider a Euclidean plane and introduce on it a
"complex coordinate" z = x + iy; as z we take x - iy. Here we shall
not concentrate on the geometric meaning of these new "coordin-
ates", but considerthemapping(x, y) _(z, z) as a formal coordinate
transformation, The Jacobi matrix of this transformation is
( _:). The Jacobian J = - 2i '* 0, so that the transformation
is regular. Since dz = dx+ i dy and dz= dx- i dy, the Euclidean
metric in thee new coordinates takes the form ds 2 =dxl+dy2=
(dx+ i dy) (dx- i dy) =dzdz. Hence, the metric of a sphere is
d-4 az+dy' -4 dzdi W l lere I 1 2_. .-_ 2 + 2=r2
- (l;-z'+V')' - (1+"1')" z -z z-x y

1.4. P8eudosph<re and Lobachevskian Gtom.try
57
Similarly, we obtain the complex form of the Lobachevskian
. 4 d. d:
metric: ds 2 = _
(1-1"")"
There exists another useful expession for the Lobachevskian met-
ric on the upper half-plane, We choose another copy of a Euclidean
Development
of the sphere
S',S
Sphere
ds'  da' + sln>ect,,'
0"8,,.
0".. < 2
d,2 + ,,,
dS'  4 - (1 .j. -,"'1'"-
o  r < go
(}" ,,< 2
d.' " 4 dr' + r'</",'
(1 -"""fTJ'
0", < 1
(}" " < 2.
Figure 1.35
plane, introduce a new complex coordinate w, and fix the upper half-
plane (i.e. the set of points such that 1m (w) > 0, where w = u +
iv, v = 1m (w)). Consider the mapping R' (w) _ RI (z) defined by
z = ::t , where a, b, e, and d are complex numbers such that
ad - be '* O. This mapping is called homographic. If ad - be = O.
the mapping transfoqp.s the entire plane HI (w) into a point, that is
why we have exc.lded this trivial case.
Our task is to nnd a mapping %= :: sl1ch that would trans-
form the whole upper half-plane into the interior of the unit
circle I z I < 1 on the pla'1e z; in his casa the real axis 1m (w) = 0
is mapped into the boundary circle I z 1=1. Any non-degenerate
homographic mapping (i.e. the mapping for which ad - be =F 0) is
uniquely defined by the image of any three ,points W" WI' and H-'a

58 1. Introduction In DiDer,ellt/.1 Geometry
not on the slime sII'aigI11 line in ihe plane w. Here we shall not
prove this genoral statement. since we do not need it in such Ii
gencral form, but demonstrat thiS property for a particular example
of the mApping of a half-plane onlo a unit circle. Let us find the
mapping z= 'c:;tbd slIch that 0-1, i-..O. 1_i (circular permu-
tation) (see Fig. 1.36). We obtain the following syslem of equations
f b d d 1 b . .+b 0 .i+b S 1 , 11 .
or a, ,c. an : =7' I= c+d ' = Ci+d ' OVIng \IS
"3ystem (verify!), we obtain z = ;i;::' . Thus, we have found a
homographic mapping which transforms the upper half-plane into
Y i
" ::I IIII
w'= u + IV
u
Figure 1,36
a unit circle. (There are many such mappings!) Let \IS prove that
the mapping z = 1 t + iw defines a'regular coordinate transformation.
-tw
Indeed, representing the mapping z= t 1 +!W in the forln -1-
-IW
lW:'1 ( = ;:= ) , we see that in order to prove the regularit'
()f the coordinate transformation it is sufficient to prove that the
mapping z =.!. is regular (because the mapping in question is the
W
following composition: z=.!., translation by a constant vector.
W
rotation, and extension). Wl'iting the mapping z =...!.. in lerms or
w
the real and imaginary parts, we obtain x= . + u. , Y= .- + ". .
u. II II. V'
( V 2 _ U2 - 2uv )
The Jacobi matrix is (up to a constant faclor) 2 2 2'
uv v - u
whence J = (u 2 + ( 2 )2 > 0, The statement is proved.
idw(1-lw)+idw(t+tw) 21d",
Let \IS find dz, We have dz= (t-Iw)" = (1-lw): '
Th d d - 4 dwd;'
us, z z= i1-fwl' : Making an appropriate transformation in

1.4. P,.udo.ph.re Qnrl Lobaclu:...ktan G.omelrll
59
. d z 4d¥d: . 4dwiw du'+dv l
Lhe expressIOn S = (1 I. I")" we obtun ds' = =--.-
- . (wl-Iw)' v
This confirms once more that points of the real axis (the image
of the IIbsolute of the Poincare fflodel) are inf1nilely disLant points
on the Lobachevskian plane (OlocJelled in this easEl 011 the upper
half-plane). Indeed, the length of a st'gmcnt of the axis OIJ from
point i to point 0 (not belonging to the Lobachevskain plane) is
I I
l = \ d: = In v I = -In (0) _ 00.
o 0
What are the curves that we obtain when "straight lines" of a
Lobachevskian plane (in the Poincare model) are mapped onto the
z
s
Figure 1.37
upper half-plane? First, we shall answer this question for elliptic
geometry, Recall that by "straight lines" of this geometry we mean
different equators on a sphere S2, Consider the stereographic pro-
jection of 8 2 onto a plane R2. What are the curves into which the
equators are transformed?
Lemma 4. Stereographic projecaon maps an equator either into a
circle on R2 or into a straight line, an equator being transformed into
a straight line if and only if the eqUator passes through the north pole
of the sphere.
Proof. We first write the stereographic projection in Cartesian
coordinates. Suppose x, y, z are the coordinates of a point P on a
sphere and x', y' are the coordinates of its image (after stereograph-
Ie projection). From Fig. 1.37 we obtain the relations x' = tz '
y' = 1 :"z ' The equation of a circle on a sphere of radius 1 can be

60 1. Introduction to DtDor.ntlal G.ome'y
written in the form {ax + by + ,cz = d, x + y2 + z = 1}, whence
z d-aJ:-l>y z- d-a:r;-I>!I' 1-z c-d
c: c ' - c-aI ' -by' . c-ax'-by'"
(x')2(1-z)Z+(y')2(1_z)2+z2=-1.
Since 1-z> O. we have (X')2+ {y' ) 2= 1+% = 1 +- = 1 +
1-0 1-.
2 (d-IU"'-by') . ( x' )  + (y ' ) 2 +  X . + 21>. Y ' = 1 ...j.. ,  . As
c-d ,I.e. c-d c-d c-d
is known from analytic geometry, this second-order equation defines
either a straight line or a circle Ion a plane, depending on the rela-
tionship between the parameters a, b, c, and d. The lemma is
proved.
A similar st!\,tement holds true for a Lobachevskian plane.
Lemma 5. A non.degenerate homographic mapping z= ;:t:
(i.e. a mapping for which ad - be * 0, where a, b, c, and d are com-
plex numbers) trans/arming a two-dimensional plane into itself maps
straight lines and circles into sbiaight lines and circles; moreover, a
straight line can be transformed tnto a circk and vice versa.
Proof, For c = 0 the statemeJiit is obvious because the mapping
z=  w+ 7 is a translation by the vector bId and multiplication
by the complex number aId ("'{tension with rotation). Let c*O.
Th a (ad - be) h .. h I
en, z= c c (cw+d) , so t at It remains to prove t e emma
for the homogrAphic mapping z=.!... Consider an arbitrary circle
w
on the plane z and write its equation in the form I z - :10 1 2 = e 2 ,
i.e. (z - zo) (z - Zo) = e 2 . By virtue of the transformation z = lIw
we obtain (1 - zoz) (1 - z) = e 2 .Z;, whence v:; (e 2 - zoZo) +
ZOZ + zoz - 1 = O. Clearly, this equation defines (depending on the
choice of the parameters eo and :1 0 ) either s circle or B straight line.
The lemma is proved,
Corollary t. Consider the mapping z = 1 1+ Iw which transforms
-Iw
the upper half-plane into a unit circle, This coordinate transformation
maps "straight lines" of the Poincare model, i.e. circular arcs orthogonal
to the absolute, either into straight lines on the plane W, which are
orthogonal to the real axis u (w = u + iv) or into semi-circles perpen-
dicular to the real axis u (see Fig. 1.38).
Proof. Suppose the unit circle is embedded in the ,;omplex plane z
and the upper half-plane represents the complex variable w. The
mapping z =  + w is then defined everywhere on the plane w.
a-lW
According to Lemma 5, this mapping sends straight lines either into

J .4. Pseurlosph.re alld Lobachevskian Geometry
61
straight lines or into circles (similarly, circles are transformed either
into straight lines or circles). Since the mapping z = +Iw has the
,-fw
'-1
inverse one w = i(-'+O (we recall that ad - bc =F 0), the same
statement holds true for the inverse mapping w = ..:..=!.. Hence
1(.+1) .
"straight lines" of the Poincare model (when completed to straight
lines and circles on the plane z) are transformed either into straight
. Affid" IIIII
.......... ........ . ..................... .........
iitf il'J
Figure 1.38
lines or ciecles on the plane w. It remains to prove that these
"straight lines" must be orthogonal to the real axis (at points of inter-
section with this axis). This is a consequence of the following lemma,
Lemma 6. Any non-degenerate homographic mapping z= ::::::
which transforms the plane w onto the plane z preserves angles between
smooth curves at intersection points.
Proof. It is sufficient to calculate explicitly the Euclidean metric
dz dz in the new coordinates w = u + iv. The fact that the mapping
z = :: I where ad - bc =F 0, defines a regular coordinate system
can be veeified exactly in the same way as the above statement about
the regularity of the mapping z = :  :: . Straightforward calcula-
tion yIelds
dz _ a du> (cw+d)-cdw(aw+b) _ ad-be dw
- (cw+d)' \cw+d)' I
- lad-bel' -
Le, dz dz '= t cw+d I. dwdw. In lcrms of the real coordinates (:I., y)
I ad-be I'
and (u. v) we have dx 2 +dy2= I cw+d I" (du. 2 +dv 2 ). Thus, our
mapping is conformal, i.e. it multiplies the Euclidean metric by
a positive (variable) factor and, as' was peoved eaclier, preserves

62 1. 1 ntrodu.clion 10 VIDermUa! Geometrv
angles between intersecting smooth curves. This proves the lemma
and, therefore, Corollary 1,
Consider the properties of the mapping w= I{-:;') in greater
detail. Point -1 goes over to infinity, hence, the diameter through
poiuts 0 and -1 in the Poincare model is transformed into a
OJ
Figure i.39
straight liue orthogonal to thel real axis u at point 0 (see Fig. 1.39).
The mapping w= I;£+} sends the circle-j z 1=1 into the real
. I d d . f ' e i 'P_1 h - chp_1 .. ,.
aXIs u. n ae . I 1\.=-,--, I. en i\.= j =1\., I.e. t'. IS
i(c''P+1) t(c 'P+i)
real.
Let us consider a Lobachevsian plane modelled on the upper half-
plane. If PI and P2 are arbitrary points of the upper half-plane,
Figure 1,40
they can always be connected by a single "straight line" in the Lo-
bachevskian plane. Figure 1.40 illustrates how to construct such a
"straight line". If the two points lit' on a straight line orthogonal to
the real axis, a "straight line" of the Lobachevskian plane coincides
with the former straight line. Let us fmd an explicit formula for the

1.4. P..u.dosph.re and LobacheV$kian C.omelrll
63
length of a "straight segment" from PI to P (in the Lobachevskian
metric), We first make a simple trasformation of the "straight line"
which does not alter the length of its arcs, that is, translation along
the axis u. Since the Lobachevski'an metric is of the form ds' =
dr' + dy. . .,. '. ..
. . It IS Invariant (preserved), relative to translations (m other
II .
words, it is "translation-invariant" relative to real displacements, i.e.
displacements along the axis x), Suppose PI and P do not lie on a
straight line orthogonal to the real axis. In particular. we may assume
that the centre of the circle represefiting the "straight line" through
PI and p. is at point 0, The paraetric equation of this "straight
line" is r = r o ' where (r, cp) are Ipolar coordinates. Let PI and
p. have the coordinates (ro. CPt) and (ro. CP.). respectively. Then the
length of the arc from PI to P is'
CJI. qJ.
l(P P)= r rodtp = (' =.!.ln ( .!:!:.!. ) I CDS",,'
I'. J 'osinq> J sin<p 2 t-t oa"",
<1>, q>,
( tan )
-In -
tan . r
Suppose now that points PI and Pa lie on a straight line orthogo-
nal to the real axis, Obviously, l (Pl' p.) = In(), where (x, YI)
and (x, Y.) are the coordinates of points PI and p., respectivel'.
Note that if PI and P.lie on a circle' with the centre on the real axis.
the distance between these points remains unchanged under similar-
ity transformation with the centre' at point 0 (we assume that the
centre of the circle coincides with the origin).
Using the formula for the length of an arc derived above. we can
prove that for any triangle on a Lobachevskian plane formed by
segments of "straight lines" the triangle inequality holds true:
namely, I (Pl' P) + I (p., P 3 );;;;;. I (PI' P a ). where PI' p., and
P 3 are the triangle vertices,
Let us prove this inequality. It is qonvenient, as before, to consider
a Lobachevskian plane as a two-sheet hyperboloid +S: defined by the
equation _(Xl)' + (x'). -1.. (x3)' = -1. Recall that "straight Hnes"
on a Lobachevskian plane are intersections of planes through the
origin with a two-sheet hyperboloid. Let e" e. E +Sr be two vectors
with their ends on a Lobachevskian plane. and Jet P be a pJane gen-
erated by these vectors. Then the scalar product in R: induces an
indefinite scalar product on P. Since (e 1 , e.> = -1, the plane P is
Isomorphic to R:, so that we can introduce the linear coordinates
(x, y) on P such that the metric taes the form -x' + y' and the
equation of a "straight line" r is written as -x' + y2 = -1. Hence,
= _ .lln (1 +cos 'P.) (t-C05 'PI)
2 (1 - Cos tp.) (t!+cos 'PI)

1. Introduction to DtDre"titll Geomelry
64
the length of il segment of the curve r between points e l = (Xl' YI)
and ,e 2 F' (XI, YI) can be calculated by formulas derived above. To
this end, we put x = cosh X. y = sinh ')t. whence XI = cosh XI,
Yl = sinh XI' X 2 = cosh X2' Y2 = sinh 1'2' and the length of the
$egment is
1 12 = I )(1-- 'k I = I COSh-I x 2 - COSh-I x I
= \ cosh-! (.:1:,:1: 2 - II1Y2) = cosh-! ( - <;" e a }).
I
Thus, for the three vectors e,. e 2 . e 3 E +S the triangle inequality
becomes In  113 + 1 32 , Substitution of tne lengths in terms of
scalar products of the vectors e.. e2. and e 3 yields:
cosh-I (- (e l . e,»  cosh-I ( - (e 2 , 83» + cosh-. ( - (e l . 63»)'
N ate that all the three scalar products - (el' e 2 ), - (e 2 . e 3 ), and
- (e 1 , 63) are negative on +S. Calculating the function cosh of
both the left-hand and right-hand si des of the i ne quality we obtain
- (el' e 2 )  (e 2 , e 3 ) (e l , e3) + V {e 2 . 63}2 -1 V {e.. e 3 }2 - 1
.or
- (e u e 2 ) - (8 2 , e 3 ) (e l , e3)V (e 2 . e 3 )1-1 -V (e l , e3}2-1.
Instead of the linear inequality. it is sufficient to prove the relation
(e l . e 2 }2+{a2' 8 3 )2(8.. e 3 )2+2{e l . e 2 ){e 2 , 83> {e l . e3)
(e.. e a }2 {e 2 . e 3 )2 + 1-(e l . e3}2 - (eu e3)2
<lr
-1 + 2 {el' e 2 } (e 2 . e 3 }{e l , e 3 ) + (el' e2)2 + (e l . e3)2+ (e 2 . ea)2O.
Since {ell e l } = -t, (a2' e 2 ) = -1 and (ea. e 3 ) = -1. the left.
hand side of the inequality to be proved is the Gram determinant
omposed of scalar products ot the vectors e 1 . e 2 . e3:
( {e l . el) {e 2 , al} (e a , e\} )
det (eu e 2 ) ('e 2 , e 2 ) {e 2 , e 2 } O,
(e l . e3> {e 2 . e a } (e 3 , e 3 >
The sign of the Gram determinant remains unchanged under basis
transformations, so that in the standard basis it is of the form
( -1 0 0 )
det 0 1 0 = -1=;60,
001
Thlls. the triangle inequality is proved.
Using similar considerations for a sphere, we can prove the follow-
i ng statement

1.4. Pseudosph.,e alld Lobachevskiall Geometry
('t:)
Lemma 7. The .\'hortest distance betwem any two points P and Q I
the segment of the ",traight line" connecting the,c points.
Let us consider a Lobachevskian plane and its model on the right.
hand sheet of a two-sheet hyperboloid +S:. The "straight Hne" COII-
necting P and Q is the arc of the se'ction of +S by a plane through
point 0 (the pseudosphere centre). Consider also an arbitrary smooth
curve "i connecting points P and Q on the hyperboloid and approxi-
mate this curve by a broken line composed of "straight" segments.

f \'--.
p C Q
Figure 1.4!
]t is sufficient to prove that the length of any such broken line is not
less than the length of the "straighd line" connecting P and Q. Any
two points on a Lobachevskian plan can be connected by a "straight"
segment. We have already proved his statement in the model on
the upper half-plane when we constr,ucted, using an arbitrary pair of
points on the upper half-plane, the circle passing through these
points and orthogonal to the real axis (at the intersection points). We
now can replace our broken line by' a new broken line such that its
length does not exceed the length or the original one. Consider two
adjacent links of the original broken line and (if these links do not
lie on a single "straight line") consruct a new segment which con-
nects the beginning of the first link with the end of the second link
(Fig. 1.41), Since triangle ABC is formed by three "straight" seg-
ments, it obeys the triangle inequality (see above), so that the
length of the new broken line does riot exceed the length of the orig'
inal one. Continuing this process land bearing in mind that the
original broken line has a finite number of Jinks, we obtain (after a
fmite number of steps) a broken line consisting only of one link, Le
we have connected points P and Q by a "straight" segment. However.
we have proved above that a "straignt" line connecting any two points
i unique. lIenee, the process of "simplifying" the broken line ceased
at the step where the broken line coincided with the "straight" seg-
ment connecting P and Q. The lerrtma is proved.
Thus, we have demonstrated why plane sections of a hyperboloid
act as straight lines in Lobachevsian geometry. It turns out that
preciely these lines realie the shor,test distance between points on a

66 1, J /Ilrodllriion in D 4!!trenlial Geomelrg
Lobache\'sldlill pi ane, We have got accustomed to the fact that on a
Euclidnan plane the molion along II straight line }TIeans the motion
along II tr<1jectory which is tho shortest distance between two points,
Since ill Lobachevskian geom4try (as well as in geometry on a
sphere) the conccpt of "rectiline'ar motion" cannot be formulated in a
simple way in terms of vectorS' wHh constant ('ompouenls (which
uniqut'ly ,reline the dircction of ' motion), it would be qnite nal ural to
call motion along .1 smooth cllr've. which is the shortest distance he-
tween the 1>tnrting and terminal points, "rectilinear motion" We lIa\'(>
found a cl ass of snch curves both on a sphere and on a Lobache /sk ian
plane, III bolh caseS the class comprises plane sections ofLwo.Amell-
sional surfaces modelling the corresponding geometries (of sphere
Md of pseudosphere). In the sequel, \\e shall deal with the elemen-
tary calculus of \'ariations and prove that this class of curve.:; can be
distinguished from all other smooth curves also by the property that
translation (a generali zalion of .ordinary translation in a Euclidel\l1
space) call easily bl! effected along slI('11 "straight Jines" (c)]ed geo-
desi(').
Problems
j. Demonstrate that thl' slim of the angles of II trial1gJi
(III II pseuclosphere composed of "straight" segments is ]E'S 'han 2n
2, Expres the slim of the angles of a triangle on a 1 seudosphere
(formel! by "straight" segments) in lerm of its area.

Chapter 2
General Topology
Topology is a branch of mathematics that studi':!s the
properties of geometric objects which remain unaltered under "de-
formations" or transformations similar to deformations. There are
many concepts in mathematical ianalysis which are analogous in
their properties and investigation D;lethods. For example, convergence
!lnd limit in analysis as (a) limit of a s..quence, (b) various types of
limits of a function of one variable, (c) limit of a function of several
variables, (d) limit of a vector-valued function, (e) convergence of
integral sums. All these concepts qased on some common methods of
investigation we understand intuitively as the closeness of points
of a set. Another important example is various types of continuity
which are very close to convergence. General topology studies the
most fundamental properties of geometric spaces and their trans-
formations related to convergence !lnd continuity.
2.J. DEFINITION AND BASIC PROPERTIES
OF METRIC AND TOPOLOGICAL SPACES
2.1.1, METR(C SPACES
"Closeness" of elements of a set is one of the fundamental
('oncepts in topology. It can be measured mo!'t conveniently liS
distance between the elements.
Consider a set X of arbitrary ell/ments. A metric on X is a numer-
kal non-negative function p (x, y) dependent on a pair of elements
x. Y E X such that the following conditions are satisiied:
(a) p (x, y) = 0 if and only if x = y,
(b) P (:t:, y) = p (y, x),
(c) for any three elements x, y, z E X we always have p (3:, y) 
!J (.1, z) + f1 (z, y).
Properties (a), (b), and (c) are called identity, symmetry, and
triangle axioms, respectively.
The set X equipped with a meric p is called a metric space; the
elements of X are called points. The value of the metric p (x, y) on
a pair of points x, y E X is called the distance between the points x

08 2. GeT/oral 7'opology
and y. A metric space should, in general, be denoled as a pair (X, p).
But if there is no confusion, I we shall fl"eqllently denote a metric
space just by X.
A simple example of a metriC space is the set of real numbers RI.
The metric on this set is deflDed by p(x, y) = I x - y I. The func-
tion p possesses all the properties of metric and transforms RI into
a metric space.
H X is a metric space with metric p and Y c: X is a subset of X,
then Y is also a metric space with the same metric. The metric space
(Y, p) is called a subspace of melric space X. Let Y c: X be an arbit-
rary subset of a metric space. Gonsider the upper bound of all num-
bers p (x, y) with x and y running points of Y. If this upper bound
is finite d = sup {p (x, y)}, the set Y is called bounded and the
x,yEY ;
number d is called the diameter; of Y.
A ball n.eighbourhood of radius. e with cen.tre x E X is the set O. (.r)
of all points y E X sllch that p (x, y) < e.
The distance between two sets Y I , Y.E X islhe numberp(Y 1 , Y.)=
inf p (x, y), where x runs Y j and y runs Y.. If Y 1 and 1'2 have a com-
mon point, then p (Y u Y.) = 0:. There exist, however, disjoint sets
with zero distance between them.
A ballllt!ighbourhcod of a set Y of radius e is the set O. (Y) of all
points x EX sllch that p (x, Y) < e.
The point of contact of a set Yi is any point x such that p (x, Y) =
O. I n particular. every point x of the set Y is the point of contact of
this set. The converse statement' is, in general, incorrect. For exam-
ple, the interval (a, b) c: RI has points of contact a and b which do
not belong to the interval (a, b).
The closure of a set Y is the set of all its points of contact. The
closure of a set Y is denoted by Y.
Thus, YC Y, but the converse statement is, in general, incorrect.
In the above example the closure of the interval (a, b) is the seg-
ment la, bl.
A set Y of a metric space is called closed if its closure Y coincides
with Y, Y = Y.
A point .r is called an interior pOint of a set Y if a ball neighbour-
hood of x is contained in Y, i.e. O. (x) c: Y. The set of all interior
poinls of 1\ set Y is called its interior and is denoted by Int Y. If 1\
set Y COiJlcides with its interior, Int Y, then the set is caUad open.
Theorem 1 . Let X be a metric space. The set Y c: X is closed if and
only if the complement X,-Y is open.
Ploof. Let}' be a closed set, Y = Y, and x E X'- Y. This means
that ;t is not a point of contact of Y, i.e. p (x, Y) =::I e > O. Demon-
strate that O. (.r) c: X,-Y. Indeed, if y E O. (.1:). then p (,j" y) <
10. If y E }', lhen p (x, y);;;' p (x,'Y), i e. p (x, y) ;;;. e, which contra.

2.1. Ba.lc Prop<rtfu of Metric and Topolo(llcal Space. 09
dirt the hypothesis. Hence, thei set X....... Y is open.
Conversely, let X.......y be an open set. Then if :r EX.......}'. lhere
exists a ball neighbourhood Oe (x) contained in X.......y. This mean<;
that p(x, y);;;;' E for y E Y, i.e. p (x. y)  p. Hence. .r is not the
point of contact of y, Thus, if :r E Y, then x E (X....... Y). i.e. :r E }".
This means that Y c: y, i,e. Y = Y. so that the set Y is closed.
Theorem 2. Let X be a metric space. Then the union of any family
<1/ open sets /s an open set and th intersection of finitely many open
.'et' is also an open .et. In dual/orm: the intersection of any family of
clo'ed set. and the union oj finitely many closed sets are cloed sets.
Proof. Let Y" c: X be a family of open sets, We now demonstrale
that the union Y == U Y" is an open set. Suppose :r E y, then
'"
lEY" for a certain index a and since 'l'a is open, the ball neigh-
I>ollrhood Oe (x) is entirely contained in y, Le, 0" (x) c: Y". There-
fore. O. (.1) c: Y" c: U Y" = Y, that is, Y is an open Sl't. Let the
"
index a. assume linitely many vah,Jes. We now verify that the inter-
section Y' = n Y" is an open set. If x E Y', then x E Y" for each
'"
value of ex, Since Y" is an open set, the ball neighbourhood Oe" (;I.)
I ies in Y". Take II = min e" > 0 (here we use the fact that the set
of indices ex is finite). Then O. (;r) c: Oe" (x) c:: Y" for any a., i.e,
O. (:c) c: n Y" = Y. Hence, Y is an open set.
"
The dual statements for closed sets follow from Theorem 1 and
the union, intersection, and complement are related as
n (X''y",)=X'UY", u (X,y,,) = X....... ny".
a. Ct c: Ct
In the previous statements we have not used property (c) of the
metric, which is often called the triangle inequality. In many pro-
positions to follow concerning me(ric spaces only the properties for-
m\llated in Theorems 1 and 2 will J>e used, We can, therefore, extend
the class of spaces for which the concepts of open and closed sets,
interior points, and points of contact are applicable. Before pro-
ceeding further, however, we prove several useful propositions about
metric spaces based on the triangle inequality.
Theorem 3, J.et X be a metric space, Then,
(a) the ball neighbourhood O. (x) is an open set,
(b) the interior Jnt Y of an arbitrary set Y is an open set,
(c) the closure Y of an arbitrary .set Y is a closed set.
Proof, Let y E o. (x), then p (x, y) = 6 < e, Demonslrate that
there exists a number e' > 0 such that 0e' (Y) c: O. (x), Choose e'
such that 6 + e' < e. If z E 0e' (y), then p (z, y) < e' and hence
!' (z, x)  p (x, y) + p (y, z) < 6 "t" e' < e, i.e. z E Or (x). This
means that Oe'(Y) c: O. (x). Thus, O. (x) is an open set.

70 2. General Topt>Zogy
We now prove that Int Y is an open set. Note thajf Y 1 c Y..
then lnt Y J C Int Y.. Hence, if. x E Int Y, we have 0,.. (x) c )-,
and by virtue of statement (a) 9. (x) = Int Oe (x) c Tnt Y. Thus,
br definitioll, lilt Y is open.
Let liS finally verify that Y isl a closed set. By deflllition, x doe
not belong to Y if ancl only if x E lnt (X,,-Y) , i.e. X,,",y =
Int (X,,-Y). According to Theore!n 1, if Tnt (X"-Y) is open, then Y
is closed. This completes the proof of Theorem 3.
2.1.2, TOPOLOGICAL SPACES
A topology is said to be defined on II >Jet X if a fami! y of
subsets or X, called open sets, is ;valid and the following conclLtions
are satislied:
(a) The entire set X and the empty set are open sets.
(b) The union of any family of open sets and the inter.ection of
finitely many open sets are openJ sets.
A set X with a topology defined on it is called a topological space,
and the elements of X are called Us points. The complement to an
open set is caIled a cwsed set. Obvously, in any topological space X
the fonowing duality properties for closed-sets hold true:
(a') The entire set X and the em'pty set are closed sets.
(b') The intersection of any family of closed sets and the union of
finitely many closed sets are closed sets.
Thus, to deline a topology on a set X, it is sufficient to define
a family of sets satisfying conditions (a') and (b'). instead of a
family of open sets, and call the former closed sets.
Theorem 2 of the preceding section implies that a family of open
sets of a metric space X induces a topology on X, that is, transforms
X into a topological space,
Many important properties of a metric space are also valid in
a topological space. Any open st containing x is called a neigh-
bourhood of point X" of a topological space X. Similarly, an open set
containing a subset of Y is calIed la neighbourhood of set y. A point
of contact of a set Y c X is a point.x such that each its neighbourhoocl
has a non-emply intersection with Y. The set of all point of contact
of a set Y is said to be lhe clo.ure of Y and is denoted by y, An inte-
rior point of a set Y is a point :t E Y contained in Y together with
its neighbourhood. The set of all interior points of a set Y is called
the interior of Y and is denoled by I nt Y.
Thc()fem 4. The set Y c X is closed (i.e. it is the complement to an
open set) if and only II Y = Y
Proof. Let Y be a closed set, i.e. X,,-1' is open. Then the set X,,-y
is a neighbourhood for any its point, i.e. the points of X,,-Yare
nol points of contact of the $et Y. Therefore Y <= Y, i.e. Y = Y:

2.1. Basic Prop.rlies o{ Metric and Topolok!cal Space. 71
Conversely, let Y = Y. Then if x E y, x is not a point of contact
of y, Le, some neighbourhood U" of x does not intersect with Y,
that is U",c: X"Y. Thus, the set x"y can be represented as a
union of open sets U '" and is therefore an open set. Theorem 4 is
proved.
Theorem 5, The closure Y of an arbitrary set Y of a topological
space X is a closed set. _
Proof. According,!o Theorem 4 we need to prove that Y = Y
The inclusion Y_ c:: Y is obviou::.. and it remains to verify the in-
verse inclusion Y c:: Y. Let x E Y. This means that any neighbour-
hood U of a point x has a non-empty intersect;on with Y. lor exam-
ple, if a point y E U n Y. then the set U is a neighbourhood of y.
Since y E Y, the set U has a non-empty intersection with the set Y.
Hence, x is a p0!:tt of contact of Y, i.e. x E Y. We have demonstrated
therefore that Y c: Y. Theorem 5 is proved.
Example 1, Consider a set X cC'nsisting of only one element, x.
Then on X we can define a unique topology such that its open sets
are represented by X and the e,"pty set.
Example 2, Consider a set X consisting of two elements x*, y.
In this case several distinct topologies are valid on X. A first topology
is represented by the set of all subsets considered as open sets, i.e.
{0. {x}, {y}, X}. A second topology is given by the following
family of open sets: {0, X}. Finally, a third topology can be in-
troduced with {0, {x}, X} as open. sets. These are all distinct topolo-
gies on the same set X, and dellrle three distinct topological spaces.
Example 3. Let X be an arbitra.ry set. Introduce on X a topology,
assuming any subset of X to be open. Then any single-point subset
is open and therefore any subsetj being the union of its points, is
also open. This topology is calleti discrete.
Let X be a topological space and Y c: X bo its subset. Then on Y
we can also construct a topology by considering any set Y f1 U,
where U is open in X, as an open set. In this case the topological
space Y is called a subspace of th.e topological space X, and the to-
pology in Y is called an induced tiff/ology. If X is a metric space and
Y is its subspace, a topology in Y can be derlfied irrespective of tlte
order 01 operations: restriction or the metric and transition to to-
pology or transition to topology and induction of the metric.
Let Y c:: X be a subset in a topological space X. The set Y is
called dense (eL'erywhere dense) if Y = X.
Theorem 6. If Y 1 , Y 2 c:: X are two open dense sets, their intersection
Y = Y 1 n y is open and dense in X.
Proof. Let a; E X be an arbitrary point and U its neighbourhood.
Since the set Y 1 is dense, we have U n Y 1 '* 0, i.e. we can lind

72 2. Gm.tal TopololY
R poinl y such that y E U n Y I . Since U n )"1 is an open set and )'
is dense, we have U n Y 1 n }' "1= 0, i.e. Un}" "1= 0, whence
it Io1lows that Y is II dense set ii1 X. The lheorem is proved,
2,1.3, CONTINUOUS MAPPINGS
The concept of a top()logical space has been so succeSF--
fully formulated that the definition of a continuous mapping can be
hOrl'owec! word for word from imathematical analysis.
Definition according to Cauchy, The mapping I: X --}' of a
topological space is said to be fontinuous at a point Xo E X j[ for
any neighbourhood , (f (ro)) of the point 1 (.rn) E Y lhere exists
R neighbourhood U (xo) of Xo E X such that I (U (xo» c: V (f (x o )).
The mapping I i called continuous if it is continuolls al each point
of space X.
Theorem 7, The mapping f: X -+ }' i. contlnuouN II and ollly If
olle oj the following equivalent conditions Is lIatlsfied:
(a) the I1IL'er.W! Image 01 aly Opel! ,yet I,y an open set.
(b) the InL'er'e image of any clc.yed set Is a clo.ed set.
Proof, Since we have for inverc images I-I (y,-A) = X'-I-I (A),
l:onditions (a) and (b) are equivalent. Suppose I is a continuous map-
ping and V c: Y is an open set. We now demonstrate that the in-
verse image 1-1 (V) is open, Let x E /-1 (V), then I (x) E V, i.e. V i.i!
" neighbourhood of point! (x). By the definition of 8 continlloll!!
mapping, there e:dsl.$ a neighbolll'llOod U of a point x such that
! (U) c: V. i.e. U c: /-1 (V), antl therefore the set I-I (V) is open.
Conversely. let condition (a) be satisfied. If V:3 I (.:to) is a neigh-
bourhood or point! (xo), then U = /-1 (V)  Xo is a neighbourhood
of ao and / (U) = f (t-I (V» = V c: V. Hence, I is a continuous
mapping, The theorem 15 proved.
Conditions (8) and (b) of Theorem 7 are especially convenient for
verifying the continuity of mappings of topological spaces. Below
we shall need the following useful theorem.
Theorem 8. Let a topological space X be defined as a union of its
two closed .yub'ets X = "\ U F 2 I!d let I: X -+ Y be a mapping 01 X
into a topological space Y. The 7,lapplng f is continuous if and only if
the restrlctlorL.Y ai/to the subsets PI and 1-'2 (denoted as I I [0',: 1"1.... Y
and I I F 2: F 2.... YJ are continuous.
Proof, Let 1 be a continuous mapping. We now demonstrate that,
Sll'. I I FI is a continuous mapping, The inverse images for I I F 1
(an be clliculated hy the rormula (! I f'1)-t (A) = F 1 n f-' (A). Thus,
if A is II closed set, I-I (A) and HI nf- 1 (A) are also closed. COII\'er!'e-
J', let! ! "'I and 111'2 be continuous mappings. We hall prove
that j is continuous Suppose A = A c: }-. Then.
I-I (A) = f- I (A) n X = i-I (A) n (F f U F 2 )
= WI (A) n F.) U (/-1 (A) n F 2 ) = (f I F,t' (A) U (f I F 2 t l (A).

2,1. Basic PropuU. / Metric and Topological Spacu 73
Si nee Ihe set (f I f\) _I (A) is closed in the snbspace FI and Jo\ = FI'
\\e have (/ I Ft)-I (A) closed in X. Similarl', the set (/ I F 2 )-1 (A)
i also closed in X, whence it follows that /-1 (A) is closed in X.
Theorem 8 is proved.
Let liS consider a continuous IIPping I: X -- Y of a topological
space X into a topological space Y. If the mapping I is one-to-on&
.illd the inverse mapping 1-1 is also ccmtinnolls, I is called A homeo-
1I10/'phi.m, and the topological spaces X And Yare called homeo-
morphic. A homeomorphism delines II one-to-one correspondence not
only between points of topological spaces X and y, but also between
I he topologies themselves, i.e. between families of open and closed
ets. A property of a topological space is saie! to be topologically in-
1I111'iant if this property is the sa1De for homeomorphic topological
spaces. Thus, while studying topologically invariant properties, we
lIeeel not distinguish between homeomorphic spaces,
Example 4. The c.ontinuity of the mapping I: X -.. Y is II topo-
logically invariant property. Indeed. if cp: X' -.. X and 1/': Y -+ Y'
Me homeomorphisms, the composHion X'  Y' is also a continuous
mapping. In general. the composition of two continuous mappings
i. 11 continuous mapping.
Example 5. A continuous flilictlon, Le, a continuous mapping of
1\ topological space X into the spaqe R I of real numbers, is an impor-
I/Int particular case of continuous mappings. The condition that a
function I is continuous can be formulated as follows: for any point
.r" E X and any e > 0 there exists a neighbourhood U of Xo such
Ihat for y E U the inequality II (xo) - I (y) J < & is satisfied. Th&
uniform Jimit of a sequence of cohtinuous functions can be defined
ror functions on a topological space l . J list Ilke in the case of a function
or one real variable, the followig statement Is valid:
111 = lim In and the sequence {In} 01 continucU8 lunctions on a
n....
topological space X conl)erge. unilormly to I, the lunction I is continuous.
Indeed, let:l: o E X and & > O. We can choose a number n such that
II (x) - In (x) I < &/3 for any x E X (uniform convergence). Fur-
Iher, since the In are continuous, there exists a neighbourhood U
of 3. 0 such that Iln (xo) - In (x) I < &i3 for x E U. Then,
II (xo)-I (x) 1< 1/ (xo)-/n (xo) I + IIn (xo)-/n (x) I + IIr. (x)-
('C) 1< 3,e/3 = e.
Example 6, Let I: X-+- Y be la continuous mapping of metric
i\paces and let Pl' P2 be metrics on X and Y, respectively. Then the
('ondition that f is continuous can be formulated as follows: for any
.1'0 E X and e > a there exists (L 6 > 0 .uch that the inequality
1'1 (x, xo) < 5 implies P2 (f (x), I (xo» < e.
It \\'ould aJso be useful to extend tho concept of the convergence
of a numerical sequence to R metric space. V{e say that the sequence
of poiJlts {XII} converges to a point xo, lim X n = Xo. provided
n"oo

74 2. General TopoloEY
Hm p (xo, x,,) = O. Many properties of spaces and mappings can be
71_010 
formulated in terms of convergent sequences of a metric space. For
example, a set Y c X is closed if for any convergent sequence
(xn} c Y the limit Xo = lim XII also belongs to Y. Further, the
condition that the mapping rX  Y of metric spaces is continuous
can be formulated following Heine: the mapping f ts continuous at
a point Xo if the equality Xo = Jim XII implies Hm f (xn) = f (:Co).
n-co rt_oo
Example 7, Consider two topological spaces X and Y and fonn
.a new topological space X X y, X X Y is the set of an pairs (x. y),
x E X, y E Y, and is called the Cartesian product of sets X and 1".
Let us now define a topology in X X Y. The set U c X X Y is
called open if U can be represented as the union U = U (V" X W",).
a
where V", eX, W", c Yare open sets. It is a trivial matter to
verify the properties of an open set. The set X X Y with the to-
pology just delined is called he Cartesian product of topological
spaces X and Y (these spaces are called factors of Cartesian product
X X Y). The Cartesian produc possesses the following properties:
(a) the spaces X X Y and Y X X are homeomorphic, and (b) the
spaces (X X Y) X Z and X X (Y X Z)-are homeomorphic. In
(a) we should choose the homeomorphism in the form <p: X X y-
y X X, cp (x. y) = (y, :c). If U = U (Va X Wit) is an open set
"
of space X X y, then <p (U) = U (W a X Va) is an open set of
"
Y xX. In (b) we should consider the homeomorphism in the form
1jJ: (X X Y) X Z -+ X X (Y X Z). Ip «.r, y,) z) = (:c, (Y. ,,»). The
projection of the Cartesian product X X Y onto one of the factors,
say X, f: X X Y -+ X, f (x. y) = x, is a continuous mapping.
Indeed, the inverse image of the open set U c X is f- l (U) = U X Y,
i.e. f- l (U) is an open set.
Example 8. Let X and Y be metric spaces, then the Cartesian prod-
uct X X Y admits a metric consistent with the topology of this
product. Let PI and P. be the mtrics of X and Y. respectivel', The
metric P in the Cartesian prodllct X X Y is defined as
P «Xl' Yl)' (X 2 ' Y.)) = ma>: {PI (Xl' X2), P2 (YI' y.)}. (1)
Let us verify properties (a). (b). and (c) of Section 1.1 for the metric.
If P «XI' Yl)' (x t , Y.» = 0. then PI (XI' x.) = P2 (YI' Y2) = 0, i.e.
Xl = :c 2 , Yl = Yt, whence (Xl' Yl) = (X.. Yl) Conversely, if
(Xl' i/1) = (X2' Y2)' then X1 =x., Yl = Y. and hence PI (XI' Xl) =
P. (Yl' Y.) = 0, that is, P «Xl' YI)' (X.. Y2» = max (O, 0) = 0.
Property (a) is proved. Further,
P «Xl' YI). (Xi' Y2» = max {PI (Xl' X t ), P. (YI' Y.)}
= max {P1 (X 2 . Xl)' PI (Y2' Yl)}
= P «X 2 , y.), (XI' YI»'

2,1, Basic Proprtitl oj Metric alld Topological Spaces 75
MId properly (D) follows. Fjnall'. let liS verify the triang1e ine-
quality (c):
II «oT,. y,), (oT" Y» = max {PI (x., It), p. (Yl. Y)}
..;; ma\: {PI (XI> .1"3) -I- PI (:r3' x.). r (Y.. y,,)
+ flz (YJ' Yt)}
..;; ma:.: {Pt (XI' x.,). fl. (YI' Y3)}
+ max {PI (X3' X,) + fit (Y3' Yt)}
= p «x" y,), (X3' Yo»
+ P «x.. 113)' (x., Y2»'
Thus, formula (1) defines a metrij: in the Cartesian product X x Y.
We now demonstrate that the topology defined by r coincides with
the topology of Cartesian product. Consider II bal neighbourhood
of radius £ with the centre at point (,r o ' Yo)
O. (.ro. Yo) = {(x. y): P «x. y), (.ro; Yo» < e}
= {(x, y); P. (x, xo) < e, P2 (y, Yo) < e} = O(xo) X O. (Yo)'
lienee, any open set in the sense df metric p is open in the topology
of a Certasian product and vice versa.
There are some other ways of defining a metric in the Cartesian
product. A rather customary method is to establish an analogy be-
lween the factors X and Y of the' Cartesian product and coordinate
axes on a plane. The metric is given by
p' «XI' Yf)' (XI' Y.» = Yp.(xp x.)" +P2 (YI' Y2)2, (2)
Verification of properties (a), (D); And (c) is left to the reader. In
order to prove that the topology dfind by metric (2) coincides wHh
that defined by metric (1), it s\iff1ces to fmd constants C, > 0 and
C. > 0 such that
e,P «XI'
YI)' (x., Yt»p' «XI' y,). (Xl' Y2»
";;C 2 P«X I Yt), x.. Y2»'
(:I)
Indeed. if Oe is a ball neighbourhood in the melric P and O i
a ball neighbourhood in the melric p', then it is sufficient that the
inclusion
O. «x o , Yo» c: O. «xo. Yo»C: Oe" «(xo. Yo»'
(4)
be valid alany point, where e, e' are chosen for an arbitrar' £," >0.
From ineqllality (3) it follows that inclusion (4) is satisfied for e' =
C 2 f:" and £. = e'IC I , and it remains to prove inequalities (3).

ill 2, a.nOTal Topology
\Ve have
p'«x l , YI)' (x, Yz»=Yrdxs. XZ)2+P'(YI' Y2)'
Y2maJC {pdxl' x 2 ), P.(yl' Y2)}
= Y2p«xt. YI)' (x,. y.»,
P «XII YI)' (x., Yz»
= m aJC {PI (XI' x.), P. (YII .I/.)}
Y PI (XI. x.)' +P. (UI' y.)2= p' (XI' YI). (:1:.. y.».
Hence, inequalities (3) hold foi C 1 = 1 and C. = y2: We ha,'e
thus demonstrated that in the Cartesian product X X Y two metrics.
(1) and (2), Induce the same topology.
Problems
1. Give an example o{ a metric on a fmite set which can-
not be induced by any embedding of the set in a Euclidean space.
2. Demonstrate that a finite set on a straight line is closed.
3. Demonstrate that p (x, Y) = p (X, 7),
4, Prove that the function f (x) = p (x, Y) Is continuous for any
subset of y,
5. Demonstrate that any metrc on a finite set induces on this set
a discrete topology,
6. Prove that an interval, a h,alf-interval, and a segment on the
real axis are not pairwise homeomorphic,
7. Prove that the metric P (x. U) is a continuous function on the
Cartesian square X X X of a metric space X.
8. Prove that a set X can be represented as the difference of t\\"o
closed sets if and only if (X,-X) is a closed Set.
9, Demonstrate that under a continuous mapping the image of
a dense subset is'dense in the image.
2.2, CONNECTEDNESS. SEPARATION AXIOMS
Among all topological spaces we may distinguish narrowt'r
classes characterized by certain topological invariants. For instance,
in the preceding section (Example 3) we have considered the class
of discrete topological spaces The following property holds true for
these spaces: any mapping f of a discrete space X into a topological
space Y is continuous, In this section we shall consider some olher,
more important classes of topological spaces.

2.2, Conn.cted""... S,p(Jr(Jtton Azioms
77
2.2.1. CONNECTEDNESS
A topological space X is said to be disconnected if it con-
lains a subset Y c: X, both open and closed, which is distinct from X
lIud from the empty set )25. For example, any discrete topoiogical
:!pace consisting of more than one point is disconnected, since any
ubsel of this space is both open apd closed, It a space X is discon-
nected, it can be decomposed intp a union of two open, disjoint,
non-emply subsels; X = Y U (X).,. Y).
A topological space X is called a connected topological space if X
cannot be decomposed into II union of two open, disjoint, non-empty
lIbsets.
Example 1, Consider a .egment [a, b] of the numerical axis and
demonstrate that [a, b) is a connected set. Suppose the segment Ca, b)
is represented as a union of two non-empty, open-closed disjoint
sets, [a. bl = A U B. Without lqss of generality. we can assume
that a E A. Then since A is open, there can be found an e > 0
:4\1ch that the half-interval [a, a + e) c: A. Let eo = sup {e} with
I> rllnning the numbers such that fa, a + e) c: A. Then for any
e < EO the segment la, a + e) c: A, Le, a + E E A for any e < eo.
Hence, the closeness of A implies that a + Eo EA. The only possi-
bilily is a + eo = b, otherwise eo is nol equal to sup {e}. ThllS,
fa, b) = A, Le. B = )25 contrary to the proposition. We can show
in a similar fashion that an open. interval (a, b) of the numerical
axis is a connected set.
We now prove several statements which may be useful in verifying
the connectedness of topological spaces.
Theorem 1. Let X = U X"" each X", be connected, and the inter-
et
section n X", be non-empty. Then the space X is connected.
,.
Proof. Assume the converse, i.e. X = A U B, A n B = 0. and
the sets A and B are open and non-empty, Then X", = (X", n A) U
(X,. n B). Since X", is connected, then either X,. n A =)25 or
X", n B = 0, Le, each set X is entirely contained either in A
or in B. Non-emptiness of A and B implies that there exist points
a E A and bE B, Let a E X",. then X",. c: Ai let bE X,." then
X,., c: B. Hence, X"" n X,.. = 0, which contradicts the con-
dition. Theorem 1 is proved.
Theorem 2. If in a topologtcal space X there exists a connected set C xv
containing any two points x, y E X. Then X is a connected topological
space.
Proof, Assume the converse, i.e. X = A U B, A n B = 0, and
the sets A and B are open and non-empty. Then we can find points
a E A and b E B. Consider the decomposition Cab = (Cab n A) U
(Cab n B) The point a belongs to Cab n A and the point b to Cab n
11, Hence, the sets Cab n A and Cab n B are non-empty, open in

i8 2. General Topology
Cab. and do not intersect. We have come to contradiction.
Theorem 3, The image of a connected space 1$ connected under a con.
tlnuous mapping.
Proof. Let I: X -- Y = f (X) be a continuous mapping. The
condition Y = f (X) means that Ian inverse image of a non-empty set
is also non-empty. Therefore. if Y is a disconnected space, Y =
A U B, A n B -: 0, and A and B are open and non-empty. Hence,
X = f-1 (Y) = f-1 (A) U 1- 1 (B); so the sets ,- 1 (A) and /-1 (B)
are open, non-empty, and do not. intersect. Theorem 3 is proved.
Example 2. It follows from Theorem 3 that any continuous real
function y = 1 (x) defined on a segment (a. bJ of the real axis takes
y
Figure 2. t
intermediate values. If the function 1 does not assume an intermedi-
ate value Yo' the image f (la, b»' could be decomposed into the union
of two open, non-empty sets: the values which are less than Yo and
those which are larger than Yo. but this contradicts Theorem 3.
A particular case of using Theorems 2 and 3 is the concept of path-
wise connectedness. A topological space X is called pathwise connected
if any two of its points x, y ca'1 be joined by a curvilinear segment,
i.e. there exists  continuous mapping I: 10, 1] __ X of the segment
10, 1] of the real axis into X sUFh that 1 (0) = x and 1 (1) = y.
Theorem 4. A pathwise conneated topological space X is connected.
Proof. Indeed, by Theorem 3 the image! (10, 11) is connected and
contains the points x and y. Thn by Theorem 2 the space X is con-
nected.
Example 3, Consider a connected topological space which is nol
pathwise connected. Let X be the closure of the graph of the func-
tion II = f (x) = sin (1/x) liS a et in a two-dimensional Euclidean
space R (Fig. 2,1). The metric in R2 is assumed to be classical, in the
sense of the length of a segment connecting two points in R. Then
the set X is a union of the graph of y = sin (lIx) and the vertical
segment r 3 = {(x. y): x = 0, -1  y  +1}. The graph of the
function f is decomposed into two subsets. each being homeomorphic

2.2, Connecledlli:... Separalton Az!om.
79
10 the interval: f l = {(x, y): 0 < x < 00, y = f (x)}, 1'z =
{(, y): -00 < x < 0, y = f (x». Thus, if X = A U B, A n B =
0, and A and B are open, non-empty sets, then each of the subsets
f l , r 2' and r 3 lies entirely eithet in A or in B. Let r. c:: 8, It is
<\ simple matter to verify that any neighbourhood of r 3 intersects
hoth r l and f z ' i.e. f l c: 8 and r c:: B. Hence, A = 0, which con-
IrAdicts the hypothesis. So X is a connected space.
We now demonstrate that X Is not pathwise connected. Consider
t\\"o points P = (-tin, 0) and 0 = (1/n, 0) in X. Suppose there
exists a continuous mapping f: !O, 1) -+ X, f (0) = P, f (1) = Q.
The mapping f is defined by two continuous numerical functions:
I (t) = «x (t), y (t», y (t) = sin (1/x (t» for x (t) :;6 O. Since x (0) =
-1/1[, the exact lower bound to stricly exceeds zero for t satisfying
J- (t) = O. Thus, the conditions x (t) < 0, y (t) = sin (1Ix (t» hold
on the interval to, to)' Since x (t) is a continuous function and there
exists a sequence t  to, t" -+ to, where x (t,,) = 0, we have x (to) =
lim x (t) = 0, In this case, however, the function sin (11.t: (t»
t.....'o
does not have a limit for t -+ to - 0, and hence the function y (t)
is not continuous. Thus, X is noU a pathwise connected space.
Example . Let us consider two topological spaces X and Y and
construct a new topological spae Z = X U Y consisting of the
points of X and Y (we assume here 'that X and Y do not have common
points). By an open set of Z we shall mean any set of the form W =
U U V, where U c:: X, V c: Yare open sets in X and Y. The space Z
is called a topological sum of the spJces X and Y. The space Z is really
disconnected, for its subspaces X find Yare open, disjoint
subsets, Using two spaces X and Y, we can construct another topo-
logical space called a connected sum or a wedge of topological spaces.
To this end, we fix two points, on in each space: Xo E X and Yo E Y.
Consider the union of X and Y and: identify in it the points:&o and Yo.
The space thus obtained is denoted X V Y. By open sets in X V Y
we shall mean sets U such that the intersections X n U and Y n u
are open in X and Y, respectively. If the spaces X and Yare con-
nected, their connected sum X V Y is, by Theorem 1, a connected
topological space. If X and }' are pathwise connected spaces, their
tonnected slim X V Y is a pathwise connected space.
2,2,2, SEPARATION AXIOMS
A topological space X, is called a Hausdorff .pae if for
IIY points x, y EX, x =P y there exist disjoint neighbourhoods
(] (x) and U (y), Le. U (x) n U (y) = 0. :'<lot every topologicaJ
I'pace is a Hausdorff one, In the preceding section (Example 2) we
have considered the space X conS1sting of two points x and y the
open sets 'of which fire {0, X, {x}}. The points x, yare then dis-
tincl, and no intersecting neighborhoods of these points exist. In

80
2. Gellral Topology
a Hausdodf space any point x E ¥ is a closed set Indeed, if y :;06 x,
we can Hnd a neighbourhood U (y) of point y which does not con-
tain x. Then the set X,,- {x} = I U U (y) is open and its com ple-
v.*"'
ment consisting of one point x is closed.
A discrete topological space X Is a Hau8dorff space. Indeed. allY
point x E X is an open set, and hence for x :;06 Y the neighbourhoods
{x} and {y} are disjoint.
Theorem 5. Let X and Y be Hausdorff topological spaces. Thm the
Cartesian product, the topological sum, and the connected sum oj X
and }' are HausdnrD topological spaces.
Proof. Consider two points (XI. y,) E X x Y and (Xl' Y2) E
X X Y. If these points are distinct. then either X, * X 2 or YI * Yl'
In the first case. since X is a Hausdorff space, there e:\.ist t\\'o dis-
joint neighbourhoods U (XI) and U (Xt). Then the neighbourhoods
U (x 1 )o X Y and U (x 2 ) X Y of the points (X1' Yl) and (x 2 ' Y2)
are disjoint either. In the second case, where Y is a HausdorlT
space, there exist disjoint neighbourhoods V (Yt) and J1 (y.).
Therefore, in this Case the neighhourhoods X X V (Y,) and X X
V (Y2) of the points (Xl' Yl)' (X2' Y2) arc disjoint.
Let us consider a pair of distinct poil\..ts x, y E X U Y. If these
points lie in the same space, say X, the condition that X is a Haus-
iorH space implies that there exist disjoint neighbourhoods U (x)
and U (y) which are at the same ime open sets in the union X U Y.
If the points belong to distinct ;spaces, say :c E X and Y E y, the
spaces X and Y themselves are disjoint neighbourhoods of these
points.
Consider a wedge X V Y and a pair of distinct points x, Y in it.
First, let the two points belong to X. Then one of them, say x, does
not coincide with a fixed point Xo = Yo, so that disjoint neighbour-
hoods U (x) and U (y) exist in x.. After eliminating Xo from U (x),
we may considel that U (x) does not contain this point. The set
U (x) is therefore open in X V Y. The set U (y) is also open in
X V Y, provided it does not contain the point xo' If Xo E U (y),
the set U (y) U Y is open in X V Y and does not intersect with
U (x). This completes the proof of Theorem 5.
Definition. A topological space X is called normal if it is a Haus-
dorff space and for any two disjoint closed sets Ft and F 2 there exist
disjoint neighbourhoods U l  Fl and U B  F 2 .
Normal spaces are the most common occurrence among topo-
logical spaces. This class is fairly wide and includes all metric spaces.
Theorem 6. Any metric space is normal.
Proof. Let X be a metric space with metric p and let FI and
F 2 be disjoint closed sets. Suppose xEF" e(x)=+p(x. F B ) and
put U 1 = U O'(x)(x). Similarly, U.= U O.'(V)(y),wheree'(y)=
. 

2.2. Connectedne.s. SepQrQtlon Azloms
81
 p (y, F,).We have obtained neighbourhoods of tho sets F, and
F2' We now demonstrate that U t and U 2 are dhjoinL Asllml'
the convers', i.e. there exists a point zEU , nU 2 , Then for certain
points zEF , and YEF we have zEO. (:x:) (x), zEO..(v)(y), i.e.
p (x, z) <  p (x, F 2 ), P (y, z) < {- (y. F f ), In pal'ticui,:Jr, p (x, z) <
; p (x, y), p (y, z) <  p (y, x). Adding tho Jast two inequalities,
we obtain (J (x, z) +1> (y, z) <  p (x, y), in contradiction with
thl' triangle inequality. TJleorcm 6 is proved,
The system of open sets {U oo} of a topological space X is caJIed an
open covering if X = U U oo. Covering is a convenient concept for
a
studyil1g topological spaces. For xample, i£ a continuous functiun
fa. is defined in every Ua. and Ue [,lor-lions foo and f coinid(' on
every intersection U a n U, there exists on X a continnOII. "II'"
tion I which coincides with fa. ill every open set U a.'
Given two open coverings {U oo} and {V} of a topologi<:al space X.
We say that the covering {V} refines the covering {Ua.} or is finer
than {U oo} if each set V  is contained in the set U a., 0; = 0; (a).
Theorem 7. Let X be a normal topologtcal space and {Ua.}1
be a finite open covering. Then there exists a finer covering {V "'}:=I
such that Va. c: Ua..
Proof. The theorem impJies the existence of a finer covering
whose clements are numbered by the same index 0; and the sets
N
are also included with the index . (:Qnsider a clospd set X,,- U U a
a.2
contained in Uj' Since the space X is normal, there exists 11. neigh-
N _
bourhood VI such that X,,- U Ua. c: V, c: VI c: U " Then the sys-
"'=2
tern of sets {V" U 2 , .,,, UN} covers X, and we can find a set
V 2 c:V 2 where the system {V 1 ,V 2 ,U S ' ".,U N } also \,;oversthe
space X. Replacing U A successivoh' by VA c: VA c.:: Uk' we arrive,
after N steps, at the covering in guestion {VI' ' . ., V N}'
Problems
t, Prove that if a Onite (countable) number of points
is eliminated from R n (n  2), the remaining space is connected,
2, Prove that if linitely many $ubspaces of dimension less than
(n -1) IS eliminated from R n , the remaining space is connected.
3. Find the maximal number of connectedness components into
which R2 is subdivided by flOitely many straight lines, What is
the minimal number of the connectedness components?

82 2. General Topology
4, Let f: X - X be II continuous mapping of i\ :Iallsrlorrr space.
Prove Lllllt tile Sl't of fb.:cci poi nls (i.e. f (,I) .... .1') i ('/os("({.
5. Prove tllilt the imag(' of a normal SPII('(' mapped continuol1sly
into 1\ IIR\lsdorff space is normal.
6. Prove Ulat X is a ltausrlorff space if and old' if the diagonal
 ...., {(.r. y): J' .,-: y} <= X X X is dosed in X 'I( X,
7, Prove lhat the mapping f: 'x ->- )' into a Hallsclorfl space }"
iJ; continuous if nnd only if the graph r, ...., {(x, f (.1"»: x E X} <=
X X }' is dosed in X X Y.
2.3. COMPACT SPACES
Compactness is one of the most important properties of
/I topological space. III particular. this property Is rundameillal in
the study of real numbers alld corlt.inuolls fllllctions.
2.3.1. DEFINITION
A Hallsdorff topologiC,al space X is called compact if
,my open covering {U",} has a finite par"- {U..h}ii"=l !hal covers X.
Thc oilly condition which can bl' IJnposed is Ihat a tillite fiut'r cover-
ing ShOl1lc1 c:'\ist.
Remark. Such 1\ spacc was !irst, caUed bicompact, whilt' the term
compact space was used for a metric IJicompact space. RCI't'nt)y.
however, a bicompact space has been called simply II compat'l sparl.>.
In what follows we shall use this, more rerl'nt, lermino]ogy.
Example 1, A rCII) segment la, III] is II rompact spar. lildeerl. if
{U",} is an open covering of fa, h], we CRn IIssume, wil hOllt los of
generality, that each element of U'" is RIl interval (c"" d..) (except
for two half-intervals (la, a') and (b', bl), Consider the set P of all
\\lImbers x E (a, bl such that th segment [a, xl is covered with
fmitely many elements of U e' THen, if ,J; E P, Y < I, the point y
also belongs to P. Since the point a belongs to the element of the
covering [a, a'), a' > a, the set P consists of more than one point a.
Let ;l'o be the eXR('tnpper bound of P, IO > a. If Xo < b, thc point xo
lies in the interval U a. = (c..., da.), i.e, c.. < Xo < da. . Suppose
y, z are s\lch that Ca. <: y <.'to -< z < da. .' Then yEP and the
segment la, yJ is covered with finitely may sets U. so that thE'
segment la, z! is also covered with a finite number of sets Up. Hencl.>,
xo is not the upper bound of P. iThus, I D = b. and b' < Io = b.
If b' < y < x o , then yEP and again the segment la, yl is covered
with firtite)y many sets Up, so that the whole segment la. bl is also
covered with fmitely many sets Up.

3.8, CDmpaet Spac48
83
2.3.2, THE PROPERTIES OF COMPACT SPACES
Theorem f. A compact topologtcal space is normal,
Proof, Let X be a compact spac,e and F be a closed set in X, Dem-
onstrate that if a point x does not belong to F, there exist disjoint
neighbourhoods U (x) and U (F).
Lemma 1, A closed subset F in a compact space X is a compact
space.
Indeed, if {V..} is an open covering of the set F, each V" =
u n F, where U" are open sets in X. Then the family of open sets
{U ", X"'-F1 covers the space X and hence there exists a finite family
{U",., X",-P} covering X, Thus, {V..,.} covers F.
Let us proceed with proving tt'huorpm 1. Sinle oX E F, for any
point yEF there exist disjoint neighbourhoods U3x and Vu3Y
betause X is a Hausdorff space. Obviollsly, the fnmil' {Vu} covers
the set F. According to Lpmma '1, we can Hnd finitely many sets
N
{V IIh}"'1 covering F, so that the neighbourhood h1 U llh of point x
N
does not intersect with the union U VIII< -:=J F.
h=1
Let us consider two disjoint closed sels F, Rnd 1<'2 in a compact
spate X, Then for any point xEF, we CRn find disjoint
neighbourhoobs UIJt and V., of t1e point x and of the set Fa, re-
spectively. The fllmUy {U,,} covets the compact set Fs; hence, there
N
exists a finite covering {U "h}:=I' Thus, the union U U"h contains
h='
N
Ft and does not intersect with the intersection n V"I< containing
1<=1
F 2 . Thl'orem 1 is proved.
Theorem 2. Let F c: X be a cO"Jpact subspace in a Haw;dorff topo-
logical space X. Then F is a closed set in X.
Proof. Let oX be an arbitrarY, point of X not belonging to F.
Since X is a Hausdorff space, there exist for points YE F
disjoint neighbourhoods UIIy and Vu3x. Then the family
{Uu} covers the set F and, as this set is compact, we can find a
N
finite family {U llh }:.., that covers F, Thus, the union h1 UUk con-
N
lains F and does not intersect with n VIII<' Hence, the point :c
1<..1
has a neighbourhood not intersecting with F, which means that
the set F is closed. The theorem is proved.
Theorem 3, Let f; X - Y be a continuous mapping 01 a compact
space X into a space Y. Then the image f (X) is a compact space.
Proof, Let {U..} be an open covering of the set f(X), Then the
family {i-I (U,,)} is an open covering of the compact space X.

84 z, C;nral TopoloGV
Helice, some filllte family {I-I (Ua.A)}f1 covers X and therefore tho
family {Ua.Jo.}f1 covers the image f (X), The thoorem is proved.
Theorems 2 and 3 imply the following general property of a con-
tinuous function on a compact space known from mathematical
analysis.
Theorem 4. Let f: X _ R1 be a continuous function on a compact
space X, Then thelunction f is bounded and attains a maximum (min.
imum) value.
Prool, According to Theorem 3, the image I (X) is a compact sub-
space in HI and, according to Theorem 2, I (X) is a closed set. If
the image f (X) were unbounded, the system of intervals Un =
(-n, n) would cover I (X) and nb finite subcovering could be found
in this system. Suppose A = sup {I (x)}. Then there exists a sa-
EA:
quence X n E X such that Urn I (x) = A. i.e. A E I (X), A = I (xo),
n-..
Xo E X, Similarly, B = inE {f (x)} is also a value of the function I.
EX
The theorem is proved,
2,3,8, METRIC COMPACT SPACES
In the case of metric spaces the compactness property
can be formulated In terms that are customary for mathematical
analysis.
Theorem 5. Let X be a metric space. The space X is compact if and
only If one of the following equiv4lent properties is 8atisfied:
(a) any sequence {xn} has a convergent subsequence,
(b) any sequence of embedded, 1UJn-empty, closed subsets {F n},
F n => F n+l, has a non-empty intersection.
Proal, Let X be a compact metric space and {F n} a sequence of
embedded, non-empty, closed sets, Fn => Fn+I' If OFn = 0, the
open sets G n = X,F n cover the space X, and G n c G n +1' Since
X is a compact space, it is covered with a finite system {G n .. . . .,
G n .}, Without loss of generality we assume that n l ::;;;; n 2 ::;;;; , , .
 n., Then GnA. C G n ., Le, X "f' G n . and hence F n. = 0, which
contradicts the hypothesis,
Let us now demonstrate that property (a) follows from property
(b). Suppose {xn} is an arbitrary sequence of points of X. If {xn}
does not have limit points, any subset F C {xn 1 is closed. We can
therefore construct a sequence of embedded cfosed subsets {F n}
with an empty intersection by.putting F n = {xn+l' X n +2' ' . .}.
Hence, by property (b) the sequnce {xn} has a limit point X D EX,
Choose a subsequence {x nh } convergent to XD and consider a neigh-
bourhood 0 1 (Xo) , There exists a number nl such that x n , E 0 1 (X o )'
Let the terms {Xn" . . ., X n .}, n 1 < na < . . , < n., be already

2.3. ComptJcI Spate.
85
constructed, and p (xo. x"A) < ilk. Let also
e=min{ .t1 ' p(x o ' XI)' ....P(xo,x".)},
Then in the neighboul'hood O. (xo) we can find a point 3:'" E
I t .+1
O. (xu), Clearly, p (3:'0' x"0+1)< e::;;;; .+1 and bencl' n.+ 1 > n., Thus,
we have constructed, by induction. the subsequence {.} with
lim p (xo' 3:'".) = 0, which means that lim x'" =3:'0'
'-Finally, we shall prove that propert;(;) implies the compactness
of X. Let {U..} be an arbitrary opbn covering of the metric space X.
Suppose there exists a number 8 such that the covering with balls.
{Oe (x): x E X} is inscribed in {U ..}. Then e is called the Lebesgue
number of the covering {U ..}.
Lemma 1, J 1 condition (0) 01 Theorem 5 is satisfied, any open covering
has a Lebesgue number.
Proof. For an' point x E X W ; select a number e (x) equal to
th pxact upper bound of the numbers B> 0 such thl\L the baU
0 6 (x) lies in an element of the covering U... If the function 8 (x)
has a strictly posiUve eo = Inf e (x), then 80/2 is the Lebesgue
"ex
number of {U..}. If inf e (x) = 0, there exists a sequence {x,,} such
"ex
that lim e (xn) = O. According to property (a), We can choose a
"-..
Mubsequence %"/1 such that lim :r;" = Zo' Suppose ko is a number
A-..
1
such that p (xo, x"A) < T 8 (xo) for k> k o ' Then, 0 I (xnA) C
T,e",)
0 1 (3:'0) C U... This means tbat 8 (X"A) >  e. (xo) and hence
'2 8 (",)
lim 8 (x",,);;;'  8 (xo) > 0, The lemma is proved,
It...
Let us consider an arbi trl\ry covering {U..} , By Lemma 1, tho c.ov-
ering with balls {O, (x): x E X} is inscribE'Q in {U e}' Suppose {U (r,)
does not have a finite subc.overing. Then tOe (x): x E Xl dol'S not have a
finite subcovering either, Fix an arbitrary point XI E){, Since tho ball
0.(x 1 ) does not co\'er the space X, there exists a point .%2 EO. (XI)'
Suppose the points x" .." X n are so constructed that XIt EO, (x.)
for n;;;;,k> $. The system of baBs to. (XI)' .... Oe (xn)} does not
COVl'r X, so that thert! exists a point xn+t E Oe (:1:1) U .., U O. (3:'n).
Thus, we have constructod, by induction, ilie sequence {x,,} with
p (x, 3:',,) e > O. On the other hllnd, property (a) implies that
thue exists a subsequollce {z"lt} such that 11m xn = xo. i.e.
It-..
11m p (x nk ' :1: 0 ) = 0, In particular. an appropriate choice of the
k...

86 2. Gentral TQPology
1
ntlmber ko Among k> ko gives p (x nh ' xo) < 2" e, whence
p(.2: nk ' .2:/ll"I)p(xn' Xo) +p {X"A+l' xo)<'
This contradiction proves Theorem 5,
As a corollary of Theorem 5. we formulate the following funda-
mental property of real numbers: the lIequence of embedded segments
of the real axis has a common point.
2,3,. OPERATIONS OVER COMPACT SPACES
ff. a Hauselorff space X is a union of a finitE! number of its
compact subsets. X is a cornpact space. InrloE!d, if X= U X A ' where
",I
X h are compact SpliceS, and {V..}; is an open covering of X, tlletl
each subsol X.. is covE!red wiLh finitely many elements {V aA , I' ..
Va }, Then the entire finite system {Va . 1kn, 1snk}
,nlt k. a
covers lhe wholE! splice X. The following theorem is very impor-
tant. _
Theorem 6. The Cartesian product X X Y of compact spaces X
and Y is a compact space.
Prool. By Theorem 5 of See, 2.2, the Cartesian product X X Y
is a Hausdorff spac!'. Let {U..} .be an arbitrary (:overing of X X Y.
Without loss of generality, we I may assume that U.. = V'i.x W...
Fix: a point :c E X and consider 1\ subspace x X Y c: X X r. The
snbspace :c X Y is homeomorphic to Y and is therefore a compact
space, Thus, there exists finitely many elements {U.. (k, x)}) covering
n(x)
the space x X Y. Put V =:: n V.. (h, x). Then the open sets {V X
h=1
Wa. (h, x)} are refinements of thei covering fU..} an d cover x X Y.
,,(x)
The sets {W.. ,", d';l) cover the space Y, U Wa. (II, x) = Y, and
11=1
the sets {V} COVIH' X. Since the space X is compact, it is cov-
ered with a finitE! family {V.}:;'f' 'J;'lum the space X X Y is cov-
ereel with a fmite family {V. X W"(k. x.): 1 sm, 1 k n (x.)}.
The theorem is proved.
Remark. If the spaces X and }1' lire metric. the proof of Theorem 6
is much simpler Indeed, consier a sequence {z,,}, z" = (x"' Yn)'
Since X is a compact space, there' exists, by Theorem 5, II con-
vergent subsl'quence {x"k}' The compactness of Y implies that in
the sequence {Y"A} we extract a convergent subsequE'nce {Ynll). Thus,
the subsequenco {z,, J is also convergent.
.

2.4. Functlonal Separabllity. ParW/on oj Unity
87
Problems
1. Prove that a union of finitely many compact spaces is
compact.
2. Let X be a metric, non-compact space, Prove that there exists
a continuous function unbounded on this space.
3, Prove that a metric compact pace has a countable dense subset,
4. Given two closed disjoint sets A and B in a compact metric
space. Prove that peA, B) > 0,
2.4. FUNCTIONAL SEPARABILI'fY,
PARTITION OF UNITY
In this section we shall prove the theorems which enable
continuous functions to be analyzed under relaxed conditions,
as in the case of functions of one real variable, We have already
noted that a continuous function' on a topological space behaves,
in many respects, like a function :0£ one real variable: namely, the
sum I + g, product I 'g, and ratio Ilg (g =1= 0) of two continuous
functions are also continuous funtioJls, A limiting process is valid
in the class of continuous functio!}s on a topological space X. A se-
quence of functions f" converges u;lilormly to a function I if for any
II> 0 there exists a number N such that for f! > N tile inequality
J f (x) - In (x) 1<8 is satisfied :for any point x EX.
Theorem 1. The uniform limit 01 a sequence of continuous lunctions
on a topological space X is a continuous lunctlon,
Proof. Let I (x) = lim fn (x), We now demonstrate that the func-
tion I is continuous. Fi a number & > 0 and a point Xo E X, Then
there exists a number n such that for any point x E X we have
II (x) - f n (x) I < &/3, Since f n is a continuous function, there
exists a neighbourhood 0 (x o ) such. that for x EO (xo) the inequality
If n (x) - f" (xo) I < 8/3 is satisfied. Then we have for x EO (xo)
If (x) - f (xo) I < I , (x) - In (x) I + If n (X) - f n (Xo) I
+ I In (x o ) -f (xo) J < 8/3+e/3 +£/3 = e,
Theorem 1 is proved,
2,4.1. FUNCTIONAL SEPARABILITY
Let FI and F 2 be two disjoint closed sets in a topological
space X. To construct disjoint neighbourhoods of these sets, it suf-
fices to construct a continuous function f on X such that 1 (x) ;;a. a
for x E Fl' , (x)  b for x E F 2. and a> b, Thus, as the neighbour-
hoods of the set;F 1 and F 2 we can select the inverse images of the
intervals, ,-I (c, 00) and f-l (-00, c), where b < c < a. The state-

88 2. General Topology
ment, converso in a certain sense, is also valid (it is called Urysohn's
lemma).
Theorem 2, Let X be a normal .topological space and po. /0\ be two
closed disjoint sets, Then there existS a continuous function f: X _ !O, 1]
such that II Fo _ 0 and f r FI s:=r1.
Proof. Conslruct a system of opon sets f r numbered by all
binary-rational numbers Or1. such that the following condi
lions are slIlislied: (1) Fo c: f o ' (2) r , C X,P p (3) Fr C f r . for
r<r'.
Then the s)'stem of open sets {f r ) can be extendecl by adding
[heels r,= U rr' where t is sf! arbitrary real number, Ot::;;;;1.
r<:r
The system of open sels {f,} satisfies the same conditions: r, c: f "
for t<t'. Inclecd. choosing rali(mal numbers rand r' such that
t <r<r' <t', we obtain f, c: f r " f r , c: r " , Helice, 1', cl'rc f r , c
f l , COlI!itruct a function f: X -+(0,1], putting f (x) =sup {t:X(; r,).
We now op.J1Ionstrllte that f it contin UOIl5. Fix a point Xo And an
E>O, and luko to=f(xo)' By Lhe delinition of f we have .ToE
f(/.-e/2)o Xo E r(lo+£/2). Consider a neighbourhood of Xo equal to
U = r U .+£/21' roo_em. If Y E U, then Y E f(t.+./2), ye r(lo+£/2)' Owing
to the dolinition of the function I, t-ne inequaliUos t o -e/2::;;;;
f(y)to+d2 lira satisfied, i.o. If(xo)-f(Y)I<s. Thus, the
function f is con tinuous,
To complete the proof of the theorem, it remains to construct
a system of open sets f r satisfying conditions (1), (2), and (3). Since
the space X is normal, for any cl,osed set F and its neighbourhood
U J F c: U, there exists another neighbourhood V such that Fe
V c: 11 c: U, For the sake of brevity we shall write V <1; U, pro-
vided V c: V c: U. Thus, Fo = Fo c: X,F1 and therefore F 0 rG
X......F 1 . Hence, there exists an open set fo such that Fo <i:: fo iG
X'Fl' Similarly, we can find an :open set r 1 such that F 0 C?;; r 0 
r 1 t& X,FI' Suppose that for all binary-rational numbers r =
p/2 n , 0  p ::;;;; 2'1, the open sets r r have already been constructed,
and f("IZn) t& f«P+1)/2"" Define the set f(2p+ 1)/2"" in such a way
that f p/2" t& f{21'+ l)/2n" (g;; f(P+I)/2"' Hence, we have constructed,
by incluction, the whole family (I'r}, If x E Fo, then I (x) = 0, and
if x E Fl' then I (x) = 1. Theorem 2 is proved,
The folJowing important theorem is a consequence of Theorem 2.
Theorem 3. Let X be a normal topological space, F c: X a closed
set, and j: F _ RI a continuous function on F. Then the function t is
e;s;tendable to the continuous junction g: X _ RI on the entire space X.
II the tunction I is bounded, If (x) :1 ::;;;; A, the tunction g can also be
chosen in such a manner that it 1$ bounded by the same constant,
J g (x) )  A,

2.4. PUIle/lanal S.parabIlUy. Parlltlon "I U"lIy
89
Prool. First, Jet \IS 1ssume that I is bounded, II (x) I ::;;;; A. rllt
o (x) = I (x) and consider two closed subsets Ao = (x: CPo (x) ::;;;;
-A/3) and Bo = {x: CPo (x) ;;;.: A13}, Since the sets AD and /J o are
disjoint, there exists, according to Theorem 2. a continuous function
10: X _!-A/3, A/31 which is -A13 on Ao and A/3 on Bo. In
other words, 1/0 (x) j ::;;;; AI3, x. E X, I !Po (x) -/0 (x) I ::;;;; 2A/3.
Put CPI (x) = CPo (x) -10 (x). Then the function CPl is bounded on
F by the constant 2A/3. Thus. repeating the procedure, we can con-
struct two disjoint closed sets Al = {x: !PI (x) :s;;;; -2A/9} and
BI = {x: !PI (x) ;;;.: 2A/9}, and a continuous function '1: X _
1-2A/9, 2A/91 which is -2A/9 on Al and 2AI9 on 8 1 , In other words,
111 (x) ! :s;;;; { . t A, I !PI (x) -/ 1 (x) I :s;;;; 4A/9. Repeating this pro-
('ess infinitely many times, we can construct two sequences of func-
tions 'n: X _ RI and CP..: F _ R' satisfying the following con-
dilions: !Pn +1 (x) = !j)n (x) -In (x).
( 2 ) " A
1/ n (x) I:S;;;; 3' T'
I CP.. (x) I::;;;; ( ; )" A,
(1)
(2)
1'\'ext,
f (x) = CPo (x) = fo (x) + CPI (x)
=." =/o(x)+ II (3")+..; + I.. (x) +cp..+dx) for :rEF.
...
Inequality (1) implies that the series  I (x) converges unl-
o
formly on the entire space X. According to Theorem 1, the func-
lion g(x)=-=  f,,(x) is continuous and g(x)=t(x) for xEF. From
,,o
inequality (1) we have
Ig(X)I::;;;; Itn(x)I::;;;; ( ; r  =A.
11._0 k'l:::lO
Theorem 3 is proved for II bounded fuootion I,
Turning to a general -case. leti us consider a homeomOfphism
h: RI -+(-1.1), Then thp compo:.ition hi: F _(-1.1) is a con-
tinuous bounded function. Applying Theorem 3 to the bounded
lunction hi, we csn construc a continuous function g: X _{-i. 11
which is an extension of the function hi. The function g assumes the
values ::1;;1 on a closed set Fl disjont with the set P. According to
Theorem 2, there exists a continuo,us function 1jJ: X __ (0, il equal
to 1 on F and 0 on Fl' Then the function 81 (x) = '" (x) g (x), x EX.
coincides with hi on F and does not assume the values :1:1. Thus

90 2. G."eraL TopologV
the function gl maps X into the interval (-1, 1). Finally, put
g2 (x) = h.-Ig i (x), Tben g 2 (x) = h -igi (x) = h -111.1 (x) = I (x) for
oX E P. This completes the proof of the theorem.
2.4.2, PARTITION OF UNITY
The support ot a continuous function on a topological space X
is the closure of the set of those points x E X for which f (x) :-1= O.
The support of a function I is danoted by supp f. Thus, a flJl"lction f
is identicaJly zero outside the -support. A convenient method in
topology is the decomposition of.a function into a sum in such a way
that tiny term has a sufficiently'small support.
Theoren' 4. Let X be a normal space and {a,,} a finite cpen covering.
Then there exist functioTl!l !p,,: Xi __ RI, 0  !p", (x)  1, such that:
(a) supp !p", c:: a". (b) Lj <p" (x) =s 1.
a
The system of functions {rp,,} is caBed a partition of unity subordi-
nate to the c('vering {a ,,}. Here we do not assume that the covering
is fmite, but only require that any point x E X should have a neigh-
bourhood 0 (x) intersecting fmitely many supports supp 'P". In
general, "Ie system of subsets {A ,,} of a topological space X is called
a locally finite system if for any point x E X there exists a neighbour-
hood 0 ,x) which has a non-empty intersection only with fmitely
many :-ets A",
EXI.'I1ple. The covering of the real axis RI with !otervals (-n, n)
is not locaIJy fmite. On the corltrary, the covering with intervals
(n, n + 2) is locally ftni te.
Thus, if the family of supports {supp 'p,,} of continuous functions
(j>" is locally finite, only finitely many functions <p" are non-zero at
any point z E X and therefore the sum <p (x) =  'p", (;7;) is correctly
'"
defined at any point x E X. The function thus ohtained is continuous.
Indeed, the continuity of 'p can be verified independently in some
neighbourhood of each point. Sirjce the system {supp q>",} is locally
finite, there exists a neighbourhood 0 (x) in which only finitely
many functions <p",. . . ., 'P"N are non-zero. Hence, in the neigh-
bourhood 0 (x) the infinite sum appearing in the expression for (j>
N
is, in fact, fmite, i.e. !p (y) :;:0  (j>"h (y) for y EO (x). The function
Jt=1
(j> is therefore continuous at every point x EX.
Proof of Theorem 4. Let us consider a finite covering {a",}. Ac-
cording to Theorem 7 of SC. 2.2; there exists a finer covering {V,,}
such that v.. c: a a.. By Theorem 2, there exists a continuous func-
tion 1/>.. on X which satisfies the conditions: 1/>" IV,,!$ 1, 1»", I (X,,-
a,,) !5l 0, 0  1/>" (x)  1. This means that SUPP 1»" c: a" and

2.4. Funcll"al Sparablltly. ParlWun I Un/ly
9\
1Pa (x) > 0 for x EVa' Put 1P (x) =  "'a (.r). The function is con-
a
tinuous. We now demonstrate that", (x) > 0 at each point x EX.
Indeed, since the system {Va.} covers the space X, a number (%0
can be fOllnd such that x E V a.' I i.e. "'a.. (x) > O. Thus, '" (x) =
f "'a. (.1') ;;;;. "'<Zo (x) > O. Let \IS finally put <p", (.7') = "'a (.7')l1p (x).
a
Then,
supp CPa. = sUPP'i><Z c:: U a' 0:;;;; <Pa (.1') :;;;; 1,
and
 <f'", (x)...  'I'a (.c)!lp (x) = ( '\j>", 1.r)/11' (x) = $(.2:)/1\) (.x) s1.
a
Theorem 4 is proved.
Remark, Theorem 4 can be extended to locaJly finite open cover-
ing. To this end. it suflices to provo Theorem 7 of Sec. 2.2 for locally
fmite coverings. If a space X is compact, we eRn always confine
ourselves to tinite coverings, while for non-compact spaces there
exist essentially inlinite coverings for which we should, nevertheless,
construct a partition of unity. Thus. given a locally finite covering
that refines a given covering {U ",}, a partition of unity can be con-
structed. The foIlowing statement is formulated without proof.
Let X c:: R n be an arbitrary subspace of a Euclidean space and let
{U co} be an open (not necessarily finite) covering 01 X. Then there
exists a partition of unity subordinate to {U a}.
Problems
1. Prove that in Theorem 2 we may require the smooth-
ness or the function f. provided X.is a Euclidean space R n .
2. Prove that in Theorem 4 we may require the smoothness of the
functions <Pa. if X = RIO.

Chapter 3
Smooth Manifolds
(General Theory)
INTRODUCTION
In Chapter 1 we have shown that a coordinate system
describing the position of a point in space Is an indispensable tool
for studying geometrical objects! Using coordinate systems, we can
apply the mothods of differential and integral calculus to solve
various problems, Therefore, an analysis of spaces which admit such
concepts as differentiable or smooth functions and differentiation
and integration has emerged as a'n independent branch of geometry.
Some examples of these spaces; may prove useful.
Example 1. Consider a unit circle SI on a plane (Le, in a two-
dimensional Euclidean space R2). To describe points of the circle by
coordinates. we define it as a set of points satisfying lhe equation
Z2 + y2 = t (1)
where (x, y) are Cartesian coordinates on the plane, Thus, each point
P E SI is uniquely defined by a pair of numbers, Cartesian coordi-
nates :c and y, For points of a circle, however, there is no neerl of
giving both coordinates. If the coordinate :c of point P is known,
the second coordinate is easil;y found from equation (1): y =
::!: V i - x 2 , i.e. the coordinate y is uniquely (to within a sign)
related to the coordinate x, Furthermore, if a point Po = (xo, Yo)
does not lie on the x-axis, i.e. Yo *" 0, there exists a sufficiently
smalJ neighbourhood U of point Po such that for all points P E U
the sign of the coordinate y of P is uniquely deermined by the sign
of the coordinate Yo of point Po. We may therefore say that points
of the circle SI are described by one numerical parameter, the Carte-
sian coordinate x, To be more exact, only the points of the upper
semicircle, i.e, the points satisfying y > 0. are uniquely defIDed
by one numerical parameter x. Similarly, the points of the lower
semicircle, Le, tho points satisfYing y < 0, are also defined by one
numerical parameter ... The parameter x varies within the same
range, namely on the interval (-1, 1) of the real axis, both for the
upper and lower semicircles. If we need to describe parametrically
points of a circle, including "singular" points Po = (1, 0) and PI =

3.1. The Concept 01 " Manllold
93
(-1, 0), we will have to interchange the coordinates % and y and
express x in terms of y through equation (1).
A question naturally arises: can ,we introduce such a parametriza-
tion of points of a circle SI that aill the points are uniquely defined
by a numerical parameter? The parameter <p, equal to the angle
between the x-axis and radius vectr ending at point P, seems to be
the most appropriate approximatipn, though this parameter is not
defined uniquely, And if we consider the values of If! only in some
interval, say 0 < If!  2lt, the function associating the angular
parameter <p with a point P E 8 1 will suffer a discontinuity at point
Po = (1, 0),
Thus, we arrive at the following' assertion. No continuous function
exists on a circle 8 1 (as a topological space) whose values uniquely
describe points of the circle.
Example 2, Let us consider in a three-dimensional Euclidean space
a two-dimensional sphere 8 2 defined by
%2 + y2 + Z2 = 1. (2)
Just like in the case of a circle, it is unnecessary to give all three
Cartesian coordinates (x, y, z) for points of the sphere S2, since one
coo rdinate, aay z , can be expressed in terms of the other two: z =
:I: V i - Xi - y2. Obviously, on the upper (and similarly the lower)
hemisphere the coordinate z of a point P is uniquely related to the
coordinates x and y.
As before, it can be shown that there exist no two j:Ontinuous func-
tions on the two-dimensional sp\1ere 8' such that their values
uniquely define a point P on the sphere.
These examples demonstrate tha!t the only possibility is to reject
the construction of a unique coordinate system for all points of a
space under consideration and usa different coordinate systems for
different parts of a space. A rigorous analysis of the situation has led
in geometry to a specific concept of a manifold,
3.1. THE CONCEPT OF A MANIFOLD
3.I,t. FUNDAMENTAL DEFINITIONS
A metric space M is called an n-dimensional manifold (or
simply manifold) if any point P of the space is contained in a neigh-
bourhood U c: M homeomorphic to a domain V of a Euclidean space
R". This condition can be formulated in brief as follows: an n-di-
mensional manifold M is 10clln,I' homeomorphic to a domain in
a Euclidean space R", in which case the dimension of M is said to
be equal to n or dim M = n. Thus, if M is an n-dimensional manifold,
we can find in M a system of open sets IUd numbered by finitely
(or infinitely) many indices i and a system of homeomorphisms

WI 3. Smooth Mantfold. (Genra! TMorv)
CP/: U I _ VI eRn of the sets U I on the domain VI' The system {Ui}
must cover the space M, i.e. M = U U , . and the domains VI may,
i
in general, intersect one anothr.
Suppose a Cartesian coordinate system (Xl, . . ., x") is valid in
a Euclidean space R n . Then for any point P E U, the Cartesian coor-
dinates of the point 'PI (P) E Vi can be considered as a numerical
parametrization of p, The homeomorphism CPI is therefore called
y
y
u.
F1gure 3.1
Figure 3.2-
Figure 3,3
a coordinate homeomorphism and the Cartesian coordinates (Xl, ' , "'
x") of cP, (P) are termed local coordinates of the point P E Uland
denoted by z" = :r!' (P)*, k = 1, . . ., n. A system of functions
z" = x" (P)* given on an open set Uris called a local coordinate
system, and the open set UI together with a local coordinate system
defined on it is called a chart of a, manifold M. Thus, a chart is a pair
(U io CPI), and we shall denote it, for brevity, only by the first symbol,
U" A set of charts {U,} covering the entire manifold M is called an
atlas, It is convenient to numbe local coordinates of a point P EM
by an additional index characterizing the chart UI: r: = x (P).
Since the point P can belong simultaneously to several charts, it
acquires several sets of local c'oordinates.
Here are simple examples of manifolds.
ExampJe 1. In the Introductidn we have considered a circle SI c::
R2 defined by the equation x 2 + y2 = 1, Let us cover SI with an atlas
consisting of four charts (Figs. 3.1 and 3,2)
U I {(:t, y) E SI: y > OJ, U 2 = {(z, y) E SI: y < OJ,
US = {(x, y) ES I : z>O}, U, = ({z, y) ES I ; x<O}.
· Slrictly speakinll. the Cartesian coordinates (ZI, . . ., ",n) in
a Euclidoan space Rn are linear functibns defined in Rn, and the correspondence
associating wilh a vector a E Rn a seti of its coordinates (ZI (a), . . '. :z;n (a» is,
in fact. a mapping of Rn into an arithmetic linear n-dimensional space. It
would therefore be better to write zll = :rll (P) "" zA (cpj (P)), but for the sake
of brevity we sball omit the symbol' CP, if no confusion arises.

8.1. The Co/ltept 01 a Manllold
95
The corresponding domains V lt V 2 , V 3 . and V. on the real axis RI
coincide and are equal to the open iterval (-1, 1). H6meomorphisms
'PI and 'P2 are constructed as projections of the circle onto the x-axis:
<1'. (x, y) = fl'2 (x, y) = x, and homeomorphisms 'Pa and 'P' as pro-
jections onto the y-axis: <Pa (x, y) = <Po (x, y) = y. In order to
prove that the mappings CPh, k = 1, . . " 4, are homeomorphisms,
it is sufficient to write explicitly the inverse mappings
cp;' (x) = (x, V 1- x 2 ) E 8 1 , cP;' (x) = (.1', - V i-x? ) E 8',
cp;'(y);::;(V 1- y 2 , Y)E5 1 , W;'(y)=(-V 1-y2 , y)E8 1
and demonstrate that they are continuous. Then we obtain on the
circle four local coordinate systems, each consisting only of one
coordinate: XI = CPl (x, y) = x, X2 = CPo (x, y) = x, X3 = CPa (x, y) =
y, x. = CPo (x, y) = y. Certain points are provided with two Jocal
coordinate systems. For instance, i for points P of the intersection
U I n U 3 the coordinates Xl (P) and X3 (P) are valid (Fig. 3.3),
There are other ways of introducing an atlas on a circle, In Chaptl!r 1
we have considered polar coordinates (r, q» on a plane. The equation
of a circle in these coordinates is very simple: r = 1. Strictly speak-
ing, polar coordinates on a plane !are not a coordinate system (see
Chapter 1). We introduce therefore two charts on a circle 51, namely,
U. = {(x, y) E 8 1 : x =p -1} a,nd U 2 = {(x, y) E 8.: x =p 1}
(Fig. 3.4), Let CPI (P) = CP. (x, y) be the value of cP in the interval
(-n, n) and CPo (P) = CP. (x, y) be the value of cp in the interval
(0, 2n;), i.e. VI = (-n, n), V. = (0, 21t). Obviously, the local
coordinates CP. = 'PI (P) and CPo = <PI (P) coincide for points of the
upper semicircle and do not coincide for points of the lower semi-
circle, that is, !PI (x, y) = CP2 (x, y) for y > 0 and 'P. (x, y) =
CP. (x, y) - 2n for y < 0 (Fig. 3,5).
Example 2. The circle 8 1 consiQefed in Example 1 is a rather com-
plicated manifold. The simplest example is represented by Euclidean
space an, We may take an atlas consisting of only one chart U = R",
the coordinate homeomorphism cP is the identily mapping q>: U __
V = R n , the local coordinate sytem is Cartesian coordinates of
points in Rn, Similarly, any domain U c: R n is an n-dimensional
manifold whose atlas also consists of one chart with a Cartesian coor-
dinate system.
Example 3. Let f: R" -+ RI be a continuous function and r , c:
Rn+! its graph, Le, the set of points (x., ' . " xn, X(,+I): x"+ =
f (x', . . ., x"), The space r l is an n-dimensional manifold with an
atlas consisting of one chart U = fl' The coordinate homeomorphism
11': U __ V = R n will be defined as a projection along the last ('oor-
dinate: Ip (Xl, . . ., x", x"+1) = (x., ' . ., x n ) E R h . Then the in-
verse mapping 'P- I is given by cp.l (Xl, . . ., x") = (Xl, ' . ., xn,
f (Xl, . . " x n » and is, apparently, a continuous mapping.

96 3. Smooth. Manllold, (General Theory)
Example 4. Let us consider an n-dimensional sphere S" of unit
radius defined as a set of points in R"+1 satisfying the equation
(X 1 )2 + (X 2 )2 + . . . + (Z"+1)2 = 1. We shall demonstrate that an
n-dimensional sphere is an n-dimensional manifold. The open sets
ut= {(Xl, ..., X"+I) E S": xi> OJ,
Uj ={(xl, ..., x"+I) E S": xi <OJ,
will be taken as an atlas, We obtain 2n + 2 open sets covering the
entire sphere sn. Indeed, if the point P = (zt, . , ., X"+l) belongs
y
Figure 3.1i
Figure 3,5
neither to charts U t nor to charts Vi for all i, then the inequalities
xi  0 and Zl  0 are satisfied, i.e. Xi = 0, i = 1, 2, . , ., n + 1.
Then (XI)2 + ' , , + (X"+1)2 = '0, that is, the point P does not lie
on the sphere sn. The coordinate homeomorphisms <pi and <PI are
defined as projections of the Euclidean space Rn+l onto R" along
the coordinate xl. In this case the domains vt and Vj coincide and
are equal to a unit ball, and the coordinate homeomorphisms are
given by
Ipt(x t , ..., X"+I) = <pj (Xl, ..., X"+I)
= (Xl, "') x i -" .:c iT ', ..., x".I)ER".

3.1. Th. COIlc<pt of a Manifold
97
The inverse homeomorphisms are defined by the formulas
(o:pt)-I (1fI, ..., y")
=(yt, ...,I/i-'. Vt-(yl)Z_..._(y")2, yi, ..., yn)ER"H,
(cpi)-I (yl, .., y")
 (yl, ..., y;-I, - V 1_(yl)2- ... _(y")2, y', ..., y") ER"+I,
and are apparently continuous.
EJC!ample 5. Let us consider a projective plane RP2. By tllis plane
we mean a space the points of which are straight lines through the
origin in Ra. Deftne the distance between two straight lines as the
least angle between them. Then RPZ becomes a metric space. We
now prove that Rpz is a two-dimensional manifold. To this end, it
is convenient to describe any straight line P E RP2 by thl'ge homo-
geneous coordinates (x: y : 1.) which admit multiplication by a
number'" =F O. Homogeneous coordinates do not vanish simultane-
ously, i.e. x 2 + y2 + 1. 2 > O. Cover RP2 with three charts: U I =
{(x: y : 1.): x =F O}, U 2 = {(x: y : 1.): y =F O}, and U a =
{(x: y : z): z =1= OJ. Let VI = V. = Va = R2. The mappings fj)k:
Uk - V k = RZ are taken as coordinate homeomorphisms, namely
o:pI (x : Y : 1.) = (y/x, z/x), fj). (x : y : z) = (x/y, z/y), CP3 (x : Y : z) =
(xlz, ylz). Thus, we have constructed three local coordinate systems
x; = ylx, x = z/x; x = xly, x = zly; x = x/z, x: = ylz,
To verify that the mappings CPk are homeomorphisms, it is sufficient
to construct the inveJ;'se mappings
q>1 (x, x) = (1: x; : x:), q>;' (x, x;) = (x:: 1: x;),
cP;' (x, r.) = (x : x: : 1)
and prove that they are continuous.
The examples presented above show that the same manifold M
admits distinct atlases. Even though the charts, as open sets, remain
unchanged, we can alter the local coordinate system in a chart by
choosing another coordinate homeomorphism. In particular, the
following lemma holds true.
Lemma 1, Let M be an I!-dimensional manifold and let U be Us
chart with a coordinate homeomorphism !p and a local coordinate system
(x'. "', x"). It u' c= U is an open subset of U, a coordinate homeo-
morphism fj)' and a local coordinate system (yl, . , ., y") can also be
defined on U', and in this case q>' (P) = o:p (P). y (P) = Xk (P) for
P E U'.
Proof of Lemma 1 follows from the fact that the homeomorphism
Cf: U -+ V maps homeomorphically any open subset U' cU. It is

tlB 3 S ",".Ch .11 all/folds {General TheorgJ
l1mci(!llt tharefora to take the restriction of <p to V' as q>' and Lhe
restriction of coordinate functions :t!' to the same subset V' as yh.
Lemma 1 shows that \Ising a given alIas IUd we can construct
a new alias ronsisling of liner charts. 0/1 the other hand, Lhe union
of two 1I1Iaes IUd and {Vi} is again an atlas of the manifold. Thus,
there exists a maximal atlas consisling of all the charts of a gi \'en
manifold. A maximal atlas may he considered as a union of "U
aliases on a manifold,
1'ow we shall prove another useful lemma.
Lemma 2, Let (Ud and {Vj} be two atlases on a manllold M. TheIl
there exists a third atlas which refines these two atlases.
T.) prove the lemma, we put WI} = Vi n Vj. According to Lem-
Dla 1, a local coordinate system can be introduced on each open
set Wi)' On the other hand, W;J C V I and Wi} C Uj, so that the
system of sets {WI}} covers M. Hence, {WI)} is an atlas refining
both {Vd and {Vj}.
3.1.2. FUNCTIONS OF COORDINATE TRANSFORMATION.
DCt'/NITION OF A SMOOTH MANIFOLD
Among metric (or, in general,-topological) spaces mani-
folds !>eem to be of special interest. For example, any continuolls
function /. 1'rf --+- RI defined on an n-dimensional manifold AI in the
neigh bOllrhoo<l of cae h point P EM can be identified with an ordinary
conlinuous renl-valued ftlnction h (Xl, . . " x") of n independent
real v!u'iahlcs (:I'L, . . ., x"), the function h heing defined in a domain
of a Euclidl'otl l'pacc R". Indeed, let U be a chart containing a
point P, let (I': V ...... VcR" be a coordinate homeomorphism of
this chart, Hld let (Xl (P), ' . ., x" (P) he a local coordinate system
in V. If x = (,1;.1,. ., x") is a vector with coordinates (Xl, . . ., x").
we pnt h (xl, . . ., x") = I (q>-I (x». Conversely, if h is a continuous
function of n real variables defined in a domain Y c R", we can
associnte with II fl continuous function j valid in the domain V of
1he manifold M: f (P) = h (Xl (P), . , ., x" (P».
More generally, let j: M 1 -M. he a continuous mapping of an
n-dimensional m<1l1ifold M) into an m-dimensional manifold M 2'
Suppose Qo = I (Po), Pa E M, and Qo E M 2 . Then in a small neigh-
bourhood V j Po the mapping I can be identified with a continuous
vector function h of , independent variables. Indeed, let U ' j Qo
be a chart of the manifold M z and let (yl, . . ., y"') be a local COOl'-
clinnte system. Since th!' mapping I is continuous, there exists, accord-
ing to Lemma 1, a chari. U of point Po such that I (V) c V'. Suppose
(Xl, ..., x") i 01 local coordinate system in U. Since points P of
the chnrt U are in a one-to-one correspondence with their coordinates
(Xl (P). ..., x" (P», ancl points Q of the chart U' are also in a ona-
to-one correspondence with their coordinates (yl (Q), . . ., ym (Q».

3 /. TM ConCtpt <>/ a M ant/old 9!1
the equality Q = / (P) means
!l (Q) = l (f (P) = y" (h (Xl (P), ..., .t" (P»
=h k (:1: 1 (P). ..., .x" (1').
The functions h k (Xl, . . ., x n ) are continuous, and the ma pping f
is uniquely reconstructed in U by these functions.
Thus, any continuous function I on a manifold M in a local coor-
dinate system can be represented by a real-valued function h of II
independent variables. If we alte.r the Jocal coordinate system, the
function h will also be modified. What is the law of modification of h
under coordinate transformation? Let (XI' . . ., x") and (yl, ' , " y")
be two local coordinat(j systems. Without loss of generality, we may
IIssume that these coordinate systems are defined in the s[\me chart U.
Suppose hand h' are functions of coordinates (Xl, . . ., x") and
(yt, . . ., y"), respectively, whic? represent the function f Then
f (P) = h (Xl (P), ..., x" (P» = h' (yl (1'), ,.., y" (fJ». (1,
Since the coordinates yl, . . ., yn are also continuous functions in U,
they can in turn be represented as functions of n. independent vari-
ables (Xl, . . " xn), i.e.
yl (P) = yl (Xl (), ".. x" (1'»,
(2)
y" (P) = y" (I (1'). ..., "(PH.
In these eqnations we deliberately use the same symbol ii' to denote
hoth the coordinate of point P and its representation as a function
of (Xl, . . ., x"); yk = yk (Xl, . . "xn). Then from equation (1)
we obtain the identity
h (Xl, . . ., x") = h' (yl (x', . . " x"), "', y" (Xl, . . " xn», (3)
The functions yk = y (Xl, ' . ., 'x") on the right-hand side of rela-
tions (2) were called in Chapter 1 functions of coordinate transforma-
tion, provided the set U is a domain in a Euclidean space. We shall
retain this term for manifolds as well.
Definition I. Let M be an n-dimensional manifold, {Ud its atlas,
'PI coordinate homeomorphisms, and {x:J a set of local coordinate
systems. In each intersection of two charts UIJ = U t nUl two local
<'oordinate systems {x} and {xq} are valid such that x (P) =
'l" (xl (P), . . " xi' (P», P E UIJ. The functions x = x (xJ . . . ..
xi') are called functions of coordinate tran.s/ormation or functions of
lrallsttion from the coordinates {x} to the coordinates {x}.
Transition functions are not denned in the entire domam Vb
but only in its part V)/ = !PI (UiJ) where it is meaningful to speak
about two coordinate systems.

100 3. Smool" Manifolds (GeMral Theory)
In Fig. 3.6 the domains VI and VI in a Euclidean space are shown,
for con venience, as disjoin t sets.
The transition functions x = xt (x:, . . ., x'J) map the domain
J!)I into Vi) in R"
XI ".... {.1: (.r, .... xj')} = {xi (XJ)} = Xi (XI) = <p;<pj' (X}) = Ifill (x}). (4)
The mappings !(Iii: VJI -+ Vi) given by equation (4) are, ill fact,
another writing of transition functions, and represent a homeo-
morphism of the domain V'I onto VI!,
Nole that if i = j, then Ui} "" VI> VIj = V JI = VI> and x (xl.
_ . ., xj') S! x,. Let us now return to the representation of a con-
Figure 3.6
tinuous function I defined on an n-dimensional manifold M as a
function h of n independent variables, local coordinates of a point
of the manifold. It is known that a narrower class of function-
differentiable functions-is of great significance in mathematical
anaIY1:'is. We now transfer this Important concept to functions de-
tined on a manifold. If a function h (xl, . . " x n ) is continuously
differentiable, we cannot say the same about the function h' (yl, . . "
y") representing f in another local coordinate system (yl, . . .. y").
Indeed. the functions hand h' are related by equation (3). Thus, the
condition that h' is also continuously differentiable is that the func-
tions of coordinate transformation 3!' = x (yl, . . ., yn) should l)e
continuouslv differentiable. If tnese functions are not continuously
clifferentiabie, there exists a funcion f such that its representation It
in the coorclinates (xI, . . ., x") is a continuously differentiable
function, while the representation h' in the coordinates (yl, . . ., y")
is not. As an example, we consider the function f (P) = xh (P),
P E U c: /11. Then h (Xl, .., x") = Xh is, apparently, a continu-
ous]y differentiable function, while h' (yl, . . ., y") = xh (yl, . . .,
y") does not possess this property.

3.1. The Concept of a Manifold
(CII
We thus arrive at the following definition.
Definition 2, A smooth n-dimensional manifold is all II-dimensional
manifold 11'1 with an atlas {V d having locRI coordinate sytem1<
{.x} satidying the condition: the transitioll fUlictions x' '-= x:' (.r;,
. . ., xi) are continuously differentiable for any pair of charts Vi
and V J in the entire domain ot dfinition,
This definition enables a class of continuously differentiable func-
tions to be distinguished among all functions valii )n a manifold Ill.
Definition 3, A function f: M -.. R' defined 011 a mooth man i-
/,,!d ,11[ is called continuolJ.$ly differentiable at R point Po EM if in
?ny local coordinate system (x:. . . "x?) (the charts U, 31'0
t,elong to a fixed atlas) the function f can be represented as a COII-
'inllously differentiable function h (xl, . . ., x\') of I independent
vRriables in a neighbourhood of the point (x: (Po)' . . ., x}' (P,,».
:'o1ote that the condition of continuous differentiability of the
transitioll fUllction in Definition 2 is essential for Delinidon 3. As
we have already pointed out, if the transitioll functions were not
('ontinuously differentiable, the condition of continuous differenlia-
hiJity of I at point Po EM would depend 011 the choice of the chart
U I containing Po.
Example 6. We now consider the following atlas on a manifold At.
Let M = RI be II real axis and let the atlas consist of two identical
(harts VI = U. = M = RI. but with distinct coordinate systems.
On U I we detine the coordinate XI = x, x E RI, and on V 2 the coor-
dinate Xt = x 3 . Then the transition functions are
xJ = x. (XI) = (X I )3, (5)
V -
x, = x, (x.) = x.. (11)
While the transition function (5) is continuously differentiable
(a polynomial), the function (6) has a discontinuous derivative, Thus,
according to Definition 2, the manifold M with the atlas {VI' V.}
i not a smooth manifold.
Remark, If an atlas on a manifold M consists of only one chart
(i.e. M is homeomorphic to a Euclidean domain), M is a smooth
manifold,
Definition 4, Let on a manifold M there be given two atlases
IUd and {Ui} such that M is smooth with respect to each atlas.
Two atlases {V;} and {Vj} are called equivalent if every function
<If coordinate transformation from any local coordinate system in
{Ud to any local coordinate system in {Uj} is continuously dWer-
pntiable.
A substantiation behind this delinition Jies in the fRct that any
function f on a manifold M is continuously differentiable in the atlas
{U;} if and only if it is continuou.ly differentiable in (U). Thus,
from the point of view of continu,ously differentiable functions on

102 3 Sm'1PtJ. Manifold, (Genral Theory)
.a manifold M equivalent atlases "have equal rights", so that an}
of the equivalent atlases can be used to represent a function as a co '1-
tinuously differentiable real-valued function of independent vari-
ables (coordinates of a point). Delinition 4 admits another formula-
tion: two atlases {U i} and {Uj} are equivalent if M is a $mooth mani-
fold with respect to a new atlas equal to the union of the initial atlases,
{U,} U {Uj}.
We often have to deal with narrower classes of functions. Re(,I\JI
that a real-valued fnnction h (Xl, . . .. x") is smooth of class cr
(r = 1, 2. . . " 00) in a neighbourhood of a point (.x, . . " .x')
if in this neighhourhood all partial derivatives of h up to order r
exist and are continuous. For r = 00 this means that the function h
has continuous partial derivati\',es of any order. Consl'quently, we
shall impose on atlases of a manifold M the conditions formulaled
in the following defmition.
Definition 2'. A manifold M with a flxed atlas {U,} is called a
I>mooth manifold of class c r (r = t, 2, . . ., (0) (or Cr-manifold)
if all the transition functions are smooth of class C' at all points of
the domain of their defmition.
Definition 3'. Let AI be a C"-manifold and let f be a continuous
function on this manifold. The function f is-ealled I>mooth of class C',
s .;;;;; r (or C'-function) in a neighbourhood of a point P n E M if any
representation of f as a function h of local coordinates (Xl, . . " x")
(from a flxed atlas) is a C'-function in a neighbourhood of the point
(Xl (Po), . . .. x" (Po»). The fun{'tion f is smooth of class C' if it is of
class C' in a neighbourhood of each point Po in the domain of de-
finition.
Example 7, Let us modify Example 6 by choosing the coordinate
x 2 = x + x-I x I in the second cllart U 2' Then M is a Cl-manifold,
but it is not C2- man ifold.
Remark. Below, if not stated otherwise, we shall consider only
C'" -manifolds and fnnctions on these manifolds.
In examples 1-4.we have considered manifolds with such atlases
that these manifolds are always of class C"",
Geometry may also deal with more strit't conditions on atlase
and their transition functions. For example, if all transitiou func-
tions are real-analytic, i.e. in a neighbourhood of each point in
their domain these funt'Lions can be expanded inlo convergenl T!1'-
Jar series, the manifold is called a real-analytic manifold. A real-
nlllytic. manifold is C""-manifold.
A more important class of manifolds is represented by complex.
analytic manifolds. Let JI[ b a 2n-dimensional manifold. {Ui)
its atlas. and <pj: U) -+ Vi <= W" its coordinate homeomorphim$.
Identify II 211-dimensional Euclidean space HZ/! with an /I.dimensional
complex linear space C", assuming that complex coordinates of

3.1. Tile Conc.pt of a Manifold
!Iii
a point (1.1, . . ., 1.") give rise to 2n real coordinates (x', . , " .T",
y', . . ., y"), 1.h = Xh + iyh. Then, 2n coordinate functions $J (P),
. . ., xj (P), y} (P), . . ., yj (P) i,n the chart U} IIrB transformed
into n complex-valued functions 1.' (P) = x7 (P) -r iy1 (P). The
functions 1.1 (P) are called complex coordinates of a point in the
chart U}. In the intersection of two charts U J fl U I the transition
functions are given by
xJ=xj{x; ..., xr, y,. 9/ 1 ),
xj=xj(xL ..., xf. y}, ..., yf).
yj=y}(xL ..., x?, yJ, ..., yr),
(7)
yj= y'J (xl, .,., x1', y\, ..., yj').
These functions can be represented as complex-valued functions
of n independent variables
%}=%j(z, ..", z?),
(8)
1.j = 1.7(1.\, .... 1.f).
Functions (8) are called transition functions or functions of trans-
formation of complex coordinates.
A manifold M with a fixed atlas {U}} and local complex coordinate
systems (1.}, ' . ., 1.;') is called a complex-analytic manifold, provided
all transition functions (8) are complex-analytic. i.e. they can be
expanded in convergent Taylor series of complex variables in a neigh-
bourhood of each point in the corresponding domain of definition.
As an example of a manifold admitting a omplex-analyticstruc-
ture, we shall consider a two-dimensional sphere 8 2 with a specially
defined atlas. In Chapter 1 we have constructed the stereographic
projection of the sphere 8! = X2 + y2 + 1. 2 = 1 fcom the north pole
Po = (0, 0, 1) onto the coordinate plane (x, y). This projection,
denoted by CPo, maps all the points of 8 2 except for the pole Po
(i.e. the open set U a = S2,,(P o » homeomorphically onto tile entire
plane Va = R2. In Cartesian coordinates the homeomorphism
</-0 is of thl! form Cpo (x, y, z) = ( 1':'" 1 z ) . We introrluce
therefore in the chart U o one complex coordinate W a =  ::"1: expressed
in terms of the Cl\rtesian coordjates on a sphere. Furthermore,
\\e shall consirler the south pole PI =(0,0, -1) and Ihe ster-
eographic projeclion <PI from the south pole onto the same coor-
dinale plane (x, y). Thl' projection !j11 maps homeomorpldcally the

104 3, Smooth Manifolds (General Theory)
set []I = S,(PI) onto the entire plane VI = R', In Cartesian coor-
dinates (j>1 takes the f01"m cpj (x, y, z) = ( 1Z ' 1 z ), Introduce
in the chart V, a complex coordinate WI = -::;. , Then, in the
, . U V b ' .r2+y 1 H
Intersection 0 n I we 0 tam WOw l =  ==, ence,
1 1
WO=WO(W 1 )=-, W1=W1(W o )=-. (9)
10 1 Wo
Functions (9) are complex-valued, so that the sphere S. is a complex-
analytic manifold. Each chart U 0 and U 1 covers the entire sphere S.,
except for one point, and is identified by virtue of the coordinate
homeomorphisms <Po and CPl with ,the complex plane Cl = R., Thus,
the sphere S is usually identified with the so-called completed com-
plex plane obtained from CI by adjoining another "infinitely dis-
tant" point.
An arbitrary smooth manifold need IIOt necessarily be a complex-
analytic one. For example, if its dimnsion is odd, the manifold
is not, by trivial considerations, complex-analytic. Yet. there
exist manifolds of even dimension which do not admit a complex-
analytic structure either. For instance, as is""demonstrated in Sec. 4.(3
of Chapter 4, a projective plane is not a complex-analytic manifold.
3.1.3. SMOOTH MANIFOLDS, DIFFEOMORPHISM
Let M) and M. be smooth manifolds and let I: M 1 - J1l 2
be a continuous mapping, As was already noted, in a neighbourhood
of any point Po E M 1 the mapping f can be represented as a vector
function h, l = hI< (Xl, .. '. n), where (Xl, ' . ., x n ) is a local
coordinate system in the neighbourhood of Po E M 1 ami (yl, . . '. y"')
is a local coordinate system in the neighbourhood of point Qo =
f (Po) EM..
Delinition 5. Tl\e mapping I: M 1 -+!VI 2 of smooth manifolds is
called a smooth mapping 01 class e' (r = 1, 2, . . .. (0) (or e'-map-
ping) if for arbitrary local coordinate system (Xl, ' . ., x") in the
neighbourhood of any point Po E !viI and (yl, ' , ., ym) in the neigh-
bourhood of point Q = f (Po) E !vi 2 the representation of 1 8S a vec-
tor function y = (y ) = (h ($1; , . " $"») = h (x) is a vector fnllc-
tion of class e'.
Note that the definition of a C'-manifold has a meaning if only
the manifolds M , and M. are smooth of class not less than e'.
Let I: M 1 -+!vI 2 be a homeomorphism of manifolds. If 1 is a C/-
mapping, the inverse mapping 1-1 need not be a smooth mapping.
Therefore, if the inverse mapping f-l: Iv! 2 -+!v1 1 is also a C'-mapping
the homemorphism 1 is called a smooth homeomorphism of class cr
or C'-diffeomorphism, Diffeomorphisms of smooth manifolds play the

3.1. Tile COllrept 01 0 M allifuld
105
same role as homeomorphisms of opological spaces. If I: M. _!If'/;
is a diffeomorphism, the manifolds ,U. and M 2 are called diffeo-
morphic. The set of all manifolds ill subdivided into non-intersecting
classes of pairwise diffeomorphic manifolds. Any general property of
smooth manifolds, smooth functions or mappings on a manifold can
be transferred to any other diffeomorphic manifold. We shall not.
therefore distinguish between diffeomorphic. manifolds.
There are however such properties of manifolds that their "iden-
tity" for a pair of diffeomorphisms is not quite obvious, In particular,
we have assigned to each manifold a numerical characteristic, the
dimension. Do diffeomorphic manifolds have the same dimension?
Theorem 1. Let I: 1'.1 1 __M. be a C'.homeomorphism (r :1) of
,mooth manifolds. Then dim lIf. = dim Mi'
Proof. Suppose Po EM} is an arbitrary point, Qo = f (Po). and
g = f- I is the inverse mapping. Choose local coordinates (x., .. '. ''')
in the neighbourhood U o of point Po and local coordinates (y.. . . .,
ym) in the neighbourhood V o of point Qo' Then the mappings f and g
can be represented as vector fllndions x = h -1 (y) and y = h (x),
with h (h- 1 (x» _ x and h- I (h (y» = y. Consider the mappings h
and h -1. The mapping h consists of m functions y' = hi (Xl. . . ., x")
of n independent variables (Xl. ' _ _, xn), Compose the matrix of
II 1\ partial derivatives of h'
( .!!!::..  )
az 1 az"
dh=: :,
iJh'" ilk'"
'..8%"
The matrix dh is a rectangular one of order (m X n), i.e. it has m
rows and n columns, The matrix dh is called the Jacobi matrix of the
mapping h,
Lemma 3, Let U o c: R". V o c:: R"', and W o c:: R h be open Euclid-
ean domains, let f: U 0 -+- 1'0 and g: V 0 - W 0 be continuously differ-
entiable mappings, and let h: U o _ W o be the composition of f and g,
i... h (P) = g (f (P», Then the relatiotl dh (P) = dg (f (P» ,df (P),
P E U o . holds true jor the mappings f, g, and h, In other words, the
Jacobi matrix of the composition h = g 0 f is the product 01 the Jacobi
matrices of the mappings g and f.
The proof of the lemma directly follows from the rule of differenti-
ation of a composite function. Let (Xl. . . ., x"), (yl, ' . ., y"'),
and (Zl, ' , ., Zh) be Cartesian coqrdinates in the domains U o , Yo,
and WOo Then
!Ii = /1 (Xl. .... x"),
oi  hi (Xl, .,., x") = III {f! {Xl,
r= gi (yl, ..., y"'),
..., x"), .... f'"(x 1 , ..., X"».

JOG 3. Smooth Manifold. (G"eral Tileory)
Differentiating the functions h J with respect to Xi, we obtain
m
ahi ( I II ) " ogj (I I ( I II' ) all ( ! " )
7jXf x, .." X ='" LJ 1!1" x,. .., x ), '" ;;;T x . ..., x .
1=1
(to)
This formula exactly coincides with the definition of a common ele-
ment of the product of two matrices, i.e. dh (x', . . ., x") =
dg (j' (Xl, . . ., x"), . . .) .df (..1, . . " x"). If a point P has the
coordinates (Xl, . . ., x"), this ,equality can be written compactly
in the form dh (P) = dg (f (P)) .df (P). The lemma is proved.
We now apply Lemma 3 to a pair of mappings hand h- I , Let eo
be the composition of the mappings h and hl and let e 1 be the com-
position of h- 1 and h, i.e, eo (Q)' == h (h- 1 (Q», Q E V o ' and e1 (P) =
h- 1 (h(P», PEU o '
The mappings eo: V o - V o and e 1 , U o -+ U o are both idmtity
mappings, so that their Jacobi matrices are identity matrices of
order m and n, respectively, In particular, rank de o = m and
rank del = n, On the other had, by Lemma 3 we find
de o (Q) = dh (h- I (Q» ,dh- I (Q), Q E V o '
deJ (P) = dh- 1 (h (P)),dh (P)-;- P E U o '
It is known from linear algebra, that the rank of the product of two
matrices does not exceed the rank of each matrix. Since rank dh 
min (m, '1) and rank dh -1  min (m, n) (the matrices dh and dh- I
are rectangular!), rank dc o ,.;;; min (m, n) and rank de t  min (m. n).
Hence, m  min (m, n) and n  min (m, n) or max (m n) 
min (m, n), that is, m = n. Theorem 1 is proved,
The concept of dimension is valid not only for smooth manifolds,
but for arbitrary manifolds as well. A question naturally arises:
do the dimensions of homeomorphic manifolds coincide? The answer
is yes, i.e. il M J andM a are two komeomorphicmanilolds, then dim M I =
dim Ma. This is a fundamental theorem of general topology, hut
i is beyond the scope of our course.
In conclusioll, we shall pro\'e several useful statements COIl-
cerning the structure of an atlas By definitioll, an atlas on a mani-
fold M consists of open sets U I homeomorphic to domains VI c R".
If M is a smooth manifold. the coordinate homeomorphisms <Pi: U I -
Vi are also smooth. It is convenient sometimes to simplify the form
of domains Vi in R", though at the e\:pense of an increased number
of charts in the atlas.
Lemma 4. In a smooth manifold !If there exists an atlas {Ud such
that any chart U I is diffeomorphic to R".
Proof. First, we shall demonstrate that there exists an atlas Hlch
that any of its charts is diffeomorphic to an open ball of radius t
in R", Let Po E!If be an arbitrary poinl. U i 3 Po, and let CPr: U I .....

3.1. The COllrept 01 a Mallil,,jd
,..
107
V, c= R" ue a coordinate homeoniorphism, Qo = rpl (Po). Since VI
is au open st't in R", we can find a number e such lhat the e-ual!
about Qo lies in V" Denote this hall by O. (Qo) and its inverse image
rri' (0. (Qo» by WI" The family of open se15 {WI') is an atlas on M
aud each chart Wl' is diffeomorph'ic to an open hall in R n . To com-
plete tho proof of the lemma, we; shall demonstrate that an e-ball
is diffeomorphic to R". It is sufficient to consider the case e = 1.
Let (x' ." x") be a point in a unit ball, i.e. (xl)2 + . . . +
{.r")2 < I Put
Jt .rk
VI-(.r 1 )2-., -(T")2'
 yk
:1: = /"1+(yl)'+"'+(!I")2 '
Functions (11) and (12) are smooth and map 8 bal! of radius 1 into
R n and vice versa.
Lemma 5. Let M be a smooth compact manifold equipped with an
atlas {fl,). Then there exists a smooth partition of unity 1J>/ subordinate
to the covering {Ud.
Proof. According lo Lemma 4, it suffices to assume that all
charts are homeomorphic to a ball of radius 1. Let <P,: U r -+-D7c=
R" be coordinale homeomorphisms.. Choose a rather small II> 0
such that <pi' (Dr,-e) cover the manifold M. Suppose there exists
on Ihe ball D7 a Coo-function f such thatsuppf=Drl_e)'O
!1. Put l
- { 0 if jJ E UI>
''i(P)= !(<p;(P» if PEU,.
Since f (<p,(P» = 0 for P E <pi' Da....). the hUlclions 1J>/ are smooth
on 111 dnd sUPP ;PI c: U;I O1jiI1. Moreover, supp 1jJ1:::> !Pi' Del_e),
hence the sum ;p;iji (P) = 2:;P1 (P) is strictly positive at each point.
i
We take Ihcm 1J>i (P) =;PI (P)/'iji (P). The functions 'I (P) form a
smoolh partition of unil subordinate to the covering {U I }.
Il remains therefore to construct a COO-function f in R" such that
ils sllpport is equal to the baJJ Dii-e). We shall seek f in the form
I (Xl, . . ., XU) = h «(x'r + . . . + (X")2). Hence, it is sufficienl
to lind a smooth function of one ;variahle It (x) such that h (x) = 0
Cor x> (1 - e)2 and h (x) > 0 for x < (1 - &)2. We shall choose
th!' funclion
(11)
(12)
{ e-I/("-Cl-<I')', x < (1- r)Z,
h(x)=- 0 (1 ) 2
, .r - .
which b k nown \0 be smooth of class Ca>. Lemma 5 is proved.
· Here D': is an r-ball about the origin.

108 3. Smooth Marl/folds (Gen.ral Theory)
Problems
1. Prove that the space of positions of a rigid segment on
a plane is a smooth manifold.
2. Prove that the group SO (3) is homeomorphic to a three-di-
mensional projective space.
3. Describe the configurationa,l space of a system of two hinged
bars in a three-dimensional space.
4. Give an example of a smooth one-to-one mapping which is not
a diffeomorphism.
5. Demonstrate that no alias consisting of one chart exists on
a sphere sn.
3,2. DEFINITION OF MANIFOLDS BY EQUATIONS
There is a conventional way, used most frequently in prac-
tice, to describe and construct manifolds.
In the preceding section (as weli as in Chapter 1) many examples
of manifolds appeared as a set of solutions of a non-linear equation
given in a Euclidean space. For instance, an n-dimensional sphere S"
in a Euclidean space R"+1 is defined by the equation <i t )2 +
(X 2 )2 + . . . + (X"+1)2 = 1; a psudospheYe S: is given by X2 +
y2 _ Z2 = -1. In general, if I (.x 1 , ..., .x n ) is a continuously di[-
ferentiable function, the set of solutions of the equation 1 (X1, . . .,
x") - c = 0 is called a manilold of level c 01 the lunction f. Thus, the
Euclidean space R" is decomposed into a union of level manifolds
for the function I. In the case. of a function of two variables, the
solutions of the equation are called level lines lor t and in the case of
a function of three variables leuel surfaces.
To justify the term the level manifold for a function I, one should
prove that the level manifold for 1 is really a manifold, However,
this is not always the case.
Example 1. Let us consider the function 1 (x, y) = x 2 - y2. Its
level lines are described by the equation x 2 - y2 = c. If c > 0,
the level lines consist of two connectedness components, each being
described by the equation x = v c + y2 or .x = -V c + y2 , i.e.
these lines are the graphs of functions of one variable (Fig. 3.7).
Similarly, for c < 0 the level lines are the graphs of two functions:
y = V .x 2 - c and y = -V x 2 ......l c . Thus, for c =F 0 the level lines
are one-dimensional manifolds (see Sec. 3.1, Example 3). Of special
interest is the level line for c = 0; it consists of two intersecting
straight lines y = .x and y = -x, and is not a manifold. Indeed,
we shall demonstrate that the point Po = (0, 0) on the level line
c = 0 does not possess a neighbourhood U 0 homeomorphic to a Euclid-
ean domain. If such a homeomorphism CPo: U 0 __ V 0 <= RI existed,
we could assume, without loss of generality, that V o is an open

.1.2. Dolint/ion of Mantfolds by Equations
109
interval and U 0 contains ail points P lying from Po at a distance less
than E. Then Uo"'-(Po) has at least four connectedness components.
while its homeomorphic image Vo"'-(q>o (Po» has only two connect-
edness components. This contradiction proves the absence of homeo-
morphism !Po for RI. In a similar way we can verify the absence of
homeomorphism !Po for the domain V 0 c: R" for n  2.
Nevertheless. a level manifold, for a continuously differentiable
function 1 is almost always 8 manifold,
Theorem 1. Let I = I (Xl, . . ., x") be a function. of class C"" defined
in the entire Euclidean space R" and let M. = {(.xl, . . " x"):
I (Xl, . . " x") = c}. II the gradient 01 f is non-zero at each point 01
Figure 3.7
the set Me, thisset is a smooth (n - t)-dimensional manifold 01 class
Coo, and we can choose (n - 1) Cartesian coordinates 01 the ambient
Euclidean space R" as local coordinates in a neigkbourhood of every
point Po E Me.
Proof. The theorem is, in fact, the implicit function theorem for-
mulated in convenient terms. Fix a point Po E Me. Po = (.xl. . . .,
x;). Since
grad p . f =1= 0, grad f = U!I . .... :!,, ).
there exists a non-zero partial dervative at point Po' Wilhout
loss of generality, we may assume :!n (x, ..., .:z:(J) *0. Let Qo=
(x, ,." X-I) be a point in R"-I which is the image of Po under
projection along the coordinate axis x". According to the implicit
function theorem, there exists a 'neighbourhood V o :1 Qo of point Qo,
an interval (x: - li, x + li), and a continuous function y =
y (Xl, . . ., X"-I) of class Coo defined in V o such that:
(a) f(x', ...,X"-I, y(x', .,', X"-I)=C in V o '
(b) x=y(x, ..., xr'),

J 10 3. Smooth Manifold$ (Gtnera! Theorv)
(c) I xg-y(x l , ..., .x"-I) I <0 in lhe domain V o '
(d) any solution (Xl, ..., x")EV o x(,1.g-o, xg+o) of theequa-
lion /(x', ..., x")=c is of lhe form x"=y(x l , ..., X".I). Let V o =
-I en (Yo y, (xR- 0, xH + 6» stand for the neighbourhood of point
Po E Me. This neighbourhood is the chart in (juestioll which
contains 1'0- Take lhe restriction of tho projection R"-.+ R"-I
to V o as a coordinate homeomorphism CPo (Xl, . _ ., x") =
(Xl, . _ ., x'l-I) E V o and defme the inverse mapping qJ1 by the
relation
cp;1 (xl, ..., X"-I) = (Xl, ..., xn-I, y (Xl, .,., x"-I»,
It follows from condition (c) that <p;'(x l , ..., X"-I) EVoX (xB-o,
x3 +6) and from condition (a) that qJ;' (Xl, _.., xn-I) E Me. Thus,
cp' (Xl, ..., xn-1) E V o , i.e. the mappings <Pa and cp;' are conlinuous
and mutually inverse.
We have proved that M. is an (n - i)-dimensional manifold and
found in the neighbourhood of eah point Po E Me a local coordinate
system formed by some Cartesian. coordinates in the Euclidean space
HR. We now prove that the transition functions are smooth. Let PI>
be also contained in another chah VI and let Cartesian coordinates
(.>.1, ..., xi-!, x'+', ..., x") be c.hosen as;o,-:al coordinates in fI,.
Then in the intersection V o n V, the coordimltes (Xl, .,., X'-I,
X'+I, .. ,x") are expressed in terms of (,'1;', ..., xn-I) as
Xt=x',
x;"'=x'-',
xi+l=xi+f,
(1)
X'h..S = Xn-f't
X"=Y(X I , ..., X"-I).
Since the function y = y (Xl, ..., x".') is smooth of class Coo, all
the functions (1) are also of class Coo. This completes the proof
of Theorem 1.
Example 2, Consider an n-dimensional sphere S" given by the
"+1
equation /(:1.', ..., X"+I) =  (:z:1')Z = 1. The gradient of f is
k=1
grad f = (2XI, 2J z , . .., 2X"+I). If ,he point P "" (:t l , ,." X"+I) lies on
S", then not 1111 of its coordinaltS vanish, that is, one of the
gradient coordinates is nOIl-ze!"9. Conditions of Theorem 1 are
satisfied, hence S" is a Coo-manifold.

3 2, Dlillilton 01 Manifolds by Eq"atlolls
111
Example 3, Consider a Euclidean space R"' of dimension 1'1 2 and
represent the points of RR' as square matrices A of order It with the
coordinates A = (ai})' Consider also the sel SL (1'1, R) of all matrices
A E R"' with the determinant equal to unity, det A = 1. The set
SL (1'1. R) is a group with respect to multiplication of matrices and
is called a special linear group. We shaJl demonstrate that SL (1'1, R)
is a Coo-manifold of dimension n - 1. Consider a function I of II
\'ariables, f (a/j) = det (a/j)' This function is a polynomial and is
therefore of class Coo. I n order to' apply Theorem 1, we have to cal.
culate grad I at aJl points of the group SL (1'1, R). Let E be an iden-
tity matrix. Since det E = 1, E E S L (1'1, R). Calculate grad I at
point E. To this end, we express det A as
uetA=a ll detAJ\-a u detA I2 +.., +(-1)" H detA,,;'lIln' (2,
This expression contains the determinants of the matrices AI"
which are polynomials of all variables (ail) except those in the
lirs!. row. Then the parti.!l derivative of I with respect to all is
of the form a dl =,!- (all clel. All) = det All' At point E we obtain
Q]1 (Ia ll
(j ol (Jk) = 1. (3)
all
Thus, I.he gradient of f at point E is non-zero. We now demon-
<;trato that at an arbitrAry point AoESL (1'1, R) grad I is also
Tlon.zero. Introduce new variables bfJ defined by (b IJ ) = B = A;' A =
.4;'. (alJ)' If A=Aol then B=E and
f(A)=/(AoB)=dct (AoB) = det Ao.detB=detAo-f(B).
Difft>rentiating the superposition ,of functions, wo obtain
...!L (E) = '" .!.L (A ). aalJ . (4)
IIb u L.J iJaiJ 0 dbl!
i}
The left-hand side of Eq. (4) is equal to unity according to formula
(:J): hence, at least one of the terms on the right-hand side of Eq. (4)
i< non-zero and therefore one of the partial derivatives d al (Ao),
au
n'l well as grad I, is also non-zero. Thus, the conditions of Theorem 1
are satisfied, so that the group SL (1'1, R) is a smooth manifold of
dimension (1'1 2 - 1),
Theorem 1 can easily be extended to a system of non-linear equa-
tions. Note that Theorem 1 admits another formulation. The grad-
ient of a function t can he represented as a column of partial deriv-
atives of f and is therefore the Jacobi matrix of f. Then the con-
dition of nontriviaJity of gradl at a point Po E R" is equivalent
to the condition that the rank of the Jacobi matrix df of the function f
i I!qual to unity, i.e. is maximal.

112 ,. Smooth Manifolds (Conoral Theory)
Let there be given a system of equations
I'(x l , '0" x n ) =c l ,
F(z', ..., x") =rfl-,
(4')
/'(x l , ..., x")=cR,
whiclt can be written in compact form as t (x) = C, where x =
(Xl, 0 . ., x n ) E R", C = (c1, 0 0 0' c A ) E R\ and f is a mapping de.
fined by the functions (fl, ' 0 "jA). The set Me of solutions of system
(4') is called a level manifold for the system of functions (fl, . . ., fA).
Theorem 2, Let t: R" _RA bl: a C""-mapping and let Me be a set
t solutions of the system of equations t (x) = c. It the rank of the Jacobi
matrix of f is maximal at every point Po E Me (i.e, rank dt (Po) = k),
Me is all. (11. - k)-dimensional smooth manilold of class COG, and
(n. - k) Cartesian coordinates 01 the surrounding Euclidean. space R n
can be taken as local coordinates in the neigkbourhood of each point
Po EMc.
The proof of Theorem 2 is exacUy analagous to that of Theorem 1
with the only difference that instead of one variable x n we choose k
variables (xi., . . 0' Xih). Denoting this group of variables by one
.symbol, say y = (xl., . 0 " xlh), we obtain the same formulas as
in Theorem 1-
Example 4, Let us consider in a Euclidean space RC with the coor-
dinates (xt, x\ x3, r') the system of equations
(Xl)' + (z2)' = 1, (x3). + (xC). = t. (5)
The corresponding functions f1 and f2 are of the form
f1 (Xl, x', :ii, XC) = (X 1 )2 + (x 2 )", j2 (Xl, x., :ii, xC) = (x 3 ). + (r')2.
In order to apply Theorem 2, we shall calculate the Jacobi matrix
.of the mapping t = (fl, I')
, ofl of l Oil of I )
I 8.<' a;?E" ""JI3 ox' ( 2X I 2x' 0 0 )
dl = t af' af' of' of' = 0 0 2.z3 2.x4 0
,8.<' o:r.- a;o 7i7
Clearly, rank dt  1 if onl, one of the rows of the Jacobi matrix is
zero, wbicll is impossible at the points representing solutions of
system (5). Thus, the solutions of this system form a two-dimensional
manifold of class COG. Since system (5) splils into two equations,
each for its own group of variables, the set of the solutions of this

3.3. Tangent Vectors. Tang.nt Space
i I.j
system can also be expressed as the Cartesian product of the solutions
of each equation, i.e. the solutions of system (5) are represented as
the product of two copies of a cirde. This manifold is called a (two-
dimensional) torus.
3.3, TANGENT VECTORS. TANGENT SPAr:E
In Chapter 1 we have sen that the so-called infinitesimal
properties of space are of great help in the study of metric properties
of curves ami surfaces and, generalJy, of metric properties of domain
in a Euclidean space, These are the properties defined in a very small
neighbourhood of a fixed point P by neglecting small quantities of
an order higher than the distance from P. In mathematical analysis
we use a similar procedure of neglecting infinitesimal quantities
while studying the behaviour of a function in a neighbourhood of a
point. In the case of smooth manifolds there is also a natural desire to
neglect infinitesimal quantities. One of such methods is 10 introduce
special concepts analogous to tangent vector to a curve and tangent
plane to a surface.
3.3,1 SIMPLE EXAMPLES
Let us consider a smooth curve x = x (t) = (Xl (t),
: (t), :iJ (i» in a three-dimensional space R3, where t is a parameter.
Fix a value to and expand the vector function x = x (t) in a Taylor
series about the point to
x (to + t) = x (lo) +  (to) t +0 (l2), (1)
The first two terms on the right-hand side of Eq. (1) may be con-
sidered, on the one hand, as an approximation of x (t) in a neigh-
bourhood of the point to by a linear vector function. On the other
hand, this linear function y (t) = x (to) + dto) ,(t) defines
in R3 a straight line through the point Po = x (t G ). Furthermore.
among all straight lines through Po the straight line y (t) "approaches
most closely" the initial curve x (i). Let us first explain the term:
a straight line "approaches closelY," a curve x (t). We say that the
straight line y (t) = a '7' bt (I b ] = t) is tangent to a curve x (t)
at point x (to) if the distance from the point x (t) to the straight line
is an infinitesimal quantity in comparison with the distance from
Po = x (to) to x (t). The point Po lies then on the straight line y (t),
We may asume x (to) = Y (to), so that y (to -;- t) = x (to) - bt.
The distance from the point x (to - t) to Y (I) is
I x(to+.it}-x(to)-b(x(loTt)-.X(to)' b) I
= 0 (I x (to.J.. t) -x (to) I), (2)

11 4 3. Smooth Manifolds (GeMral Theorf/)
Suppose  (to) '* O. Expanding x (I) by formula (1), we obtain
instead of Eq. (2)
I ; (to) 6t- b ( : (to) !!.t, b)! = 0 (6t 2 )
for 6t -I- 0 or, dividing by !!.t,
I  (to) -b (  (to), b) 1=0 (M). (3)
Since the left-hand side of Eq. (3) does not depend 011 !!.t, we
obtain by a limiting process for 6t -+ 0
 (to)=b(  (to)' b), (4)
which means that the vectors band : (to) are collinear, Thus, the
linear part of the Taylor formula (1) for the vector function x (t)
1
x
I igure 3,8
z
x
Figure 3.9
defines a parametric representation of a tangent straight line at point
Po (Fig. 3.8). Consider now a surface M in a three-dimensional space
R3 given in parametric form as a v,ector function x = x (u, II) of two
independent parameters u and v. The surface x (u, v) is called non-
degenerate if at each point the partial derh'atives :: (u, v) and
 (u, v) are linearly independent as vectors in R 3 . Fix parameters
(uo. v o ) and a plane TI, x = x (uo, v o ) + au + bv, through point
Po = x (uo. 1)0) on the surface. The pi ane n is called tangent to
the surface !If at point Po if the distance from TI to tho point x (u, v)

8.8. Tangenl V«c1ors. Ta"g«nt SptJtt
115
is an infinitesimal quantity in comparison with the distance from
x (u, v) to Po' Expanding the function x (u, v) in a Taylor series
about the point (u o , vo)
x (uo+ u, 110 + II) = X (u o ' va) + : (u o . 11 0 ) u
{Ix
+-a;-(u o , vo)v+O(6.u2+v2), (5)
we nnd that the linear part in (5) defmes a two-parameter repre-
sentation of the tangent plane IT 'to the surface M at point Po =
x (uo, vo) (Fig. 3.9). It is natural to caIl any vector emerging from Po
and lying in the plane n a tangent vector to the surface M at point Po.
I t can be seen from Eq. (5) that parametric representation of a plane
11 tangent tQ a surface M at a point Po is of the form
) ax ) 
x (6.u, v)=x(uo' va +a;;- (u o , 1.'0 u+8V(uo' v o )6.v. (6)
Hence, any tangent vector S can be decomposed into a lin('ar com-
bination of the vectors :: (uo, vo) and  (un, v o )
ax ax
s=au-(uo' Vo)ll+a;(Uo' vo)l.I (7)
for an appropriate choice of the parameters u and v. Thus, the
vectors :: (u o . vo) and : (Uo, v o ) form n basis in the <angent
plane n, and u and 6v are linear coordinates of the tangent vector S
iu this basis.
Let us draw a smooth curve x = x (t) through point Po on a sur-
face 11l. Since the curve x = x (t) lies on the surface M, iL can be re-
presented parametrically as the composition
x (t) = x (u (t), v (t» (8)
of functions u (t) and v (t). In other words, the functions u (t) and
t. (t) define parametrically a curve in a local coordinate system (u, v)
on the surface M. Then, the condition that the curve passes through
point Po can be expressed as the condition for the coordinates:
1J 0 = u (to), va = v (to)' Calculation of a tangent vectOI' to a curve
(or, as it is frequently called, the velocity vector of a curve) yic1ds
:: (to) = :e (x(u(t), V (t)))I/=I,
b ) du b 
= 8U (u (to), v (to) dt (to) + iiV (u (to), l.I (to» dt (to)
ax du ) Ox ) dv (
="';);:;""(u o ' v o ) dt (to +a;(uo, 110 di" to).

116 3. Smo o!h Manilolds (Gtnr(l.1 Theorll)
Hence, a tangent vector (0 a curve on a $Ilrface "'I lies in the tan-
gent plane.
O L 1: ax ax
efinltion. et!:> ="7fu" (u o ' VO);I + 7jU (uo' vo) 2 be a tangent
"ector to a surface M at a point Po. Then the numbers (1, ) arc
called coordinates of the tangent t'ector S to fir! at point Po il! a local
coordinate system (u, v) on the .urlace M.
This definitiQn works not only for the coordinates (u, t.) dest"ribing
parametrically the surfllce M, but also for IIny coordinate system
(u', v') in II neighbonrhood of point Po. Indeed, if (u', ") is some
other coordinate system, the coordinates u IInd v lire expressed liS
smooth functions of u' and v': u = U (II'. v'), V = V (It', v'), 110 =
U (u, v), 'o = v (u, t')' Considering then compositions of the
functions, we obtain a new flarametric defmiUon of a surface "1
x = x (u (u', v'), v (u', 1"». (9)
The parametric equation of II tllngent plane n al point Po for
the new parameters (u'. to') is
( ( ' ' ) ( ' ' »+ ,ax 6. ' + {Ix 6 '
x = x u u o ' v. . v 110' v. all;' u _&,.;, v
ox all, ax ( ,1'"
=:\ (uo. v o ) + a;; (u o ' '0)"FU'!!.1l + tji;"" liD. vo) 7h7' 6(1
+ : (uo' vo) ;, 6.v' +  (u o ' vo) ;,:' 6v'
= x (uo' v o ) + :: (Ilo. v o ) ( ::' (II, v;) 6u' + ;!, (u, v.) 6.L")
+ ax ( ) ( av ( ' ' ) '" at' ( ' ' ) "' )
--;J;;' uo. V o 7};7 u o ' V o c.>U + 8i7 110' V o c.>V .
Assuming
.. au ( ' ' ) '" Oil. ( ' ' ) '"
uU= W 110' v o uU +w Il.. V o uV.
.. 0" ( ' . ) '" au ( ' . ) '"
uV = 7ii7" u.' L'o uli + a?" Ii.. v. uV .
we arrive at the parametric definition of a tangent plAne for the
in i tia I paramt!ters (Ii, v)
x=,,(u o . v o )+ : (u o . v o )6u+ ;:v (u o , v o )6v.
( 10)
TIIt1. formula (H) is auothl!r parametric dl'rmition of the urfRcl' .\f
with the snme langent plaut' n. Curve (8) can therefore be written
in terms of the functions u' (I) and /" (i) in slIch a \\'a' that x (t) =
x (u (II' (I). I" (I», L' (u' (I), 1.' (i»). Then. according to the dl+
initiol! of the coordinatl!S of a tangent "I!clor to a curve in a local

9.8. Tangent Vectort, Tangent Space
117
coordinate system (u', v'), the numbers ( . (to), ' (to») are the
coordinates of a tangent vector to a curve Differentiating the com-
posite functions u and v, we obtain a relation between the coordinates
of a tangent vectOr to a curve in different local coordinate systems
au ( t au ( ' ' ) du' ( t ) + au ( ' ' ) dv' ( )
1ft 0) = a.? u o . va TOw u o ' V o T to ,
du ( t ) OV ( ' ' ) du' (t \ + Ov ( ' . ) dv' ( )
di 0 = w Ua' V o dI Of 7iiT U o ' V o (it to .
(1 j)
Comparison of (10) and (11) shows that this relation coincides with
lhat for parameter transformation in the definition of a tangent plane.
The above definition of a tangent vector to a curve on a surface M
is convenient in that the vector coordinates depend only on a local
coordinate system (u, II) on M rather than on the way the surface M
is embedded in a three-dimensional space R 3 . This statement can be
formulated as the following lemma.
Lemma 1. Let M be a non-degenerate surface in R3, Po E M, (u, v)
a local coordinate system in a neighbourhood of point Po on M, and
(u (t), v (t)) a smooth curve on M. Then the tangent vector to the curve
at Po has in the local coordinate system (u, v) the coordinates ( :: (toJ,
 (to»). If x = x (u, II) is parametric definition of M and x (t) =
x (u (l), v (t» is a curve, then
 b   
di(t o ) =8j£ (uo, "0) iit (to)+av (uo, vO)(ii" (to).
3.3.2. GENERAL DEFINITION OF A TANGENT VECTOR
The examples considered above show that it is more con-
venient to study the infinitesimal properties of a curve on a manifold
\Ising a local coordinate system, In particular, very important con-
cepts in geometry are a tangent vector and a tangent space of an
arbitrary smooth manifold, in complete analogy with the corre-
sponding concepts for a surface in an ordinary three-dimensional
space R3.
Definition 1. Let M be a smooth n-dimensional manifold and
P a E M an arbitrary point. A tangent vector t at point Po to the mani-
fold M is a correspondence which associates with any local coordinate
system (:1:\, . . ., xr) a set of numbers m, . . ., r) satisfying the
following relation fOf each pair of 10(,111 coordinate syslems:
It n o,zt I
Si =  iJz t . (Po) j.
11 J
(12)

118 8. Smooth Mallitoids (General Th.orl/)
The numbers (I. . . ., £r) are called coordinates of the tangent
vector S in the local coordinate system (:r, . . ., xr> and relation
(12) is called a tensor law of coordlnale transformation for the tangent
vector f,.
Definition 1 is a generalization of the concept of the coordinates
of a tangent vector to a curve on a surface. The law (11) of the trans-
formation of these coordinates is a particular case of the tensor law
(12) of the coordinate transformation for a vector tangent to a mani-
fold. Moreover, any smooth curve on a smooth manifold is endowed
at each point with a tangent vctor in the sense of Definition 1.
This important property can he formulated as the following prop-
osition.
Proposition 1. Let M be a sm(foth manifold and y: (-1, 1) -- M
a smooth mapping of the interval (-1, 1) Into M. Then the corre-
spondence which associates with each local coordinate system (Xl, . . ., x")
in a ntiighbourhood of point Po = Y (0) the set of numbers
( dZ 1 dO:" )
dt (y (I)), . . ., at (y (t» t is a tangent vector in the sense of Def-
Inition 1.
To prove Proposition 1, it is sufficient to verify the tensor law of
d.1: I
coordinate transformation ( 12). Assume  J  = _ d ) (y (t)) ,where
t 1=0
(xj, . . ., xj) is a local coordinate system on M in a neighbourhood
of point Po' Then we obtain for two local coordinate systems
£1= 1t x (,\,(t»I,=o= :t .x1(x} (1' (t», ..., xj(y(t)))I,o
"iJ% dIn 8z I
=  iJz (V(t» iUxj(y(t»ll=o=  iJo:'. (POn;,
1=1 1 1=1 )
This is the tensor law (12), The proposition is proved,
It is therefore natural to calI the correspondence tlsed in Pro-
osition 1 a tangent vector to a curve y or a velocity vector of a curve y.
d '
The tangent vector to a curve I' 'Will be denoted by aJ (to) or y (to)'
3.3.3. TANGENT SPACE Tpo 1M)
The set of all tangent vectors to a manifold M at a fixed
point Po is called a tangent space to the manifold 111 at potnt P".
This set Is denoted by T P. (M). Each tangent vector t E T p . (111)
is uniquely defined by its coordnates in a fixed coordinate sstem.
Indeed, suppose we are given a set of numbers (11'. . . ., 11 ) and
assume this set to be the coordinates of a tangent vector in question
I n a fixed local coordinate system (xl., . . ., x?,), i.e. '1]" = t.

3.3, Tangent Vectors. Tangenl Spae<
11'1
In order to defme It tangent vector, we have to find its coordinates
in any local coordinate system (x:, . . " xn. Let 1IS put
 n az' I
61 =  (Po)T).
ax.
l.lC:lt 10
These coordinates must satisfy the tensor law of coordinate trans-
formation (12). To verify this, we substitute s and ;} into formula
(12)
n az I n az "Dzj
 a;r-(Po)T) =  -.;7(Po) 7(Po)'I]1
lc:t \0 ,=1 1 lClt 10
=  ( , ax! (Po) d} (PO» ) 'I]I.
lr;:tl .$C= ox} 8x t ,o
a", n Ozh liz
Since  (Po) = :z -+ (Po) -+ (Po) (the law of transformation
az l , .<=1 Ii"'j IiZ j .
of the Jacobi matrix under triple coordinate transformation), rela-
tion (12) is satisfted identically.
We have demonstrated that the set of all tangent vectors to a
Inanifold M at point Po is uniquely determined by the coordinates
of these vectors in one fIxed local coordinate system. Hence, the
entire tangent space Tp, (M) can be identified with the arithmetic
vector space Rn. This means that T p . (M) can be endowed with the
structure of a linear space. Seemingly, the structure of a linear space
in T p . (M) should depend on the local coordinate system in a neigh-
bourhood of point Po' This is not the case, however.
Proposition 2, Addition of vectors and multiplication of a vector by
a number in a tangent space Tp. (M) do not depend on the local coor-
dinate system on M in a neighbourhood of point Po.
Proof. Let S. .... be two vectOrs in T p . (M) and (x. ..., x),
(x}, .... x1) two local coordinate systems on M in a neighbour-
hood of Po Suppose (61. ..., !). (11;' .... 1]f) are the coordi-
nates of the vectors S. 1'1 in the system (xl. ..., xi') and (s;. .,..
1), (',,), ..., 111) are the coordinates of the same vectors in the
system (x}. .... xi). By the tcn'sor law (12) of coordinate trans-
formation for tangent vectors we have the relations
k  oJ: I
61 = L.J o,l (Po) j,
II I
k n Ii% I
1')1 = :z ---y (Po) 'l]j,
1=1 aZ j
(t3)
(14)

120 3. Smooth Man.jold8 (GeJleaZ TM.Y)
Termwise addition of these relations yields
h h n az I I
(j +111)=  a;\(P o ) (£i+fli)'
,! }
which means that the set ml +fll). ..., (£r + fJr» obey the ten-
Sor Jaw of coordinate transformation, i.e. derIne the same tangent
vector without any reference to a local coordinate system.
Simi1arly, multiplying (13) by a number A, we obtain A£t =
 oz (
L.J """8T (Po) ("j), i.e. the set of numbt'r (") also obeys thl' ten-
rc:::1 Xj
sor law (12). The proposition is proved.
This tensor Jaw of coordinate transformation may be considered
as a method of identifying arithmetic spaces of the coordinates of
tangent vectors in any local coordinate system. The method lies in
mulliplying the coordinate column aJ) by the lacobi matrix of the
transformation from (x}, .., xj') to (x:. ' , ., x)
ax: az 1
-a;f" . . . --o:;.!
} )
I
,]
=
,7
az i1z
az! ... i1z'!
} )
,7
or(£) = ( ::; ) (;j).
Hence, the tangent space T p .. (M) is a space isomorphic to all
arithmetic spaces of the coordinates of tangent vectors.
3.3.4. SHEAF OF TANGENT CURVES
In previous subsections we have introduced the formal
algebraic definition of II tangent space to a manifold. This definition,
however, does not reflect an obvious geometric property of a tangent
vector-linear approach to a curve. How can we define this propert'
without any reference to the position of a manifold in II linear space?
A vector may correspond to many curves which it touches. Thus,
we have nothing to do but choose such classes of curveS which have
a common tangent vector in ]inear spaces.
Definition 2, Two curves 1'1 and 1'2 on a manifold M intersecting
at a point Po are called tangent jf in each local coordinate system

3.11. Ta"gent Vectors. Ta,ngent Space
12f
(,tl, :z;") in a neighbourhood of Po we have
"
 (:z:'I ("I', (t»_Xk (Ys (t»)2= o (t-t o )2 for t-to. (15)
k=1
As before, it is sufficient to verify the tangency condition (15) only
in one local coordinate system. The tangency condition is closely
related to tangent vectors. In particular the following theorem justi-
lies the term "tangent curves".
Theorem 1. Two smooth curves "1'1 and "1'2 on a manifold M are tangent
at a point Po if and only if their tangent vectors at Po coincide.
Figure 3,10
Proof. Condition (15) can be rewritten liS
"
lim  ( zk ('VI (t» -x k ('Vs (t)) ) 2= O.
I-I. k..l t-t o
After elementary manipulations we obtain
" 2
 (  xl' (1'1 (t» - :t :z;k ('Is (t))) L=" = 0
k=1
or
:e xk (Yo (t» II=t. = :, xl' (Ys (t»lt..t.,
The last equality exactly means that the tangent vectors to the
curves 1'1 and Ys coincide, 1 (to) = .vB (to)'
Conversely, if;'1 (to) = "S (to), wa have
lim  ( Xk ('i' (l))-zk ('is (t» ) 2
I-f. k=' 1-'0
=  ( :, Xk (Ydt» - :t :z:'I ("1'2 (t»)2 =0.
k=1

122 9. Smooth Manlfold$ (General Theory)
Theorem 1 gives an alternative dermition of a tangent vector 10
a curve. The set of all smooth clin'es through a given point Po on
a manifold JI[ splits into disjoint classes of pairwise tangent curves
Let the class of tangent curves through 1\ point Po EM be called
a tangent vector. We have then 1\ one-to-one correspondence between
tangent vectors in the sense of Definition 1 and in the sense of classes
of tangent curves (Fig. 3.10),
3.3.5. DIRECTIONAL ERIVATIVE OF A FUNCTION
There is one more way of representing a tangent vector
on 1\ manifold M. We shall start, as Ilsual, with a simple example.
Let f (x, y) be a smooth function of two variables, Po = (.To, Yo)
z
Figure 3. tI
a point, and  = (I, ;2) a vector in Q plane R2. In mathematical
analysis we often deal with the derivative of a function f with re-
spect to n vector., which is defined as
t(t) =  (xo. YO)SI + g: (x o ' YoH 2 .
(16)
The derivative with respect to the vector S can also be defined in
terms of II smooth curve. Let y (t) be II smooth curve through a point
Po on R2. Suppose the tangent vector to y (t) at Po is i,. \'\'e have
then (see Fig. 3.11)
 (/) = , f (y (t)) !II.' 'V (to) = Po' (17)
Indeed, if l' (t) = (x (t), Y (t)), then S1 =  (to), 52 =  (to). Substi-
tuting the coordinates of the curve y (t) into the arguments of f

9.9. T(mgent Vclor8. rangenl Space
123
Rnd differentiating with respect to the parameter, we obtain
d d
(it f (y (t» I," = dT f (x (t), y (t» I h...,
af ( ( ) ( » d" ( ) + af ( ( dll
= 8% x to , Y to dt to a; x to), Y (to» dt (to)
:= ; (x o ' Yo) Sl+ : (X o , Yo) S2= (t).
We now define differentiation of a smooth function at a point Po
with respect to each tangent vector  to a manifold M at Po EM,
Definition 3. Suppose Po E M, % E T p , (M), Y (t) is a smooth curve
through po. y (to) = Po,  is its tangent vector at Po, ;. (to) = ,
and f is a smooth function on M. The derivative
d
di f (i' (t» 11=1, = S (I)
(18)
is called the derivative 01 the lurn:tion 1 with respect to the tangent
Ilector \. Calculation of the derivative is called the differentiation of
the function with respect to the vector .
Theorem 2. Let (xt, . . .. x") be a local coordinate system in a neigh-
bourlwod 01 a point Po = (x:, . . ., xl;) of a rrw.nifold M, let S =
(Sl, . . .. sn) be a tangent vector to M at Po, and let f = f (xl, . . ., x n )
be a srrwoth function in the neighbourlwod of Po represented as a function
of the local coordinates (xl, . . '. x"). Then
n
% (1) =  a, (x. ..., x&n i .
1_1
(19)
Hence, the definition (18) of the derivative dces not depend on the choice
of a curve in the class of tangent curves, and the right-ha(<ll side of (19)
does not depend on a local coordinate system, If g is arwther smooth
function in the neighbourlwod of Po, the product of f and g obeys the
Leibniz formula
s(fg}=f(x, ..., xH(g}+s(f)g(x. .... x), (20)
Proof. We represent the curve i' (t) through Po in the coordinate
form: y (t) = (Zl (t), . . ., x n (t». By the definilion of a tangent
rb;1 I
vector to a curve, we have dt (to) =  , whence
S (f) = :1 f (y (t» Ilcl, = :t f (Xl (t), ..., x n (t» 11""1,
n n
=  at" I (x, ... I x) ' 11=1, =  :1, (xA, ... I .x) £1,
f-I I-I

i24 9. Smooth M(mlfoliU (Gentral Theory)
Formula (19) is proved. To prove formula (20), we use (19) and <lii-
ferentiale the product of functions to obtain
n
(fg)= 2} iI' (/(xl, .oot xn)g(x l .
1=1
x" ») ki
xR!I:'xo
=  ( ::, (x, .... xo) g (.r, oo', xo)
1=1
+ J (:c, .... :Co) ::1 (x. -... xo) ) ;'
= (I ::' (.rA. .. -, xo) ;i) g(x. ,.., xo)
-I- 1 (x. ,.., x'O) (  ::; (x. .... xoH' )
1=1
= i (I) g (xA, ..., xo) + J (xA, .." xo) i (g),
Thus, formula (20) is also proved.
Differentiation of a smooth function 1 with respect to a tangent
vector \ is characterized by certain properties which do not invoh'e
any local coordinate system, This, way of presentation is most con-
venient to demonstrate the independence of our geometric construc-
tions of a particular local coordinate system. There are two such
properties:
(8) differentiation with respect to a vector S is linear, i.e. if j
and K are smooth functions, and A and jL are arbitrary numbers, then
s ('AI + ILK) = 'Ai (I) + ".\ (g),
(21)
(b) differentiation with respect to a vector \ satisfies the Leibniz
formula (20).
We may give a general definition.
Del'anltlon 4, The operation A which associates with any smooth
function I of class Coo on a smooth manifold M the number A (f)
satisfying (a) and (b) is called differentiation at points Po E M.
In the Leibniz formula (20) the values of functions are calculated
at a single point Po, so that differentiation at distinct points Po
and PI need not coincide. Obviously, differentiation with respect
to a tangent vector S is a particular case of differentiation in the
sense of Definition 4. and there are no other differentiation opera-
tions. This means that for each differentiation in the sense of Defini-
tion 4 there exists a tangent vector with respect to which the func-
tion is differentiated.

3.3. Tangent Vector,. rangent Space
125
Theorem 3. Let II-! be a Coo-manifold, Po E M be an arbitrary point.
and let A denote differentiation in the sense of Definition 4. Theil there
exists a unique tangent t'ector S at Po such that A (f) = s (I) for allY
smooth function f in a neigkbourhood of Po.
Proof, The tangent vector will be songht as a column of its coordi-
nates in some local coordinate system (Xl, . , ., x") in a neighbour-
hood of Po' Then any smooth ftlnction will be represented as A ftlnc-
tion of the variables (xl, . . ., x n ).
Lemma 2. Any Coo-function f (.2'1, . . ., x") can bt> represented in
the lorm
" I ( I ,, {Jf ( I
(Xl, .... x)= XD. .... xO)+£.J 8%1 xo,
i1
xl]) (Xi_X)
"
+  hI} (Xl,
i.,=1
x n ) (Xl_X:) (xi -xt),
(22)
where ho (Xl. ..., x") are COO-junctions.
Proof. Write the identity
f (Xl, .... x") = f (xA. ..., xo)
I
+  ;t I (xA+t(xl-x), ..., x'+t (x"-xo}) dt
o
and differentiate it with respect to t nnder the integrAl sign
f (Xl, ..., x") = f (x. ..., xo)
1 "
+   :, (x+t (xt-xA). .... xo+t (x"-xo» (xl-x) dt
o i=1
n
=f (xA. ..., :I:) + 2.: (xi-x)hl (Xl, ..., x").
1=1
(23)
In the last equality the functions
I
II, (Xl. ..., x") = r 4(x+ t (xt-x),
,) iJ.
o
are smooth of class Coo. Subslitution of xi = x: yields
Xo +t (x"-xo» dt
I
hdx:, .. , xo) =  :;1 (xb, .... xl]) dt = :!f (xA, ..., x'J). (24)
o

126 9. Smooth Man/tola. (Gerural Thoorll)
Applying formula (23) to the functions hi (Xl, . . ., .:In, we obtain
"
hi (zl, , . .. x") = hi (x, .... x) -I-  (xi -) hi} (xl,
J=I
.. -,
.:r;"),
(25)
where hI) (x1, , . " x") are Coo-functions. Substituting (25) into (23)
and taking into account (24), we arrive at the initial representa-
tion (22).
Lemma 3, Let f and g be smooth junctions on a manilold M such
that f (Po) = g (Po) = O. Then jor any differentiation A at Po the
equality A (fg) = 0 is 8ati8fied,
The prool of Lemma 3 directly follows from the Leibniz formu-
la (20).
Let us now turn to the proof of Theorem 3. Represent I in the
form of (22) and apply differentiation A to the left-hand and right-
hand sides, Since the operation is linear, we have
"
A (j)=f(xA, ,.., x) A (1) +  :;1 (x.
1=1
.. -.
x) A (x l - x)
"
+  A({XI_X) (xi-x) h'j (Xl, ..., x"». (26)
I,J=I
Note that for a constant unit function we have /. (1) = A (1-1) =
A(t).t + 1-A(t) = 2A(1) = O. In the last sum of (26) each term
can be represented as the product of two functions (.xi - xJ) and
(xi - xi) hi} (Xl, ' . ., x"), each vanishing at Po E M. Thus, b
Lemma 3, A «Xl - x ) (xi - .xt) h ij (Xl, . . ., x"» = O. Hence,
"
A (I) ""  :;1 (x. .... x) A (xt-x),
it:::ll
(27)
Since in formula (27) the function f is arbitrary. by putting t =
A (Xl _ x), we obtain the vector t = (L, .., s") sLlch that
A (1)=  :!f (x. ,.., x) Sf =s(f),
i-I
We now prove that the vector S is unique. If we could find two
distinct vectors t and '11 such that A(I) = t (I) = fI (1), then assum-
ing b = t - 1] :;i= 0 we would obtain  (I) = 0 for any Ceo-function
j in a neighbourhood of Po EM. This cannot be the case, however.
Indeed, in the local coordinate system (Xl, _ _ ., x") the vector 
has the coordinates (b l , . , ., b") not all of which are zero. Let

8.8. Tangent V.clOT.. Tl2nBent Spat.
127
k =;6 O. Then we obtain for the function I (Xl, . . ., x") Iii" x"
n n
b (j) =  :11 (x, .... ,1:H" = 2} ::: b" = b" =;6 O.
il:r1 {:::::Ii
This completes the proof of Theorem 3,
The theorem establishes a one-to-one correspondence between
tangent vectors to a manifold M at a point Po E M and differentia-
tion of a smooth function at Po' We can therefore formulate a third
equivalent definition of a tangent vector: a tangent vector is a diffe-
rential operator applied to a smooth function at point Po of a ma-
nifold M,
Calculation of a partial derivative in a local coordinate system
(.1:1, , . ., x n ) may serve as an example of differentiation of a smooth
function. By Theorem 3, the operator a:,. is a tangent vector with
the coordinates (0, . . ., 1, . . .. 0) where 1 is the kth coordinate.
Thus, the tangent vectors { a:" } 10rm a basis in the tangent space
T p (M), and any tangent vector f. = (Sl, . . ., n) can be decom-
posed into the linear combination t = ;1 a:' + . . , + n iJn .
This convenient representation will often be used below,
3.3,6, TANGENT BUNDLES
We bave already seen. that the set of all tangent vectors
T po (M) to a manifold M at a point Po is a linear space of the same
dimension as that of M, In geometry it is sometimes useful to study
the whole set of tangent vectors to a manifold M, which can, ap-
parently, be represented as the union U T po (M). This space (not
PoEM
yet topological) is denoted by T (M) and called a tangent bundle of
M. The term bundle means that T (M) consists of fibres-tangent
spaces T po (M) to distinct points Po of the manifold M. A tangent
hundle is by no means a vector space, for it is meaningless to add
vectors belonging to different fibres, If, for example, a manifold M
is a two-dimensional surface in R3, then T (M) represents the union
of all tangent planes to M. It should be noled that tangent planes to
a surface usually intersect, that is, they have common points Accord-
ing to the defmition of T (M), however, these points in each fibre
cleline different vectors, since they originate at distinct points.
Let \1S consider a circle 8 1 c: R2. Figure 3 12 depicts two tangents
\0 the circle at points Po and Qo and two tangent vectors G and 1)
with common end. Therefore, in orller to describe the tangent bundle
T (8 1 ) as a topological space in a, Euclidean space, we have to go
over to a three-dimensional space R3 and "fotate" the tangents through

128 3. Smooth. Manl/old$ (Genera' Tktorv)
.an angle relative to the plane (x, y) in such a way that they
do not intersect (see Fig. 3.13). The tangent bundle T (st) become,
then a one-sheet hyperboloid (geometric cylinder). Under such an
<Iperation the propert' of "close approach" of T PQ (SI) to the circle is
y
Figure 3,12
x
Figure 3.13
lost. The situation is as if we have "torn off',1..the tangent to the circle
and have disregarded that the fibres must be tangent to Sl. We may
s1ightly modify the embedding of the tangent hundle in R 3 , so
that T pQ (8 1 ) remains tangent to 8 1 and different fIbres do not inter-
Figure 3,14
Figure 3.15
sect. Tl\is cannot be done on the entire circle 81, but only on its
part, an arc. Let us embed an arc in R3 as a helix. Then tangent, 10
the helix do not intersect (Fig, 3.14).
Here are some examples from mechanics which demonstrate that
non-trivial manifolds and tangent hundles to them are convenient
in the description of mechanical systems.
Example 1. Consider the motion of a plane pendulum, i.e. a rigid
bar hinged at a point (Fig. '3.15). The posilion of the bar is deter-

3.4. Sbma"lfoltU
129
mined by one parameter, the angl!! q> between the bar axis and the
vertical. Thus, the set of all positions of the bar is a circle 8 1 . Such
a set is called a configurational space.
Consider a two-link compound pendulum, i.e. two bars pivoted
together (Fig. 3.16), The position of this pendulum is determined by
two angles q>1 and Ip!l' so that the set of a11 positions represents a two-
dimensional torus T2 = 8 1 X S1. Figure 3.17 depicts another
system whose configurational space is a torus embedded in R3.
Example 2. In mechanics the motion of a mechanical system is
usually described by the parameters characterizing the position of
FIgure a,16
FIgure 3.17
the system and by the v(.locities of its parts. The set of all these
positions and velocities is called a phase space, which can naturally
be identified with a tangent bundle to a configurational space. For
instance. if a particle moves along a two-dimensional sphere at a
constant velocity, the phase space is a subset in a tangent bundle
consisting of tangent vectors of constant length.
Example 3, There exist more complicated configurational and
phase spaces. Let us consider, for instance, a three-dimensional rig-
id body with a fixed point Any position of this body in R3 can be
described as follows. Choose in the body three orthogonal unit vec.
tors e 1 . e 2 . and e 3 emerging from the lixed point. Any position of
the body is given then by the position of these three vectors in R 3 ,
Thus, the configurational space can be identified with a connected-
ne$ component of the set of all orthogonal unit bases in R 3 .
3.4. SUBMANIFOLDS
We now turn to differential calculus on a smooth mani-
fold. Many important concepts of mathematical analysis, such as the
differential of a function, critical points, implicit functions, can

I.JO 3. Smooth Mant/olds (Gerlero./ Theory)
lIaturally he extended to IIn arbitrary smooth manifold. Within
thE' framework of a generaltheory'of smooth manifolds we clln easily
interprN many concepts of differential caknills: the differential of
II fllnctiOIl. the gratiient of a function, Hw implicit function theorem,
regular points of II function, ('te.
3,4,1. OlfFERENTIAL OF A SMOOTH MANIFOLD
Thl' roncept of the differential of a smooth manifold can
r(,oHiily be lr.\I\ferred to an arbitrary smooth mapping of manifolds.
DefmiUon 1. '...et I: M) ->-111 2 be a smooth mapping of smooth mani-
fold!;' The differential dl po 01 a smooth mapping I at a point Po E
!If) is II linear mapping of a tangent space T po (M I ) into a tangent
pace T Q. (iI-I ), Qo = I (Po), dl'fmed ill local coordinate systems
hy the Jacobi matrix of I. .
Recall that in Sec. 3.1 we defined the Jacobi matrix for II system
of function>: yl '--= P (xl. . . ., xi'), ..., ym -"" 1 m (X t , . . ., .:t n ) as a
matrix of parlial derivatives
ill. Ill.
a;;; ilx'"
df=
iJlm : Ilfm
<1%, i):r;n
Then. it (.c l , . . ., xn) is a local coordinate system on M I in a neigh-
bOHrhoocl of point Po and (yl, . . ., y.n) is a loca] coordinate system
on M 2 in a neighbonrhood of Qo. the mapping I can be represented
as a set of roordinate functions yk = I h (xl, .., XIL), and T p (M I )
and T Qq (M) can be represented as arithmetic spaces of columns
of lcnglll nand m, respectively. Let the tangent vector S E T po (M)
have coordinates (I, ' . ., n) and the vectOr 11 E TQo (M) coordi-
nates (11\ . . ., TIm). If 11 = df'po (S), then
ilf' a/I
rJl. (Po)"'"""JXiI(Po) 1;1
=
(1)
" ilfm (P ) aim (P ) n
1']  o... .I.r'. 0 ...
SC(,l\1ingly, l)elinition (1) shQuld depend on the choice of II local {'oor-
dilHltc sy!otl'm ill the neighbourhoods of Po E M I and Qo E !If 2' This
is not the ca$C, howcver. Indi\ed, we rail ('hoose another pair of
local coor,!inale sv!otems in the neighbollrhoods of P" E Al) and
Q" E M and \'erify, \Ising the tensor law of coordinate tram;forma-
lion for t8ngl'nt vectors, that DelinitiolL 1 is invariant. Yet, we
!'1Iall proreed difiorentJy, Wo have introduced three delinitions of a
tangent vector to a manifold !vI, One of the deflllilions. namely I hnt

3.4. Sub';umi!old&
131
employing difierentiation o[ a (unction, does not involve local coor-
dinates. Thus, if we reformulate Defmition 1 in terms of differentia-
tion of II smooth function, we automatically arrive at the independ-
en('e o[ the differential of the mapping I of local coordinate systems
in manifolds M l and M 1 .
Lemma 1. Let I: M. l ..... M 2 be a smooth mappinf(. I (Po) = f;o,
s E Tp. (Md be a tangent L'ector to M l at pOint Po. and 1'\ = dip (s)
be a tangent /lector to M 2 at point Qn i/l the sense 01 Definition 1 Then
for a/l.ll smooth lunction g on !vi 1, the lollowing relation is satisfied:
f) (g) = s (gol).
(2)
Proof. According to Definition 1, we have to choose local coordi-
nate systems (Xl, . . ., x n ) and (.Ill, . . .. .11 m ) in M l and M 2 in the
neighbourhoods of points Po and .Qo = I (Po), respectively. In this
rase I is represented as a set o[ [unctions .II" = I" (Xl, . . ., ,2."),
Ihe function g is replaced by a smooth function g = g (.III, ' . ., .11 m ),
the vectorS sand "1 arquire the coordinates (Sl, s") and (1'}I, . . -
. . " 11 m ), respectively. We have Ulen
m
f) (g) =  ::1 (Qo) Il',
II
n
 (gof) =  0:' (gof) 5 j ,
J=I
n
=  (g(fl (Xl, .,., x"), ..., r (Xl.
Oz1
j-=t
n  og ofl .
  01/' (Qo) w (PoH'
j""l '-=1
.. ".
x"» s'
=  ::' (Qo) (  ;: (Po)S' ) '
''!;;I 1. Jc:l
.\pplying formula (1) to the last equality, we obtain
m
S (gof) = 2} ;:1 (Qo) 1']' = 1'\ (g).
1...1
Lemma 1 is proved.
Considering relation (2) as the defmition of the dii£crential of a
mapping j, i.e. assuming
dJp. (s) (g) =' s (gof), (3)

1>2 3. Smoothh Ml1.nijolds ,(Gonoal Thoory)
we can infer from Lemma 1 that the operation g -+ dlpo () (g) is
differentiation of a function on ;M 2 at p=n' Qo, i,e, dip. () is II
tangent vector to M 2 at point Qo; this cOT.clusion coincides with
Definition 1 by formula (1). Hence, Definition 1 does not depend on
the choice of a local coordinate system.
Let us fmally consider the last definition of a tangent vector in
terms of a sheaf of tangent curves.
Lemma 2. Let I: M. -- M 2 be a smooth mapping, Qa -= I (Po), 'I'
be a smooth curve in M. through Po, and g be a smooth lunction on M 2'
Let also  = Y (t a ) and T) = dlp (;), Then
d
1} (g) = dI g (/ (y (t») 1,=1.' (4)
It is most convenient to proue Lemma 2 in the local coordinate
systems (,'/;.1, . , .. x") and (yl, . . ., ym) of manifolds M I snd M2'
respectively. Suppose that all the functions and vectors han the
same coordinate representations a10 in Lemma i, and y (t) =
<x' (t). . . ., :L.n (t», h .". d;t (to)' Then,
m
TJ (g) = 2} i} (Qo) 1f.
1=1
d '>:\ DC 1)/ ' dzi
d/g (f(1' (t)))lt=I.= L: ¥(Qa) DJ (Po) 7 (to)
I,]
m n m
'>:\ Dg Q '>:\ &1' ( J  r)g ,
= LI iijiI ( 0)  &",1 PoH = L.J "'iiY'i" (Qo) TJ =" (g).
,=t 1:';" 11CJ1
Lemma 2 implies that if a tangent vector  is represented as II tan-
gent vector of a curve y (t), .; (to) -= s E Tpo (M), the image T) -=
dip. (;) is represented a., a tangent vector of the curve I (V (t» on
manifold M 2' Furthermore, if follows from Lemma 2 that if the
('\In'es '\'. (t) and Y2 (I) are tangent at point Po E MI' their images
I ('\'1 (t» and I ('\'2 (t» are tangent at point Qo E M 2' The differential
of the mapping f can therefore be defined as a mapping which to a
sheaf of tangent cur\'es {,\, (t)} at point Po associates a sheaf of
tangent curves at point Q.. of manifold M 2. the shea[ containing all
the CIlC\'I!S {f ('I' (t»}. This defmition is not so clear as to revea1 why
the diHerential dl po is a linear mapping of tangent spaces. Figure 3,18
shows the mapping I (x, y) -= (x. y). I: R - R. and it can be see II
that the sheaf of tangent clines at point PI! -= (0, 0) doos not cover
under the mapping I the entire shesf of langont curves at point Qo.
Example 1. Let us consider a smooth function y = I (x) as a smooth
mapping of manifolds f: Rl -+ R i . Then, according 10 Delinition 1,

8,4, Submanlfold.
t33
the diflerential is a linear mapping of T.. (RI) = RI into T/(T) (Rt) :::::
RI delined by formula (1), i.e. rj = f' (x) . In mathematical analy-
sis by the differential of a function f (.r) we mean the linear part of
the increment In f as a function of two independent argumenl.:
dy = f (.r) dx, Thus, assl1mlng dx =  and dy = 1}. we Hnd that the
concepts coi ncide.
y
y
x
Figure 3, t 8
Example 2, Consider a smooth function I of n independent vari-
ables y = I (Xl, . . .. x n ). .rust like In the case of one variable, we
hall represent I as a smooth mapping of manifolds f: R"...... RI.
Then the diHerential of the mappiI"1t I is a linear mapping of tangent
y
Ie +Ax Ie
Figure 3,19
paces dll'o : Tpo (R") = R"...... T Q . (RI) = Rt, where Po = (.1', . , .
x). Qo::::: f (.r, ' , '. :r) The differential dlpo in local coor-
n {If
dlnates is given by T] = :2} w(Po) h. On the other hand, the
kcai I

134 3. Smoolll. Man!fola.. (General Theory)
differential of a function / is he linear part of the increment in I as
a function of two groups of independent variables, (Xl, .., x"\
and (dx l , . . ., dx") , i.e.
"
d df ( IX" ) = " ..!l.. ( Xl
y= x, ...,  iJzh '
"=1
. ., x") di' .
Assuming h = d:t!' and T] = dy, we again fmd that the concept of
the differential of a function coincides with the concept of the differ-
ential of a f\lRction represented as a mapping of manifolds. Further-
more, the matrix of dl Po is the Jacobi matrix of the mapping / and, on
the other hand. it is the gradient of the function /, that is, grad I is
the matrix of dl po in a fixed coordinate system. Obviously, if a coor-
dinate system is modified, the components of grad f will also change.
Example 3. Let us consider a smooth function / = / (xl, . . ., x")
and let Po = (x, . . ., x) be an extremal point of f. Then, by a
(P,)
Figure 3,20
x
Figure 3.21
theorem of mathematical analysis, grad fpo = O. In our terminology
this means dfpo = O. This statement can be extended to an arbitrary
manifold: if a smooth lunction I has a local maximum at a point
Po E M, then dfp = O. This circumstance finds a new interpretation
in the theory of manifolds. In the case of a Euclidean domain there
always exists a smooth function I such that grad f =1= 0 at each point.
This is not however the case for a smooth manifold. For example,
for any smooth function I on a two-dimensional sphere S2 the differ-
ential df vanishes at least at two points: at the maximum and at
the minimum of f. In general; if M is a compact smooth manifold, the
differential df of a smooth funciion cJanishes at lea.t at two points. Fig-
ure J.20 depicts Lha behaviour of the differential of the "altitude"

3 4. Sllbmanl{ld.
13;
function on a two-dimensional sphere S", and Pig. '.21 sllo\\s the
"altitude" function on a torns P.
Example 4. Let I: R" -- R m be a linear mapping which is e).-
pressed in coordinale" as the matri'C. Y = A X.
y_ ( I ) X- ( I ) A- ( !I 1n )
- m' - "t - ml" mn .
Then. obviously, the Jacobi matrix pI the mapping 1 (and hence the
differential dlPo) coincides with A. In other words. dl po m = 1 (s).
And it is no surprise. If we consider the dilierential dIp. as the Ii neal'
part of the increment of the mapping. just like in the case of a fun/:-
lion of many variables, and assume 6X = S. we ohtain 6Y =
A (X + 6X) - AX = A 6X. Hence. the increment I1Y coincides
identically with its linear part. which means that the differenlial of
a linear mapping does not depend on point Po E R".
3.-G.2. DIFFERENTIAL AND LOCAL PROPERTIES
OF MAPPINGS
From mathematical analysis we know the important prop-
erty of a smooth function: using differential properlim, of a 11111('-
lion at a point, we can deduce analogous properties in a neighhClIII"
hood of the point (though the neighbourhood can be small). fM
example, if the derivative /' (xo) of a function I is positive. r (.I,,) >
O. the function f increases in a small neighbourhood of Xo. A qu!-.tioll
naturally arises: what is the analogy hetween the positive sign of the
derivative f' (xu) and an increase of I in a neighbourhood of point
:Xu? This analogy becomes even more ohviolls if the condition f' (3" u) > ()
is formulated as follows: the dirlerential of a function 1 at a point
xo is an increasing (linear) funclion. The corresponding theorem of
mathematical analysis then reads: il the different,al of a IUTI(;t/'on I at
a point Xu strictly increases, the junction I itsell also strictly increase.
in a ncighbourhood 01 Xo. As a second eample, we may consider
the implicit funclion theorem. the equation I (3", y) -::: 0 has a soilltioll
y = Y (x) if :VI (:Xo. Yo) '* 0 at a point (xu. Yo) for which ft.r o , Yo) = u.
Let us now formulate this theorem in terms of di£ferl1ntials. Tile
diiferential of a function I at a point (xo' Yo) is
aj at
dl = 7% (xo. Yo) dx+ ay (xo. Yo) dy.
Replacing in the equation I (x. y) = 0 the funclion 1 hy its differen-
tial, we arrive at a new equa.tion dhx..v,) = 0 which in the new ,'ari-

136 3. Smooth Manl/old, (Ge'lerol Theory)
abies (d,x, dy) is linear and has the form
{}/ 8/
 (.r o , Yo) dx+ "T" (xo, Yo) dy = O.
"z vy
(5)
The condition f (x D . Yo):;6 0 is then equivalent to the existence
CJY
of a solution of Eq. (5). ThIlS, the implicit fllnction theorem can be
formlliated as follows: If the equation in terms of differentials (5) has a
solution, the initial equation I (:r, y) = 0 also has a solution in a neigh-
bourhood 01 potnt (.r o , Yo)'
This leads liS to a principle Which proves to be useful in an analy-
is of smooth manifolds and their mappings: II a property 01 the
differential 01 a mapping 01 smooth manifolds is satisfied at some point,
the analogous property holds true lor the mapping itself In a neighbour-
hood 01 the same point.
This princi pie is usually called the principle 01 general position
or the principle of linearization 01 a mapping. The first term implies
that the property of R linear mapping is preserved under small non-
linear distortions of tho mapping. Naturally, this principle does not
holc! for each mapping, and we shall not at-tempt at elucidating the
ox .lct region of its applicability. We shall concentrate here on the
most important theorems relating the properties of a mapping and
its diflerential.
We shall start with the Implicit function theorem for manifolds.
Definition 2, Let I: M 1 _ M 2 be a smooth mapping, The point
Po E M 1 is caned a regular po/ntlolf if the differential of the mapping
dl po ; T po (M) - TQ. (M 2 ), Qo = I (Po), is an epimorphism, i.e. a
mapping onto the entire space T Qo (M 2)' The point Qo E M 2 is cal.
Jed a regular point of the mapping I if any point Po of the inverse
image 1-1 (QD) is a regular point of I.
This definition is, in fact, tHo implicit lunclion theorem formulat-
ed in ter.ns of the differential of a mapping, Indeed, in the local
coordi nate systems (Xl, . . "x") and (yt, . . ., ym) in the neighbour-
hoods of points Po and Qo, respectively, the mapping f is written
as the following system of funclions 11' = fA (xl, . . ., x"):
fl (Xl, ... I .x n ) = y
(6)
1 m (:r;I, ..., ;L") = y,
Since f (Pnl = Qo' Po = (x:. ,., x'l,), Qo = (y, . . ., y;,), the
point Po is a solulion of system (6). Then the implicit lunction

3.4. Subnutlli/oldl
.1
theorem (Theorem 2 of Sec. 3.2) gives the condition for the existenr
of a solution in Iho form: rank dl =, m. Hence, the rank of the J arobi
matrix of f or, what comes to the same thing, the rank of the
matrix of the differential dl po is equal to the dimension of the tan-
gent space T Qo (M a)' This means that the linear mappIng dip. i$
an epimorphism, We corne therefore to a generalization of Theo-
rem 2 of Sec. 3.2.
Theorem 1. Let f: M] ->- M be a smooth mapping 01 .mooth malli-
Jold., Qo E M a regu.lar point 01 f. Then the inverse image M R =
1- 1 (Qo) is a smooth manifold, dim M3 = dim M I - dim M., a/1d,
moreover, some of the local coordinates in M I can be chosen as local
coordinates in Ms.
Proof. To prove that M3 is a manifold, it suffices to apply Theo-
rem 2 of Sec. 3.2 in a neighbourhood of each point Po E /I1.J' We ob-
tain that each point Po E M3 admits a neighbolJr!Jood U 3 Po
homeomorphic to a domain in a Euclidean space a n - rn , where n =
dim M1' m = dim M.. Furthermore, some (n - m) of the local
coordinates (Xl, . . ., x n ) of M 1 in' a neighbourhood of Po can be
('hosen as local coordinates in UI If these are (:1: 1 " ..., x ln - rn ),
the remaining local coordinates (xi) are expressfod on M3 in
terms of smooth functions of (x h . ., '. x ln - m ), It follows that Ma
is a smooth manifold. Indeed, lilt (yt, .,., y") be another sys-
tem of local coordinates on M I anel let (yJ., .,'. yln-m) form
a local coordinate system on Ms. Then,
yik = y;k (Xl, ., " x n )
= yik (Xl (Xl" ..., X In_m) , ..., x n (.xl" ..., a; In_m))
is a smooth function, Theorem 1 is proved.
It remains to find an analogue ot another property of a linear map--
ping, the monomorphism property.
Definition 3. Let f: M 1 ->- M. be a smooth mapping. The mapping
I is called an immersion if at each point P E M 1 the differential
dip: T p (M 1 ) -I- Tf(P) (M.) is a monomorphism, Le, a one-to-one
mapping onto its image, If moreover f maps bijectively M 1 onto
its image t (M 1 ) and this image is, a closed set, the mapping I is
called an embedding. The image f (M]) (as well as M I ) is called in
this case a submanifold in M..
Example 5. The inverse image of a regular point of f: M 1 ->- M'1:
is, according to Theorem 1, a submanifold. Indeed, since some of
the local coordinates of the ambient manifold M 1 can be taken
as a local coordinate system in Ma, the Identity mapping cp: Ma-

138 3. S moolh M ani/old. (General TMorg)
Pt'. in the local coordinates takes the form
XI=XI,
x n ... rn = .:z:,"-m.
xn-mH:::::I. xn-mtl (xt, ..., xn-m),
x n =x n (xl, ..., xn-m).
Therefore, the J scobi matrix dip includes an identity square matrix.
Hence, rank dip = n - m, i.e. dcp is a monomorphism.
Example 6. Let us consider a mapping f: S' _ R2 given by
J (Ip) = {cos q>, sin 2cp}. The velocity vector'¥- = {-sin q>, Z cos Zip}
does not vanish at any point, that is, the rank of the 1 acobi matrix
y
x
Figure 3.22
is equal to unity. Thus, f is an immersion (see Fig. 3.22). The curve in
Fig. 3.22 is a L;sajous' Hgure; it can be obtained on an oscilloscope
if sinusoidal signals are applied to the horizontal and vertical de-
flecting plates.
3.4,3. SARD'S THEOREM
We have shown in Sec. 3.4.2 how the local properties of
a smooth manifold can be deduced from the properties of the differ-
ential. Conversely, in certain cases we can fmd the properties of a
differential from the properties of the manifold itself. For example,
if f: M 1 -I- M z is a smooth homeomorphism, then the differential
df: Tp (M) -I- T/(p) (M z ) is, by Lemma 3 of Sec. 3. t, an isomor-
phism. Consider now a problem which is inverse, to a certain extent,

3.4. Submantfolds
139
to that solved in Sec. 3.4.2 Let I: ,MI - M 2 he a smooth mapping
of M J onlo the entire manifold M 2: i.e. f (M,) = M 2' Such a map-
ping can be considered as an analogue of an epimorphism for a linear
mapping. A question then arises: is the differential df: Tp (M J ) -+
Tf(P} (M 2) an epimorphism? Unfortunately, the answer is no. Con-
sider the following example. Let .II1J = M 2 = RI, I: RI -+ RI,
f (x) = X3, X E RJ. In this case f is a smooth mapping and 1 (Rl)=RI.
But at point x = 0 the differential df equals zero and is not there-
fore an epimorphism. At other points the differential dl = 3.rdx
is an epimorphism. This example suggests the general answer to
the question. We shall formulate it as the folJowing statement.
Theorem 2 (Sard's theorem). Let f: M 1 - M. be a smooth mapping
of compact manifolds. Then the set G 01 regular points Q E M 2 01 I is
open and everywhere dense.
Before pror.eeding with the proof of Theorem 2, we shall consider
several examples.
Example 7. Let f: Rl_ R I , I (x) == a = const. In this case the
differential dl is not an epimorphism at any point, but the image
f (RJ) consists of a single point a, i.e. by definHion, any point
y =1= a is regular (since I-I (y) = 0). Hence, the set of regular points
is open and everywhere dense.
Example 8. Let f: Rl - RI be a nnite smooth function and F =
{x: f' (x) = O}. The set F is closed The image I (F) is a compact set
and consists of aU non-regular points. We shall demonstrate that
I (F) is nowhere dense. If this is not so, we can find an interior J>Qint
y E I (F). i.e. y is contained in I (F) together with a neighbourhood
y E U c I (F). Since f is finite, the image 1 (F) lies in the image of
the interval I (la, bl). In other wods, it is suflicient to prove that the
image I (F'), F' = F n la, bl is nowhere dense. Let V::') F' be a
neighbourhood of the set F', V c (-2a, 2a). Then f (V) contains U
and hence there exists a point x E V such that f'(x) exceeds, by
modulus, the number E = diam U/4a. Diminishing the neighbour-
hood V, we obtain a sequence of points In -+ F'. We may assumE!,
without loss of generality, that X n - Xo E F'. Then, f'(Xn)-
t' (x o ) , i.e. \ f' (xo) I  e, which contradicts the condition f'(xo) = 0
at point Xo E F' c: F.
It is more convenient to formuhte Theorem 2 in more general
terms: if F c:: M 1 is a compact set consisting 01 non-regu.lar points,
then I (F) is nowhere dense. Demonstrate that it is sufficient to prove
Theorem 2 for the case where M 1 is a neighbourhood of a closed disk
in a Euclidean space. Indeed, cover M 1 with a flllite atlas U", and
choose a covering V", c V", c U", surh that Vex is homeomorphic to
a disk in a Euclidean space. Let Ga. c M. be a set of regular points
for the mapping 1 on 9:. Then the intersection G = nGa. consists
of regular points of the entire mapping I. If G", are open sets which
are everywhere dense, then G is also open and everywhere dens!'.

140 3. Smooth ManiloUI (General Th.orv)
Choose a sufficiently fine atlas U" such that the image I (U ,,) lies in a
single chart W of M 2. Then Theorem 2 can be proved for regular
points of the mapping I lu : U" -+ WII on ii ". Indeed, if G c:: WI!
" -
is the set of regul ar poin ts of I 1[1,,' then G U (M 2 '-I (V,,)) is the set
of regular points of the mapping I: U" __ M 2 on V". Thus, let U
be a neighbourhoorl of a disk Dn in R n and 1et I: U -+ R m be a
smooth mapping. We shall demonstrate that the set of points y E Rm,
for which D" n I-I (y) consists of regular points, is open and every-
where dense,
Lemma 3. Theorem 2 is !laUd lor m = 1.
Prool. Let F c:: D" be a set of non-regular points of a [unction I.
Then I (F) is compact and contains all non-regular points of f. De-
monstrate that RI'-I (F) is everywhere dense. If thi:; is not the
case, there exists an interval V  f (F). Fix k > n and consider the
set F  of those points for which all partial derivatives of I of order
up to k inclusive are zero, Expanding f in a Taylor series in a neigh-
bourhood of an arbitrary point y E F, we obtain I I (y) - I (x) 1<
c 1 x - y I, where C does not depend on the choice of y E Fk and
x E D n . This means that if Fk is covered with cubes of side 1/N
(the number of these cubes does not exceerl N n ), the image I (F k )
will be covered with intervals, the length of each interval not exceed-
ing 2yiik CINk. Hence, the sum of the lengths of all the intervals
does not exceed ZY;';;CINk-n and vanishes for N -+ 00, that is,
f (FA) is nowhere dense,
The remaining part of the set F, i.e. F........F k, can be represented
as a union of a finite family of'subsets, each lying on a submanifold
defined by one of the equations
{JII
I I '" 0, 1 1 + . . . + In = 1 < k,
{Jz,' '" {Jznn
Indeed, let Fl.,.. .,171 be the set of those points in F at which
{JI I {JII
1 . 1 =0, grad I 1 =pO, (6)
{Jzl' ... {Jznn {Jx..... {Jz..n
Obviously, F"F A = U F,...... I... On the other hand, the
1.+.,..jo/n<A
set F"... I.. lias on the submanifold M " ...,.. of those points
where conditions (6) are satisfied, The dimension of M" ." I.. is
Jess than n, and we may conclude, by induction, that Lemma 3
is va lid for M" ,., I..' Thus, f (F k) does not cover the interval V.
Hence, there exists a neighbourhood Uk  F k such that f (U )
does not cover V, Let ll+ ..,+In=k-J. Then F" ...I"........U A is
a -zompact set on M" ... 'n' so that t (F" ... /71 "Uk) does not cover
V ,,'TflJ, Le, I (F k n F".. ./,,) does not cover V, Helice, there

$.4. Subma"'lfold.
141
exists a neighbolJrhood U. ::> F k U U Fi, ... I" such that
,<1'+.0. +',..<h
f (U.) docs not cover thp interval V. Thus, for II + ...
... + In =s the sets F".. 0 I,.. ,-U. do not cover the remainder
V'-f (FI< U U Fl. 00. I,..) under the mopping I, And there-
.<1,+. 0 0 +I,..<
fore f(Fk U U Fl.... ',..) does not cover V, Aft<or a finite
.E:;I,+. 0 ,+,,..<"
number of steps we find that f (F) does not cover V.
We now apply Lemma 3 to prove, by induction on m, Theorem 2
for a system of functions I: U -+ JIm, D" c: U, I (P) = (f1 (P). . . .
. . " r (P»), Since 11 is a smooth function, we find by Lemma 3 that
the set G 1 of regular values of 11 is open and everywhere dense in
Rl. Let y E G 1 , N = U I )-1 (y). By't'heorem 1, the set N is a smooth
suhmanifold mapped by I into a hyperplane Rm-I. Then the
point (y, . , 0' yr) is regular for the mapping II N if and only if
the point (y, . , ., y) is regul'ar for I. By induction, the set of
points (y, . . .. ym) which are regular for the mapping II N is every-
where dense in am_I; hence, the set of regular points for I is also
everywhere dense in Rm, In order to demonstrate that the set of reg-
ular points is open we note that ihe inverse image of a regular point
DR n ,-1 (y;. . . " y) is a compact and the minor of the matrix
of the differential dl is non-zero at each point of the compact. There-
fore, for any sufficiently close point (y:, . . ., y) the inverse image
DR n ,-1 (y, . , " y) lies in quite a small neighbourhood of D 7I n
,-1 (y, . . ., y), that is, the saJ:!le minors are non-zero. This means
that the set of regular points of f is open, Theorem 2 is proved.
As an application of Sard's theorem, we shall consider a smooth
mapping I: M 1 ....... M. for dim M 1 < dim M 2 . Then, none of the
points P E M 1 can be regular, and this means that the image j (M J ) is
nowhere dense in M.. In particular, the image of f does not cover M..
Remark. Sard's theorem can be generalized to non-compact sepa-
rable manifolds. In this case the set of regular points need not neces.
sarily be an open set, but it should only be the intersection of a
countable number of open sets which are everywhere dense, Such
sets are called Go.sets. It is known from general topology that the
intersection of a countable number of sets, which are open and every-
where dense in R 7I , is always non-empty and everywhere dense.
Hence, in the case of non-compact manifolds the set of regular
points is non-empty and everywhere dense.
3,4.4. EMBEDDING
OF MANifOLDS IN A EUCLIDEAN SPACE
In conclusion let \IS stale the Whitney theorem which
asserts that any compact manifold is a submanifold of a Euclidean
space of a sufficiently large dimension.

12 3. Smooth ManIfolds (General Theory)
Theorem 3, Let M be a smooth compact manllold. Then lor a proper
dimensIOn N there exists the embeddill 11': !of _ RN,
Prool. Let {U(1}(1J be a hnite atlas and (.!, . . ., .T) a local
coordinate system in the chart U (1' We may assume without loss of
generality that the U u. are homeomorphh' to a ball D" c: RR of ra-
dius 1 and the coordinates (x. . . " :r) perform a homeomorphism
<i'ex of U u. onto the ball D n . We may also assume that D n lies in R n
and does not contain the origin (this can be achieved by a Iransla-
tion in RR). Further, let D c DR be a ball of smdIJer radius with
the Same centre as Dn, let the manifold M be covered with open
sets U;" = rpJ (D) c:: U (11 and let I be a smooth function in RB
such that it is identically unity on D and supp Ie:: DN. We put
then
Y(P)= { 1(<p(1(P) O )X:(P) if PEUa'
if PEtU",
Obviously, y (P) = X (P) if P E U;". We have obtained a system
of N = 11. M smooth functions, {y (PH. This system defines the
mapping g of M into II Euclidean space R N , g (P) = {y (P)} E R N .
We now riemonstrate that the differential of g is of rank 11 at each
point. Let P E M he an arbitrery point, If;'':;) P, and (x, ..., x)
a local ('oordinate system. The Jacobi matrix of g at point P in
the local coordinate system (x. .... x) consists of partial deriva-
a/'
tives {  (P) } . In particular for /} = ex,
ar
ay o R
--,- (P) = -'- (P) = 6j,
a.r iJ;r
i.e. Ihe Jacobi matrix of g contains an identity matrix of order n.
Hence. rank dg = n.
fhe- mapping g is thus an immersion. In order that g be an embed-
ding, it is necessary that distinct points P /llId Q be mapd into
distinct points g (P) and g (Q). Construct a new mapping g (P) =
{y (P), I (q;o. (P))} E RN..\t Owing to the same arguments as
for g, thi$ mapping is an immersion. Let P ::1= Q be two points on
a manifold. Consider a number C!. sll('h that I (erex (P» = 1. If
! (ipa (Q» < 1, thcn i (P) ::1= i (Q); if I (<fo. (Q») = 1, then y (P) =
.r (P), y (Q) = .T (Q). and ror a. l'ertain number k \\"e have
.! (P) '* .!?: (Q), i.e. g (P) ::1= i (Q). ThIlS, the mapping g: !If......
RN+M is a ClIW-lo-one imnH'r<ion, Lt'. 1\1\ !?mbedding, Theorem 3 is
proved.
Hence, any smootb compa('t IMnifold .\1 may be considered as
embedded (in the form of II submanifold) in a Euclidean space R-"I

3.4. Su.bmanifolds
143
of a rather large dimension N. I practice, however, the dimension
of RN can be reduced significantly. For example, a sphere sn can be
embedded in Rn+l and a torus Tn in Rln. The projective plane RP2
cannot be embedded in R3, but it can be embedded in R5. Indeed,
let (Xl: X 2 : :1:3) be homogeneous coordinates of a point P in RP2. Put-
ling
t __ .1:1
Y - zl+"I+z .
%1%2
y = z+z)+z '
z_ .%1
Y - zl+z¥+z '
5_ Zt%3
Y - "'r+z+z1 '
z'
y3- ,
- z!+"'!+zN '
t1_ -=,2'1
Y - z ! +:ri+:r1 '
we obtain the mapping g: RP2 -+ 1\6, g (P) = g (X t : X 2 : :f3) = (yl, y2,
g3, y', y, y6). It apPl!ars however that the image of g lies in the linear
subspace R5 <= Re defined by the equation yl + yl + y3 = 1. Veri-
fy that g is an immersion, i.e. the differential dg is a monomorphism
at each point P E RP2. In other words, we have to prove that in
any local coordinate system the rank of the Jacobi matrix of g is
equal to 2, All the coordinate fl;lnctions of g are symmetric with
respect to the permutation of homogeneous coordinates (Xl: X 2 : x 3 ).
Without Joss of generality we can tllerefore assume that Xl =f= 0 in a
Il(Hgltbourhood of point' Po E RFI"so that the remaining coordinates
X2' X 3 (for Xl = t) can be chosen as a local coordinate system. The
Jacobi matrix of g is
-2x -2X3
2x 2 (1 + .1:;) - 2X;X3
- 2X.J:2 2:1 s( 1 + x)
dg= (1-x:+ x) -2x 2 x 3
x3(1-x+:t;) x2(f+x-x:)
-2X 1 %3' (1+$-x)
to within a proportionality factor. If X 2 X 3 '* 0, the min'Of of the
flr.t two rows does not vanish. If XI = 0, I3 = 0, the minor of the
fourth and sixth rows is not zero, and if X 2 = O. x 3 =f= O. the minor
of the first and fourth rows is also non-zero. Thus, rank dg = 2.
i.e. g is an immersion. Demonstrate that g is a one-to-one mapping.
Without Joss of generality we may assllme that the homogt>neous
coordinates are so chosen that x:+ x + x;= i. Let P = (Xl: x 2 : x:,).
Q = (x;: x : x;). Then
g (P) = (x, x;, .J:, XII2' X 2 X 3 ' X 3 X l ),
g (Q) = (x;'. :f;'. .r;', x;x;, x;x;, x;X;).
If g (P) '-= g (Q), then x:= x;. If x, :/= O. then Xl = :f::x;. and we
C,jll take Xl = X;, sinr.e homogeneous roordinates can be multiplied
by :i:1. Since the fourth and sixth coordinates are equal, we obtain

144 3. Smoolh MimI/old, (G.nerol Th,ory)
.2: 2 = .2:;, .2:3 = x;, i,e. P = Q. Since all the coordinates XI' Xt, and
.'1:3 are equivalent, we always ha.ve,P = Q. Thus, g Is an embedding of
a projective plane RP2 in a five-dimensional Euclidean space R6.
An analogous statement holds true for any compact manifold M.
Theorem 4, Let M be a smooth compact manifold of dim M = n.
Then the embedding 19: M _: R(2 n +1) does exist.
Proof. We shall use Theorem 2 and try to reduce the dimension
of RN. Let e E RN be a non-zero vector and Pe: HII' _ RN-1 an
orthogonal projection along e onto a subspace orthogonal to e. Some-
  .
times, the composition M -+ RII' -+ RN-l remains an embedding.
Analyse the conditions under which the composition Pe'P is an em-
bedding. We have to verify two conditions: (a) that the differential
is a monomorphism and (b) that the mapping is one-to-one.
We first consider condition (a), Let P EM, Vp = d{j)p (TpM) C
RII'. In order that the differential of the composition d (Peep) at a
point P be a monomorphism, it is necessary and sufficient that the
projection pe should map the subspace V p monomorphically into
RN-1, which is equivalent to eE; Vp. Fix a local coordinate system
(Xl, . . ., x") in a neighbourhood U of point P and a basis in R n .
Construct the mapping h: U X R n -+ Rerr- I , assuming h (.2:\ ' . .
. , ., .2:1>, 1, ..., sn) = (l : 2: . . . : N), where Ii =
 ay: s"'and (y1, . . ., yN) = cp (Xl,. , .,xn). The mappingh is a
«=1 8z
smooth mapping of the 2n-dimensional manifold U X Rn into the
projective space RPN-I. Then the condition e e V p exactly means
that the straight line generated bye, being a point in RPN-" does
not belong to the image of h. By Sard's theorem, for 2n < N - 1 the
:set of such points is open and everywhere dense in the entire space
RPN _1. Considering a finite atlas in M, we find that for an open dense
set G in RPN _I the vector e generating points in G satisfy e E:: V p
for any point P E M. This means that the set of e such that the pro-
jection Pe embeds'", (M) in RN-1 is open and dense, Let us now turn
to condition (b). The absence of bijectiveness means that e is paral-
lel to a straight line through a pair of points P =1= Q on <p (M). Just
like in the case of condition (a), we shall consider the mapping h':
(M X M,,) _ RPN-I which to a pair of points P :1= Q associates
a straight line through (j) (P) and q> (Q). According to Sard's theo-
rem, for 2n < N - 1. the set of points not contained in the image of
h' is an open dense set G', The vector e should therefore be so chosen
that the straight line it generats lie in G n G' :1=0. We have dem-
onstrated that if 2n < N - 1, there exists a projection p such
that the composition Pe'P is an embedding. Hence, step by stepwe
can reduce the dimension of the surrounding Euclidean space RN
unless the equality 2n = N - 1. is satisfIed, whence N = 2n + 1.

8.4. Submanl/old,
145
This Is the minimal dimension of the Euclidean space R(ln+11 which
admits an embedding of any n-dhnensional compact manifold M.
Theorem 4 is proved.
Now we are able to give the general concept of a Riemannian met.
rie on a manifold (particular cases of this concept have been consid-
ered above),
The Riemannian metric on a smooth manifold M is the family of
positive definite scalar products defined in each tangent space T p (M'
If a coordinate system (Zl. ' , '. :en) is valid in a neighbourhood '-'
of a point P, a coordinate system (1, . . " gn) is also valid in
T p (M). The scalar product in (Ij , , .\ gn) is defined by II point-
dependent non-singular symmetric matrix G = (gl/)' In a new coor-
dinato system (yl, ' . " yn) the matrix takes the form
. n 8:tft. IJ:t P
go (P) = 2: gd (P) 8T (P) 8;T (P),
ft., IIc>I 1/
where G' = (giJ (P» is the scalar product matrix in the new coordi.
nate system (yl, . . ., Uti),
Definition 4. The Riemannian metric on a smooth manifold M is
the family of non-degenerate positive definite scalar products in
each tangent space T p (M), P EM" such that in a local coordinate
system the scalar product matrix is a smooth function of the locnl
coordinates.
The metric can be defined as a correspondence which to every coor-
dinate system (z:" ' . ., z) in the chart U a: associates a matrix-
valued smooth function in the chart U c: Gft. (P) = (I':J (P» such
that: (a) the matrix G'" (P) is symmetric and positive definite, and
(b) the coefficients gr, (P) are transformed as
n ozh {}zl
gf; (P) = 2: g1 (P) -f (P)  (P)
h, It 0%", IJz a
for points P E U", n u . If s = (it, ..,. in) and 1} = (tit, ,.., 1]") are
two tangent vectors, tho scalar product S, 1\ is written as
(;, 1)) = t g'f; (P) I'
j,;1
Problems
1. Prove, using Serd's theorem, that any smooth compact
manifold AI of dimension n can be embedded in RIIHI and immersed
in R2".
Hillt. consider the projections n s _ RN-I and choose an ap-
proprhte one.

146 $, Smooth Mani/olds {G.neral Th.ory}
2. Construct an embedding of a torus Tn in Rn H.
Hint: represent Tn as a hypersurface of rotation Tn-l about an
axis.
3. Construct the embedding of 8 1 X 8 2 in R5.
Hint: in R6 fmd an open dOQ1ain homeomorphic to 8 2 X R3.
4. Prove, using Sard's and Whitney's theorems, t.hat any smooth
mapping I: Mil -+- R2 n .... of a compact manifold Mil can be ap-
proximated as dose as possible by an embedding.
Hin.: complete f to the embedding RN = RN-(2 n +l)R(2 n +1) and
consider the projection along a, subspace close to RN _(2 71 +1).

Chapter 4
Smooth Manifolds (Examples)
4.1. THE THEORY OF CURVES ON A PLANE
AND IN A THREE-DIMENSIONAL SPACE
4.1.1. THE THEORY IOF CURVES ON A PLANE.
FRENET FORMULAS
We shall consider a Euclidean plane R2 referred to Car-
tesian coordinates (x, y). A smooth curve y (I) on R2 is defined by
two functions z (t) and y (t), i.e. we deal with the radius vector
r (t) = «z (t), JJ (t» of a smooth curve y (t) emerging from the ori-
gin, point O. Recall that the velocity vector v (t) of a curve y (t) at a
point t is the vector with the coordinates ( d;') , d:?» . The straight
line defined by this vector is tangent to the curve y (t) at a point t.
Here we assume, of course, that v (t) 0/= O. This assumption is sup-
posed to hold throughout this section (at the points where the velocity
vector vanishes a smooth curve may suffer a cusp; the examples have
been given in Chapter 1). The derivativ of a radius vector is some-
times denoted by
; (t) = v (t) = (; (t), 11 (t»
and the symbol I dl) I = I v (t) I will stand for the magnitude of
velocity vector (in the Euclidean metric). Let & denote the length of
a curve from a fixed point to a variable point; since the length of an
arc monotonically increases as the variable point moves aJong the
curve, we may take this length as a parameter along the curve. This
parameter is called a natural parameter, and the quation of the
curve r (&) = (x (s), y (s» written as a vector function of s is caUed
natural parametrization of a curve.
Lemma 1. The magnitude of the velocity vector of a curve written
in the natural parameter is constant and equal to unity.
b
Proof. Wt' have for the length of arc l ('Y) =  I prd') I dt. It
a
foHows that the identity dl = dt I d:t) I holds at each point which
j:; what was required.
Thus, the motion along a curve referred to the natural parameter
occurs at a constant velocity (only the magnitude of the velocity

148 I. S ",oath M arlliaid. (Ezaml'le.)
vector is constant, but not its direction). Corollary: at each point of
the curve the velocity vector v (8) is non-zero, Here we have used the
fact that the relation dt) ::/;: 0 IS satisfied for the initial parameter
t, With each point on a curve we can associate, besides the velocity
vector, another vector smoothly dependent on a point, q (8) =
d;') , This vector is orthogonal to the velocity vector, which is a
direct consequence of the following lemma,
Lemma 2. Let p (t) be a vector functton 8uch that I p (t) I =.1.
Then the vector dPd') f8 orthogonal to p (t) (for any t 8uch that
d') ::p 0) ,
Proof, The condition of the lemma implies that (p, p) == 1. Differ-
entiating this identity with respect to t, we obtain
(  . p)+(P."*)=O, I,e. (  . p)=O.
The lemma is proved.
Thus, at each point of a smooth curve i' (8) referred to its natural
parameter there exist two orthogonal vecH;ors: the velocity vector
and acceleration vector. The latter need not have the unit length,
1t is convenient to introduce the unit vector n (8) = d;8) II d') \,
Hence, as the parameter 8 changes, there arises along the curve II
smooth coordinate frame field, i.e. the family of two-dimensional
frames (v (8), n (a». The vector n (8) is called a norma! uecror to a
curve at point 8, Each frame, after being translated to the origin,
uniquely defines a rotation of a plane about the point O. The frame
field along a curve defines therefore a smooth mapping of i' (8) into
a group of orthogonal matrices\ i.e. a group of rotations of a plane.
In other words, we can say that each curve i' (s) generates a smooth
curve whose points are represented as orthogonal 2 X 2 matrices.
We shall study the properties of this curve for a multi-dimensional
case, i,e, for a smooth curve in a multi-dimensional Euclidean space
(see below),
Definition 1. Let a smooth curve be referred to the natural
parllmeter. The curvature of a curve at a point 5 is the number
k (5) = ! d;.) I,
Since tIll' curvature (by definition) is the magnitude of the
acceleration vector, we have d5) = k (5) n (8), here n (8) is the
normal vector at the point 8 and
k (8) = 1/ (  ) 2 + ( : ) 2 ,

4.1, CI/.rw, Dn II PIIIM <Inri In a Thru-Dtmon,IDna! Space t49
If el and e. are orthogonal unit vectors defining Cartesian coordi-
nates. on the plane Rt (z, y), the normal vector can be written ex-
plici tly as
I
( dlz dlll ) -1! ( dtz dill )
0(8)= asr+(iii" . "d,i"el+-;r,;-e. .
Definition 2, The curvature radius of a smooth curve at a point 8
is the number R ($) = ilk (s),
Before proceeding further, we shall consider simple examples which
underlie the choice of the terms "curvature" and "curvature radius",
The simplest curve on a plane is a straight line defined parametrically
as a linear vector function: x (s) = x (0) + as, y (8) = Y (0) + s,
where s is the natural parameter, This imposes restrictions on the
numbers IX and : obviously, the equality 'V ct. t + a = 1 must be
satisfied, since v = (01:, ) and I v (8) I Ea 1. Then the acceleration
vector rI;') is identicaIJy zero, and the curvature of a straight line is
also zero. Hence, the curvature radius of a straight line is equal to
infinity,
Let us consider a circle of radius R on a plane. The parametric
equations of the circle are
x (s) = z (0) + R COB (  ) I Y (8) = Y (0) + R sin (  ) ,
The parameter 8 varies from 0 to 2n, The velocity vector is
v (s) = ( - sin ( ; ) , cos (  ) ) ,
Hence,
d,(') = ( _  cos ( ; ), _  sin (  ) ) .
Thus, the curvature of a circle is constant and equal to ilR and the
curvature radius is ilk = R.
In many specific problems it appears, however, that the equations
of a curve are referred to an arbitrary parameter t, rather than to
the natural parameter. It would be useful therefore to have formulas
for calculating the curvature of a curve referred to aD arbitrary
parameter,
Theorem t, Let a smooth -:urve y (t) be referred to an arbttrary param-
eter t, not necessarily nat"raZ, and let the lJeloctty vector v (t) be non-
zero at point t. Then ne lormu:a
I dl. (,) I I .y' - y'' I
k ($)= ""'iiT, :;;= [(Z')I+(I/')IJ"'I
is valid, where z', x", ' , . denote dertlJattlJes with respect to t,

i50 4. Smooth MaTlifold$ (Ezampls)
Proof. Let r (t) = (.7: (t), Y (t)) be the parametric form of 1 (t)
and v (t) = (x'(t), y'(t» the velocity vector. If s is the natur.\! param-
eter, we have for an arbitrary vector function q <t)
-!!... ( t ) =.
d,q dt d,'
Tp.ke for q
q (t) = v (t)/Iv (t)l "'" drd) .
From the definition of curvature we obtain
k _ I d'r I I d ( dr ) I _ I d ( V (I) \ I
- dii' = a; a; - as I
and further
d ( V (I» ) dl d ( V (I) )
d8 ""fV(tIT = CIS' iii  .
Let liS find  , Since ds=1 d'} jdt, we have
dt 1 i
as = I drdl) I = l7ill1 '
whence
1 d ( V (I) )
k = TV'Tt)'1' . dl T"VTtIT
1 ( dv (I) v (I) d )
= I vII) I" """'dI'-T'V(l)T'di'lv(t)1
1 ( . r'd l ' l)
"""'i7l' r -m'di" r
t ( . r' d I ' f 2 )
= j7'"jT . r - 2j"7"j'i df r .
Since
;1 Ir'16= :t (r', r')=2(r', r'),
we have
k I d'r (s) I 1 ,. , (r'. r") I
=  =:""j7"""fi . I r - r '""T7'""'f2 '
or in greater data i I
1 ( . , (r', r') )
T?J2 r -r .!
1 ( . , z 'z'+g'Y' ) + 1 ( . , T'''''+q'Y' ) e
= x -x . (,,;')'+(y')' 8j "i7"P y -Y . (:t')'+(y')' 2
1 ( z'lg')'-T'q'y' t ( Y'(Z')'-y'Z''''' )
= j7""jV (z')'+ (y')' e l + "'i7l"" (:t')'+ (g')' e 2 .

4.1. Curves on a Plan and ll a Tliree-Dlmens/onal Space 1St
Hence,
k 2 = «(X')2+ (y')2t 4 . «y')2 (xy _ x'y.)2
''2. ",_i,,, 2 _ (zlt y '_y"'z')'1
+(x) (yx yx»- «x')'+(y')')"
t %lIy' -.%'y" t .
and k= «z')'+(y')')'J" The theorem IS proved.
Let us return to the motion of the frame (v (s), n (a» when the
parameter a varies. It appears ;that the derivatives of the frame
vectors satisfy simple relations, called the Frenet formulas.
Theorem 2. II a smooth curve is relerred to the natural parameter, the
relations
dY (s) dn (s) k
=k(s)n(s)i ----a;-=- (s)v(s),
al'e satisfied.
Proof. The first Frenet formula directly follows from the definition
of curvature k (s). It remains to verify the second formula. Consider
the vector function n (s), By definition, (n (s), n (s» = 1, Bnd ac-
cording to Lemma 2,
( d ) 0 . dn (.) , ( (
n (s), di n (s) =, I.e. ---ciS -=,., s) v s).
where,," is a smooth function of s. To fmd this function, we dif-
ferentiate the identity (v, n) = 0 with respect to s to obtain
( : ' n) + (v.  ) = O.
whence k (n, n) + (v, ,,"v) = 0, Le, k = -,,", The theorem is proved.
The vectors v and n can be written as a column to give the fol-
lowing form of the Frenat formulas:
( dV/dS ) ( 0 k )( V ) ( Ok )
dn/ds = -k 0 n' where X= -k 0
is a skew-symmetric matrix, This relation has a clear geometric
meaning. Let us consider a coordinate frame (i) (s) = (v (s), n (s»
at a point s and move along the curve r (s) from a to point s + As
(Fig. 4.1). After translating the frame (i) (a + /1s) to the point s we
obtain two frames at s, (0 (s) and (i) (s + /1s), where the frame
(i) (8 + /18) is obtained from (i) (s) by rotation t.hrough an infinites-
imal angle /11p. Hence, we may assume (i) ($) and (i) ($ + Aa) to be
related by the orthogonal transformation (i) (s + /1$) = A (/1$) (i) (s),
h ( cos <p sin Alp ) E d . A d . A
ware A (6s) = . A A' xpan log COS alp an SID uq>
-SIll u<p COS uq>
in the small increment 6q> and neglecting terms of the second order

152 4. Smooth Manltola. (Ezampltlj
in !jJ, we get
A(tuP)=( )+(_q> ofj))+.."
( 0 cp )
Le, (O){S+8)=(O)(S)+ _fj) 0 £11 (s)+ ...,
( d!9 (s) )
i,e,  W(s)=;: 1s (_cp ofj))(o)(S)= -(S)  (0) (s),
where cp (s) is the angle of rotation of (0) (s) relative to a fixed
coordinate frame on tho plane (say, rolative to (0) (0», A t the same
v(s)
V(8 + AS)
Figure 4,1
time, it follows from the Fronet formulas that :. (j) (8) =
( _  (8) k 8») (j) (s), Comparing these matrices, we see that
k (8) = dip (s)/d8. Thus, the curvature of a curve at a point 8 can be
interpreted as the rate of the change in the angle cp (s) at this point.
In the case of a plane curve the function k (8) completely defines
the curve, provided k * 0 for all s. To be more exact, the following
theorem is valid,
Theorem 3. Given a smooth function k (8) not vanishing for all s
such that a  8  b, Then there exists a plane smooth curve r (s) unique-
ly defined to within a translation and an orthogonal transformation
and such that for this curve k (s) Is the curvature and s is the natural
paramtlttlr.
Proof. Let us consider the system of differential equations
( dV (1J)/dS ) ( Ok (8» ) ( V (8» ) .. .
dn {s)/d8 = _ k (8) ° n (s) where k (8) 1S a gIven functIOn.
Since k (s) * 0, it follows from the existence and uniqueness theorem
for differential equations that this system has a solution (unique for

4.1, CUTIIU on " PII>M and In I> Three-Dlm.n,jonal Space 153
fixed initial values) which can be continued smoothly to the entire
interval a < s < b. Hence, we can consider the equation  =
k (s) v (8), According to the argumnts just mentioned, it has a uniqU&
solution (fo1" fixed initial values) which can be continued to
the entire interval a < 8 < b, Since any two initial values on a
plane are congruent relative to. a translation and an orthogonal
transformation, the unique solution is defined to within these trans-
formations. The theorem is proved,
Special emphasis should be given to the condition k (8) =# 0, Fig-
ure 4,2 shows two curves on a plane, which are described by the
,...-...
 ....
, ,
I \
, .---
\ ."
\ I \ '
'...... pI \
\ /
,..........--"//
Figure 4,2 Figure 4,3
same smooth functions k (8) but cannot, apparentiy, be transformed
into each other by a translation or an orthogonal transformation,
Both curves are composed of circular arcs. Distinct behaviour of
these curves becomes especially clear if we consider the vector func-
tion of the normal n (8), The normal nl (8) to the curve r. (8) is
always directed to the point 0, the centre of the circle, The curve
'1 (8) is smooth, the curve rg (8) is not, for the second derivative of the
radius vector "2 (8) suffers discontinUity at the point p, This is be-
cause the curvature k (8) is non-zero and constant at all points of
both curves, Yet, we can make two curves of the same curvature to
he incongruent and smooth, The procedure is clear from the above
example, Let us consider an arbitrary smooth curve whose curvature
is a smooth function, vanishing at a point 8 = 8 0 together with the
derivatives of all orders. The existence of such a curve follows from
Theorem 3 applied to a smooth function k (8) which has a zero of
infinite order at one of the ends of the Interval a  8  b. Conju-
gating two such curves at their common end point. at which'the
curvatures vanish together with all the derivatives, we obtain two
smooth incongruent curves with the same curvature functions
(see Fig. 4.3).

1:>4 4. Smootl. Manl/oid8 (E;&ampl..)
4.1.2, THE THEORY OF SPATIAL CURVES.
FRENET FORMULAS
Let us now consider a smooth curve r (t) in a Euclidean
space R n referred to Cartesian coordinates Xl, . . ., xn, i.e. r (t) =
(Xl (t), . . ., x" (t». Just like in the planar case, with each point of
r (8) we can uniquely associate a frame which is smoothly
varying along the curve as the naiural parameter 8 changes. Before
constructing this frame, which we shall call "the Frenet frame", we
shall prove, by analogy with the planar case, an auxiliary propositi-
on about differentiation of a matrix-valued function.
Consider a smooth curve in a linear matrix space, i.e. a one-param-
eter family A (t), where t varies on the interval -a < t < a, and
A (t) is an n X n matrix whose coefficients are smooth functions of
t. Suppose all the matrices A (t) (for all -a < t < a) are orthogonal
and have det A (t) = + 1 and A (0) = E, where E is an identity
matrix.
Lemma 3. Let X = A (t) 11.'0 denote the derivative of this one-
parameter family of orthogonal matrices A (I) at t = 0, i.e. X is
the matrix composed of the functions da d lJ(l) I , where A (t) = (alJ (t».
l 11-=0
Then X is a skew-symmetric matrix.
Proof. Each orthogonal matrix A (t) can be represented as a linear
operator in an n-dimensional Euclidean space, the operator
preserving the Euclidean scalar product. This means that for any
two vectors x, y E R n the identity (A (I) x, A (t) y) = (x, y) is
valid. Since the left-hand side of the identity is a smooth function
of t, we can calculate its derivative with respect to t at t = O. We
have (..4 (t) x, A (t) y) + (A (t) x, A (t) y)lt=o = 0, i.e. (Xx, y) +
(x, Xyj = O. Taking the basis vectors ei and eJ as x and y, respectiv-
ely (e l . . . " en is an orthogonal basis in an), we obtain (Xei> e,) +
(elt XeJ) = 0, which means that X is skew-symmetric in (e1> . . ., en)'
The lemma is proved.
Remark. Since the set of orthogonal matrices is embedded as a
subset in the linear space of al1 n x n matrices, A (t) can be consid.
ered as a smooth curve in an n 2 -dimensional Jinear space. From this
point of view the matrix X = ..4 (t) II=:0 is natural1y identified with
an ordinary velocity vector of a smooth curve A (t) at t = O.
A matrix-valued function A (t) can be expanded at t = 0 in the
powers of an infinitesimal increment,
6t:A(t)=E+ dA d (I) 1 ,at+....
t '=0
The matrix X appears then as a "coefficient" of at. We now turn to
constructing the Frenet frame.

4./. Cur. On a Plane and In a Three-Dlmen.ional Space 155
Let r (s) be a smooth veclor function which donnes a smooth
trajectory yes) in Rn. Suppose that for a$b all vectors : '
cf2: ' .." d d : are linellrly independent for each s, Then at every
c/. .--
point of r (s) there exists a coordinate frame (not orthonormal!)
., d f h dr (.) dnr (.)
wll/ch 18 compose 0 t e v{)Cto --;II"' ..., -;wr and smoothly
varies from point to point. Since tho derivatives of the radius
vector are linearly independent. aJI :: are non-zero, 1 kn.
ProposiLion 1. Let r (8) be a smooth vector function in Rn and
. . dkr d Z ' dr d 2 r dA-I r
let the kth denvatwe""dsii' epend mearly on Iii' ds"..., d."-'
at each point of the interval asb. Let also  *0 for a
dr dk-'r
s b and all Iii' . , ., (fsii=1 be linearly independent for asb.
Then the curve r (s) lies entirely in the (k-1)-dimensional plane
dr dlt-'r
panned by the vectors dS' . .., dsll- I , and this plane does not
change its position in Rn as s varies from a to b.
Proof. The hypothesis implies that there exist smooth functions
It-I
. , dRr 2; dlr
'I",(s), 1k-1, such that at each pOInt S- d k= ",(s)-,
S dB!
!I
dr dlt-'r
Since the vectors dS' ..., (iiii=l are linearly independent, they
can be taken as a basis in the plane Rn-I spanned by those vectors.
In order to prove that the curve r (s) always lies in the same k-
dimensional plane, it is sufficient to demonstrate that the position
of the plane R"-I (s) in Rn rema:s unchanged as s varies, Since
: ' ..., : are the basis vectors in RR-I (s), it suffices to show
that the derivative of the basis consists of the vectors which can
be decomposed in the basis vectors, But this is obvious from the
hypothesis, so the proposition is "roved.
Thus, if the vectors : ' ..., d;; are linearly independent, the
curve I' (s) is not contained in any (n-1)-dimensional plane Rn-I c
nn whose position remains unchanged when s varies. We now
construct at each point s an orthonormal basis whose vectors are
denoted by 't., ..., 't". Put : /1 : I=TI and consider a two-
dimensional plane spanned by 't l and =: in which we choose a
vector 't2 orthogonal to 't.. Since, by assumption, : and =: are

156 4. Smooth M(Jnlfold, (E:z:ampZ.,)
linearly independent. "1'2 has a noTlzero projecti(\n on :: ' We !Ohall
choose the next vector 'fa in a three-dimensional space spanned by
cl'l' . b d3r d 3 r Ill' d k r
(j;i"' "ra. 1'1' I.e. y the vectors 0i3 ' di' . d8' an ta e or 13
the vector orthogonal to the plane spanned by "I' and 1', (i.e. by
 and : ). Continuing the procedure. we obtain the orthonor-
mal frame 1'1' ..., 1'n' Apparently. when s changes, the vectors
of the frame 't (s) also vary smoothly from point to point. As 'in the
planar case, it can be shown that the derivatives of the frame vectors
(1'1 (8), ' . .. 't n (8» satisfy simple relations similar to the Frenet
formulas for a plane curve,
Theorem 4. Let a smooth curve r (8) in an be referred to the natural
dr dnr 1 ..J - d
parameter s and let the !lectors d,' ' , .. d," be linear 11 £nU<ipen ent
at each point a  s  b, Then there e:rist functions ka (s), ' , ., k n (s)
sUt:h that the following relations (Frenet formulas) are satfsjied:
I dT, k
Cii'" = ir,
.!!!!.
d, ="I'3-k2'f1'
dTn_1 k k
c: n1'n- n_ t "t'n_21
dTn
"""'ii8" =  k n 1'n_I'
i,e. ; =k1+I1'I+I-k,'CH, where kl=kn+l=O,
Proof, Since the vector 11 (1in) is contained in the linear
I f ell' dlr. d . , d"l'I' t . d . h I .
c OS\1re 0 di"' '"  d;i"' Its erlvlltlve CiI' IS con ame In t e meal'
closure of the vectors
dr
T'
dtr
...,,
as'
df>'r
d. i t-l .
j,e, there exist functions a/,I (s), "., a" 1101 (8) stich that
1+1
0;" =  all (8) 'fJ (8).
jt=J1
Henca, the n X n matrix X, whic.h represents tho vectors
d"I'l dTn . f tl d . r
T' .. 'I -;iI in lerms 0 10 COOl' mate rame
'(($)= (t l . .... 1'n).

4.1. Curuel on a Plall8 /l1'd III /I TIIrIM-D!m.",!onal SP/IC8 157
is of the form
4 11 au
ala au aS 2 *
a 23 a33 au
X=
an_i. n-2
o an_f, n-t an. n-1
an_I, tL al\n
The matrix X is known to have some additional properties. Indeed,
as in the two-dimensional case, the translation of the frame '(8)
along the curve,\, (8) can be interpreted in terms of a family of orthog-
onal matrices such that '(8) = A (8)" (0), This one-parameter
family uniquely defines the evolution of , (8) when 8 varies, Then,
apparently, X will coincide with the derivative dI) at s = O.
According to Lemma 3, the matrix X 1s skew-symmetric, j,e.
0 -a12
at! 0 -au 0
au 0 -a 3 4
X=
o an-a, n-I 0 - a"_I, 1\
an_I,1\ 0
Taking the functions a," 1+1 (8) as k/(l), we complete the proof of the
theorem.
We now turn to three dimensions, n = 3. Here the Frenet formu-
las become
d't,.. d'tt k k d1:s k
"'(j$=n;a'fu di= 3'f 3 - atl' /iB=- 3'1'
The vector 'f l is a unit velocity vector of the curve r (8) and is usual-
ly denoted by v (8), The vector 'fa (8) coincides with the derivative of
v with respect to s, since :: ..L v and J v (8) I = i = const, Here
we have used the fact that 8 is the natural parameter. The vector '3
is orthogonal both to v and n = f, i.e. we may assume that it
coincides with the vector product of v and n, The vector n is called
a Ilormatvector to the curve r (8) and the vector b = [v, nJ a binormal
vector to r (8). (Here [v, n) stands for the vector product of v and n,)

18 4, Smoolh Manl/ola (Examples)
In this notation the Frenet formulas take the form
dv dn db
"""iiS=kn, (ii'=1(b-kv, Cfi= -xn,
where k (s) = k (s) is called the curvature of a curve and
x (s) = ka (s) is the torsion of a curve. (The vector n (s) is sometimes
called a principal normal t.o r (s), rat.her than simply a normal.)
For the sake of convenience we shall always assume that
't 2 coincides witb : /1 : I; then k (s) coincides with tbe magni-
tude of d:;8) and is therefore a positive number (the correspond-
ing derivatives of the radius vect.or are supposed to be non-zero),
In the case of a plane curve the binormal vector is constant, i.e, it
does not alter as a point moves along the curve; in particular, the
torsion of the curve is zero. Hence, zero torsion may be considered
as a characteristic of a plane curve in Ra. We now discuss in greater
detail the properties of a spatial curve with non-zero torsion. Consid-
er a translation of the coordinate frame (v, n, h) along a curve and
project h (8), n (s), and v (s) onto the pla1ie spanned by b (so) and
n (8 0 ), where So is a fixed value of the parameter s and the value of 8
is assumed to be infinitely close to so. In this case the velocity vector
is projected as an infinitesimal vector, and we may assume, in the
first approximation, that this vector vanishes. Then we have a mo-
tion of the vectors h (8), n (s) projected onto the plane spanned by
b (8 0 ) and n (8 0 ), It follows from the Frenet formulas that this motion
is described by the relations  = xb, <!}; = -xn, i.e. the motion
is determined by the skew-symmetric matrix (_ ), which governs
an infinitesimal rotation of the coordinate frame b, n. Hence, the
vectors b, n are rotated about the velocity veetor of a curve, the
speed of this rotation being uniquely determined by the torsion of
the curve (incidentally, this is the reason for the term "torsion").
And if a curve was initially plane, it becomes "twisted in space".
Thus, a spatial curve can (locally) be obtained from a plane curve
if we move along the latter at a constant velocity and "twist" it at
every moment by the torsion x (Fig. 4.4).
Remark. As in tbe case of a plane curve, there exist invarilmts-
curvature and torsion-which uniquely (to within the isometry of a
three-dimensional space) define a smooth curve, Since we shall not
use this assertion (which can be proved exactly in the same way as
Theorem 3), the proof for the three-dimensional case is omitted.
Let us consider a smooth curve in R" and let the velocity vector
of this curve be non-zero at each point.

4.1. C..rve. on a Plane ana in a Three-DlmensiDnal Space 159
Proposition 2, A smooth curve with non-zero verocity vector is a
smooth one-dimensional manifold srrtoothly embedded in RI!.
Proof, That the curve is a smooth manifold is a straightforward
consequence of the definition of a smooth manifold. Thus, it remains
to verify that the curve is a smooth submanifold in R IC . To this end
Figure 4.4
we have to consider the differential of embedding i, i.e. the linear
mapping di. Since this mapping is completely defmed by the ve-
locity vector, the Jacobi matrix has a maximal rank at each point
of the curve, so that the cllrve is a submanifoJd.
Remark. The plane curve shown in Fig. 4.5 is a smooth manifold,
hut it is not a smooth submanifold in a plane. Obviously, this curve
Figure 4.5
can be defined by II radius vecLor with smooth components or (t)
and y (t). but at the vertict's of the triangle the velocity vector must
vanish in order that the curve may sharply change its direction,
Apparently, anyone-dimensional smooth, closed (without bound-
/lry) manifold is diffeomorphic ci,ther to a straight Hlle (non-
compact manifold) or to B circle (compact manifoJd). Thus, the

160 4. Smooll. Ma.nlfold. (Ezample,)
dass of one-dimensional mauifolds consists onl)' of two manifolds
considered to within a diffeomorphism. They are not diffeomorphic,
for a straight line is non-compact while a circle is compact,
Problems
1. Prove that if the curvature of a plane curve is identi-
cally ero, this curve is a straight line,
2, Prove that a plane curve in a three-dimensional space is char-
actefied by the condition x Ea O.
3, Prove that a straight line in a three-dimensional space is charac-
terized by the conditions: k tEiI 0, X 51 O.
4. Describe the class of those CUfves fOf which k = const and
ox = const.
5. Prove that the trajectory of the motion of a particle in a central
field is a plane curve,
4.2. SURFACES. FIRST
AND SECOND }o'UNDAMENTAL FORMS
4,2.1, T1IE FIRST FUNDAMENTAL FORM
Let us consider a Euclidean space R" and let vn-t be a
smooth manifold of dimensIon n - 1 (or of codimension one) embed-
ded in R". In this section we shall mainly concentrate on the local
properties of this hypersurface, leaving aside its global structure.
We may assume therefore that we deal with a smooth embeddIng of
a disk D"-l in R". We have already discussed different ways of
defining a manifold. including a hypersurface. For the sake of con-
venience, we shall use the parametric definition of V"-I, that is,
we shall assume V"-I (or embedded disk D"-I) to be defined by a
smooth radius vector r = r (u l , . , '. u'l-}) c: R" where the pa-
rameters (coordinates) u l , . . ., U"-I run some disk in a Euclidean
parameter space R'H (u 1 , . . ., un-I). Since the radius vector is
or or
assumed to defme a smooth submanlfold, the vectors 6Ui' ' ,., 
are lineady independent at each point of the domain of their defini-
tion. Recall that these vectors afe tangent to the corresponding coor-
dinate lines through a point P on JIll-I, As was shown above, a
smooth embedding of V"- 1 in R" indtlces on V"-I a Riemannian
metric. We now remind the construction. I t!t Xl, ' . .,,:r;" be Carte-
sian coordinates in R", then the fadius vector f is given by the
set of smooth functions xi(U I , . . " U U - I ), 1  i  n, Let ds 2 =
 (dx i )2 b the Euclidean metrk in R"; then we have the follow-
j1

4 2. Surfac8. First and Second Fundamental Forms 161
ing form:
)1 n n-1
ds1 =  (dx i (1.1. 1 , ..., u"- I »z=   { ::: dU k )2
V"-1 i=1 ic:::i c:ll.1
" 71-1 n-1
"V ";:\ az l a",1 A '"
=  LJ auk ' au p du du P = LJ gAP (f.(.) du A du P I
1=1 h, p=1 k, p=1
CAP= (::k ' a: )'
where (,) stands for the sc.alar product in R",
Definition, The first fundamental form of a hypersurface V n - I in
R" is the form dsz j :=: '" gkpduduP, where the functions
yn-l LJ
k,p
Phil (1.1. 1 , ,." un-I) have been defined above,
These functions dept'nd on the I'adius vector of the hypersurface
(Illd vary, in general, on the change of the radius vector, i.e.
upon the deformation of the hypersurface, The first fundamental
form is defined on the vectors tangent to Vn-Ij to be more exact,
if a, bE T pV"-1 are two arbitrary tangent vectors, there is defined
11-1
till' scalar product (a, b)d..(y1l-1)=C"pa"b P = }j C"pakbP, Note that
It, p-=1
10 Slmplify the notation we omit the symbol Z; if summation is
performed over the same upper and lower indices. It follows from
the geometric meaning of gloP that the scalar product (8, b)".t(Vn-l)
coincides with the ordinary scalar product of the vectors a and b
considered as vectors of the ambient space RI>, The metric tensor
matrix @j composed of the functions g"p (1.1. 1 , . , " u n - 1 ) is of the
form
(!I_ ( I . . . I . KI, n-I. )
- Kn-I, I Cn-I, 2 Cn-I, n-I .
It should be noted that the Riemannian metric has appeared earlier
as a convenient tool for calculating the length of a smooth curve on
a surface. From this point of view, the length of an arc 'Y (t) in VR-I
b,r-
is given by l (1' (t» = .. (y, y) dt. In terms of gkP (u) this
a
expression takes the form
l'i' (t) =  11  I gkP (u (t» du;,c/) du:,c/) dt.
cz It. pc::lt

162
4. Smooth Manifolds (Ezamples)
This formula is more preferable in that it contains the functions
gAl' (u) which are independent of the choice of a curve on V"-\
and only depend on the surface V"-l and its parametrization. Fur-
thermore, these func,tions do not vary under isometrics of a surface;
for example, when a plane is bent into a cylinder.
We now study how the first fundamental form is expressed for
various definitions of a hypersurface. Let V n - 1 be given as a graph x" =
I (Xl, . . ., x"-I). We have
11-\
ds? I ' = h (dXi)2+ (dx" (x', ..., ,7'''-1»2
yh-l
1=1
n-S n-I
=" (dxl ) 2+ "" ...!L...!L axil dx1>
L.J LJ ozll 8:tP
ic1 k.p=1
n-I
= h ( 011 P + :!k 0;1' )' dx il dxP
lI,p_i
= gAP (X t , ..., X"-I) d:r!' dx".
( The derivlltivl:...!L will sometimes be denoted !"u..) Let now
Qza.
V"-I bo gi \'\'[1 as an implidt function, i.e. in the form
F (Xl, . , .. x") = 0, where :::. =F O. Then, it follows lrom the
impJit;; t function theorem thet the eq11ation F (Xl, . ... x") =0 a
(locally), the solution x"=1(x l , ....X"-I). with of/8z'=
- (8F18x')/(iJF/8:r;"). Substituting (8F18x tt -)i(8F/8:r;") for I"a.. we
obtain @! = (CAp), where
gilp= [( :: . ::r. ) / ( :::. )2] + 0111"
Let us consider a particular case: embedding of a two-dimensional
surface in three-dimensional Euclidean space. Let V2 be given para-
metrically: r = r (u. v). Then the first fundamental form is usually
written as ds 2 (V2) = Edu 2 + 2Fdudv + Gdv 2 , where E = (r u . rlt),
F = (r u , ru), and G = (r u , r.) are the coefficients of the Lorm in
terms of the components of the radius vector r:
E = x + Y, + z, F = xux. + guY. + zuzu. G = x + y; + z;.
The metric ds 2 (V 2 ) is called conformal-Euclidean if E = G and
F = O.
Here is an example of the first fundamental form for a surface of
rotation embedded in R3. Let the cylindrical coordinates (r, cp, z)
be valid in R3 and let a two-dimensional surface be defined para-
metrically, (<p = u, z = v, r = r (v». Calculation yields
ds 2 (V2) = dv 2 + r 2 (v) du 2 + (dr (v»2 = (1 + (r)2) dut + r 2 du 2 .

4.2. Surfaces. First and Second FUlIdamental Forrru 163
Here F (u, .) = 0, E (u, v) = r 2 (v), G (u, v)' = t + (r)2. That
F = 0 means that the coordinate lines v = v. = const and u =
U o = ('onst <Ire orthogonal at cach point (Fig. Q,6).
)(
figure .{ 6
Lemma 1. Let yn-I C R" be a smooth submanifold and let (!J be the
first fundamental form. Then (!J is non-singular.
Proof. By the definition of the radius vector r = r (u l . . . " U"-I),
all vectors ru, 1  k  n - 1, are linearly independ'ent at the
points P belonging to yn- l . Since the matrix (!J is composed of the
scalar products of ru and r up ' i.e. (!J = ({ru' r up ». (!J is non-sin-
gular. The lemma is proved.
This fact is easily seen from geometric considerations. Indeed, if
the symmetric matrix @! were singular, it would have at least one
zero eigenvalue, and then the initial Euclidean metric would also
have a zero eigenvalue, which contradicts the delinition of this
metric.
4.2.2. THE SECOND FUNDAMENTAL FORM
Let yn_ 1 be a hypersurIace in R" given by the radius
vector r = r (u l , . . ., u'l-!) and let n = n (P) be a unit vector or-
thogonal to yn-l at a point p, We introduce the fundamental form
Q (a, a) by defIDing its values for an arbitrary vector a E T p (y"-l).

164 4. Smooth Mantfold4 (Ezamples)
To this end, we consider an arbitrary smoo'.h curve y (t) on v n - I
through the point P such that y (0) = P, Y (0) = 8. Such 0. curve
always exists, though it is not defined uniquely (Fig, 4.7). Since the
radius vector r is 0. function of t along the curve y (t), the n a =
 f (u (t» 1/=0' We consider a vector ftlnction; =  f (u (t» and its
.. d" ..
derivative with respect to t, i.e. r = d;' f (u (t». Let r. stand for
o
Figure 4.7
o
Figure 4,8
the value of .... at t = O. This is the second derivative of the radius
vector r along the direction of the vector a.
Definition. Put Q (8, a) = (, n). ..
The number thus defined is the projection of fa onto the normnl
vector n at point P (see Fig. 4.8). Calculate Q (8, 8) explicitly in
terms of the coordinates of r. We have
dr duk (t) d'r dUR du" d'u k
di = rul< , di2 = Fuku" "dj"'"' dI + ful<""'di" '
< ;,: r ((tH, n) = <fuR"" d;t d; , n) + 0,
< d'ul< > d'uk
F"I<""(iii"" ,n = '(jj'r (f,,'" n) = 0,
f"k E T p (Vn-f), n..L T p (V n - I ),
< :: 11=0' n> = (r"l<"p 11=0' n) d: 11=0 d:e" It=o = Q (8, a).
It should be rec11Ied that the vector a has the coordinates
( du l ( 0 ) dun-I ( II J )
---.JI' ..., dt ' that is, we finally obtain
Q (a, a)= {n, rukul' 11"'0> aka",

11.2. Sur/actl. First and Suond Fundamental For",. 165
This form nniquely defines a bilinear form Q (a, b) whose value on
a pair of arbitrary vectors It, bET pV"-1 is Q (8, b) = qkp (P) akb P
where
qkP (P) = (fukuP /,=0' nj.
Lemnla 2. The expression Q (II, b) = 9kpa k b P , where a, bET p V".I,
defines a bilinear form.
Proo!. We have
auk <iul' < <i'X' (0) ) auk !JuP
qk'p' = (r"k'up'!t=o' n) = auk' itu P ' auk au!' ,n =  '{;;;j1 9kp'
i.e. the functions 9kp are transformed under coordinate transformation
as the coefficients of a bilinear form, which is what was required.
Definition, The bilinear for'1l Q (a, h) is caJled the second fundamen-
tal form of the surface y>'-1 c: R".
Obviously, the formQ depends on the way V"-I is embedded in R",
that is, Q will, in general, vary under smooth deformation of V"-I.
1
:iill

Figure 4.9
This form is not invariant under isometries of V n - I in R". For in-
stance, on folding of yn-l, i.e. under such a smooth deformation of
R" that the first fundamental form does not vary, the form Q will,
in general, change. Let In he a two-dimensional plane in R a , tIlen
the radius vector r (u, v) can be considered as a linear function of u
and v. Hence, the first fundamental form is the Euclidean metric on
a plane duo + dv2 (Fig. 4.9). We now consider an isometric transfor-
mation of V", folding of a plane R2 into a cylinder with the axis
parallel to the x-axis (Fig. 4.10). Apparently, the second fundamen-
tal form of a cylinder is not zero, since the number (r.., n) is nol
zero. At the same time, the second fundamental form of a plane R"
is identically zero, hence folding just mentioned (i.e. isometry)
rcsul ts in the change of this form,
Let us consider a fixed submanifold V"-l c::: R" and Jel P E V"-J.
A pair of forms, @! and Q (the ftrst and second fundamental forms),
is defined at each point P. This pair is characterized by a set of
numerical invariants which permit the study of V"-l irrespective of

166
I. m(lOlh Manltolth (J.;:r:ampl..)
the coordinate system. Denote the matrices of the forresponding
forms by.@j and .Q and fonsider the .polynomial in A, det (Q - ,.@j),
Since @j is non-singular (see Lemma 1), there exist a matrix@j-I
inverse to @j, so that the equalion det (@j-IQ - AE) .: () is equiv-
alent to det (Q - A(!j) = O. Let A" . . ., A"_I stand for the eigen.
values of the matrix @j_IQ, i.e. for the rOOIS of the equation
II i ! i j-W
"I [0'.) "n . ,
Figure "i.10
Figure 4,11
det (Q - A) = O. We shall provc in the sequel that they are all
u-1
real. Write the characteristic polynomial F (I..) as L Ok,.k, where
/(.....t l
Ok are symmetric functions of Alo . . ., An-I'
Lemma 3, The functions Ok (Ai> . . ., A n -1) are im'ariants of the
forms @! and Q. i.e. they are preserved under an arbitrary regular COOl"
dinate transformation in a neighbourhood of point P E V"-I,
Proal. The rt'guJar coordinate transformation x...... x' in the neigh-
bourhood of point P in the tangent space TpV n - J induces a ]illellr
regnlar transformation through the Jacohi malrh:: J; T p - T p , so
that the matrices (!j and Q are mapped as foJlows: @! __ JT@jJ =-
' and Q...... JTQJ -. Q'. lIenee,
det«@j')-IQ' - J.E) = det IJ-I. (@!-IQ-AE) JI = det«!j-IQ- '.E),
which is what was required.
Special attentiun will he given to lite in"ariant
not
(1) 0'1 =  A k .:. spur (-IQ).
b,!
n-I
(2) O'n_I::'''' n A = det (@j-IQ).
h=1
The remaining Ok' 2  k  n - 2. describe more suhtle properties

4.2. Surjaces. First and S4cCII!d Flllldamental Forms 167
of 1'''.1, which are now beyond our scope. Since @!-IQ * UfoI' Q zr= 0,
III leasl one of (J is non-zero. The invariants 0', and O'n.1 are "c".
'rme" invariants (Fig. 4.11).
Definition. The function H (P) = (JI (P) = 0') (A.,. . . ., A. n _I) i
('lIed the mean curvature 01 a surface V n _( C HI< at a floint P E V"-I
Tllp function K(P) = (In_, (P) = 0'''_1 (A" . . .. 1..".d is ('Itlled the
GflIt,siall curvature of a surface 1''' -I c: R" at II poillt P.
If II =:1 and V c: R3, then H (P) = A., T A.; K (P) "'= At' Az.
Theorem 1. All eigenmlues 1.. 1 , . . .. A" -I 01 the pair 01 fOI'm @j,
Q are real J 1 all AI' .. , An _, are pairwise distillct. all eigenvecton
c i . . . ., en -I 01 the matrix @j-IQ are mutually orthogonal relati 'e to
bofh the ambient Euclidean I1Uitric in R" alld thl' Riemal/niall
metric indllced on v'n -I by the embedding V" -I ->- R".
Proof. According to the well-known algebraic theorem. thp eigen-
\'alues of a symmetric matrix are real and all ils eigell\"(>(.tors are
mutually orthogonal for distinct eigenvalues. Thi$ theol'l'lI\ Clillnot
he applied directly to our case, for the matrix @j-IQ is 1101. in geller-
,tI. symmetric. [t would be symmetri(' if @! and Q were commutati\'c.
:-:inre thp form @j (P) is symmetric lit ('lIeh p, there e"its, ill a
('pl'tain neighbourhood of P, a regular coordinatl' tram;formntion
.1 -- x' such that @j(P) can be diagonalized at a single poiut P.
That is why a linear coordinate transformation appears to be sum-
dent. Aftar @j is diagonalized, it can be reduced to an identity matri;\:
tlll'ollgh thc e.\tension along the principal axe!'; of the form. Let A
be a linear operator A: TpV"-' which turns @3 illtn all identity ma-
trix, then @j ... ATEA... AT A , where the matri;., E deli lies au or-
thogonal basis !PI' . . ., <Pn-l in TpV"-', Thus, we obtain det (Q-
A<!J) = det (A T {(A T) -'Q (A -I) - '}.,E) A ,. Consider t he form Q =-=
BTQB. where /J = A -I. The initial equation det (Q - A.@j) = 0 is
written in tbe basis <p), . . ., <Pn -I as det (Q - 1.1:.')  0, siuce
c\l't A =;6 O. We also have QT = Q. since QT =- Q. Hence, the form
Q and matrix @3-IQ have the same eigenvalues and eigenvectors.
Since Q is symmetric, all its eigenvalues (i.e. At' . . .. An_I) are real,
and if they are pairwise distinct all cigenvectors of Q are mutually
orthogonal. This follows from the familiar algebraic theorem (Ihe
orthogonal basis e l , . . ., 8 11 _ 1 need 1I0t lIecessarilr coincide wilh
<p,. ... CP"_I)' Theorem 1 is proved.
The proof implies that if all the eigenvalues are distinct, the
orthogonal basis el' . . ., 8 n _ 1 in the plane TpV"-l is derllled unique-
ly. If some of A" . . ., A. n _, coincide. the relation will hold:
ei..L oJ for Ai =;6 AJ. If some eigenvalue is of m,dtiplicily k, there
e)(ists a k-dimensional invariant suhspllce such that its vectors are
multiplied by the same number A.; in this subspaee we call also choose
(not uniquelyl) a k-dimensional orthogonal basis.

168 4. Smooth Mantfolds (I,:"omplesj
Definition. The vectors C l _ ' , '. 1.'''_1 (defmed uniqnely if AI :f::.
Aj for i * j) are cal!ed principal directions (or principal a.rl',.) of a
hypersurface 'II-l at a point P.
Th1ls. with each point of the hypersnrface '"-I we have associated
a uniquc (if AI * Aj for i :;= j) orthogonal basis cJ. ' . '. enol smooth-
ly dependent on P The vectors (ed are orthogonal both in the
x n nIP)
Figure 4.12
ambient Euclidean metric and in the induced metric on V"-l c: R".
Since (!! becomes an identit,\' matrix in the orthogonal basis
1.'1' ..., en_It the nllmbers AI'.... All_I coincide with the eig!1n-
values of the form Q written in el' ,." e n _,.
Lel us consider a particular case, Let V n - I c; R n be given as the
graph Z" = f (3;1, ,... 3;n-l) and let at a point PE V"-I the plane
T pV"-1 coincide with the plane of the variables :1:1,.", Z'H
(Fig. 4.12). Then the Jlormal n (P) to V"., at the point P has the
coordinates (0, ' . " O. 1) and the radius vector r describing V"-I
is of lhe form r=r(x l , .. _, X"-I) =(x l , "" x n - I ; /(%1, ., o. %".1».
Since the hyperpJaue (Xl, ., " -tn-I) = T pV n - 1 is tangent to V"-I
at P, the relation 8fl8xilp=O, 1in-1. is satisfied. whence
(P)=E, since Ci/=("d«j+fJlj' Consider the matrix Q=(qu),
where
iJ2/ (P)
qlj = (I' xi,.). n} = iJz1o"j ,
Thus. Q -= (f,lyJ (P)) coincides \\ilh till! Hessian of t at the point P.
The mean curvature H (P) is
n -1
H (P) = Spill' (-IQ) = sJlllr Q = 2: f.,k,ki
11=1
and the G<lnssinn cUl'vatu!'!? is
I\. (P) = del (-IQ) = det Q  det (fyi,l (P)),

./.2. Surlacts. First alfd SUD>ld Fundamental Forms J6f1
For a two-dimensional surface (n = 3) we obtain
z= f (x, y),
a'l a'l
H(P)=+ Oy' =M,
a' 11'
where t:. = _ a , + _ a ' is the Laplacian on the plane RZ;
3: y
( /,,"
K (P) = det /"'11
;:;) = fx:cfIlY- fu'
4.2.3, AN ELEMENTARY THEORY
OF SMOOTH CURVES ON A HYPERSURFACE
Let us consider all arbitrar' point P E y"1 and let
n (P) be II normal to the hypersurface Y"-l in R". Consider also an
arbitrary two-dimensional plane R" through the point P which inter-
sects y..-t along a smooth curve y (t) = R' n V"-1 (as previously,
we are only interested in a small neighbourhood of P).
Definition. The smooth curve y (t) = R' n yn_l is called a plane
section oj a hypersurface V'I-I c: R".
Inlinilcly many plane sections pass through the point P. At the
soame time, not every smooth curve y c V"-l is a plane section of
"'-I. For the curve y c: V"I not to be a plane section it is su[-
fldent that y have a non-zero torsion. Let 'I' be a plane section of
t'n-I (the plane R. containing y is fixed) and let the point 0, the
origin of the radius vector r = r (u', . . ., un-I) describing V"-I,
belong to R', Let m (P) be a normal vector to the plane curve y =
R. n V n - I contained in R. (Fig. 4.13). Since we consider an arhit-
rary section, the normals nand m may not in general coincide.
Introduce on y = R2 n J!II-I the natural parameter s: y = l' (s}
(. is the arc length). Tlum the cun'e y ($) in R is given by the radius
vector r = r ($), where r ($) = r (u 1 (&), . . ., u n . 1 (8». According to
Ihe Frenet formulas for plane Cllr\'!'s, we have k (.) = I d:.) I.
where k (s) is the curvature of y ($) al the point P. Recall also that
d'r(s)
"ds' = mk (s), wh!.'re III = m (P). P E l' (.). On the olher hand,
j r a =-  r (&) is the velocity \'ector of I he curve )' (s) at p, we ha, e-
(by the definition of Ihe second fundamental form Q)
Q(a, a) =<  ' n)=(km, n>=k(m, n}=-kcose,
where e is lhe lingle bel\\een III and n <It the point P (se!.' Fig. 4.13).
On thl.' olher Iwnd,

170 4. Smoolh Manifolds (Es:amplts)
( dr dr )
Q(a. a)=Q tii, di
( dr dr ) ( dt ) 2 ( dt ) - 2 Q ( dr dr )
=Q dt'dt di =- dI ' di'di"
where f is an arbitrary smooth parameler along the curve y=
R2 n V"-I. Here : = a is an arbitrar)' tangent vector to the curve 'V
at the point P. If IX -;: (a. I . ..., a. n -I), then
k cos 6 = Q (a. a! qiJa.itx i
l:tja 1 a 1 CiJa.ia J
Since Ihe curve 'i. which is a plane section, can be drawn along any
langent vector a E TpV"-\ thl! following theorem is proved.
Vr.- I CRn
Figure 4,13
Theorem 2. For a,ny tangent ('e£"lol' a E T pV" -I and any plane section
l' (such that;' = a) the ratio of the se£"ond to the first fundamental form
ls k. cos e, i.e.
Q (a, a _ ql Ja.ia: J = 1.' cos a
()) (a. a) g,jafa) ,
The curvature k is called the curt'ature 01 a plane sectioll.
Among plane sections, of special interest is the class of normal
sections.
Defmition. The plane sectioll V = R n V"-I at a point P is called
.normal if n (P) E R Z , i.e. e = O. 1'1.
Thus, every normal section y at a pOLnt P E V"-I is uniquely de-
fined by the tangent vector a E TpV"-I, i.e, the two-dimensional

4.2. Surlac... Fird and S..ond Fundame"tal F,mru 171
plane R2 defining this section is spanned by two vectors. he normal
n (P) and the vector a E T p V"-l. Ratating R2 about n (P), we obtain
all normal sections of the hypersurface- V n - 1 at the point P.
For the normal section y = R2 n V"-1 the formula. of Theorem 2
lahes the form
k= Q (a. 0:) -= 9'J0: 1 a 1 .
1M (0:, IX) HI Jal",j
hersnse e = O. Since the curvature of a plane section (along the
"rctor a) making the angle 6 with the nonnal n (P) depends on 6,
j his function should be written as k (8, a). From the statement proved
nhove we have k (a, a) cos 6 = k (0, u) where k (0, u) is the
('\II'"alure of the normal section (along u).
Thus, if the curvature of a normal section k (0, u) is known, the
rUI'"atllre or any plane section (along a) making an angle e with n
1"1111 he found from the relation k (6. u) = ....!... a k (0. a.).
cos
Let us recall that in the tangent plane T p V n - 1 there always exist
principal directions el (P), . . ., en -1 (P) uniquely defined if 1.., *'
Aj for i *' f. Consider these "principal axes" and construct for each
a,xis the corresponding normal plane section YI = R n V"-1, where
Ihl! plane R is spanned by n (P) and ej (P). All the principal di-
tE'rlions e, (P) are mutually orthogonal in the Euclidean metric 011
T p l'n-l. Let k, (P) be the curvature o[ a normal section '1'1 (such a
!,p<,lion is sometimes called a principal normal section).
Theorem 3. The eigenvalues AI"'" 1I.n_1 coincide in modulus
with the curvatures k l . .." k rH of principal normal secttons,
q,p.laj
Proof, From Theorem 2 we have k.CQs6= IS ' Since [or a
8.1J(1. IX
qlJa.to: J
normal section 8 = 0, 11: we have :I: k= I" whtlre IX is the
8'Ja 0: '
vector defining the normal section, Fix in T pV n - 1 an orthogonal
bRsis e 1 , ...,e n _ 1I then glJ=8IJ, QI}=8 jJ A,. and hence
n-I
 )" (a l )"
:i:k= i1
n-I
 (ex ' )"
1=1
If a E T pY"-' coincid<.>s with one of the el' then :I: k r = A." which
is what was rrquired.
Let us consider in T pyn_ 1 an arbitrRry vector a. and normal
section along a. Let 'PI (1';;;; i';;;;n -1) denote the angles between IX
and the principal directions ej> ...,8,,_1 (Fig. 4.14).

172 4. S/lllh Ma/ll/ld. (Ezamples)
en _ ,
cr
Figure 4.14
Propoaltlon t. For the normal section along an arbitrarl/ vector
a E T pV'>-1 the followtng relation (the Euler formula) is valid:
,,-I
k=k(a)=  A/coszcp/.
.=\
Proof. By Theorem 3 we have
"-I
 AI (Ctl)" /I-I
k(a) =,:1_ 1 =  AI (
 (Ct l )' ;=\
,...1
ct., ) 2 = I
,,-I
V  (Ct l )" .==1
1=1
A/ cos z (Pi'
where
COS!PI =
Ct l
.. /"-1
V  (Ct')"
il::t
-
-I Ct I '
and lal st!lnrl for the length of a (it is obvious that
Ct' )
COS!PI = TiZT '
The Euler fOl'mula permits an analysis of the so-called "etre-
mal" properties of the principal curvatures AI' ..., An_I' Consider
the curvature of the normal section k (a) ab a function of u E T p1'''-I.
Since k (a) = (p,a), where p:;<= 0 is areal number, the curvature k (a)
depends only on the direction c;)sines cos CPl' .... COS !Pn-I' whcl'e
n-I
the angles 'PI' ... I CPIl-l were introduced above and  COS 2 !P1 = I..
it
Since {cos!p,. h:;;;in-1} may be considered as the coordinales
of the unit vector allal, by putting z!=coscp/> 1in-1, we

,1.2. Surface.. First <1M Secnd Fu"damental Form, 173
,,-I
may assume that k(IX) = k (Xl, ..., x'l-I) =  AI (.1'1)2 15 a smooth
II
function on the sphere S"-2 given in T pV"-1 by the equalion
(XI)2 + ... + (X'I-I)2 = 1. From the relation k (<<) = k (- IX)
we observe that the curvature k (IX) is a smooth function
on the projective space Rpn-2 = S"-Z/Z2 (Rpn- z is a smooth
manifold). Thus, k (a) is a (unction of (n- 2) variables
,,-I
(XI, ..., X"-I,  (X')2 = 1 ) . Since S"-2 and RP"-2 are smooth
,=1
mnnifolds, we can introduce local coordinates yl,. . ., y"-2
in a neighbourhood of each point ;z; E 8"-2 (for instance,
if X"-I:;;;'O, we may assume Xl, .... X"-2 to be coordinates on
8"-2). Let us call Xo E 8"-2 a crWcal point for the function
f(x) given on 8"-2, if ...!.L I =0, 1';;;;i';;;;n-2. The equality
{Jill ""
Ii} of oyi. I . h . f . t . . t ' I .
 = _ I ---r-- Imp les t at I a pOIn x D IS cn Ica In one COOI'-
;)y' {Jy iJy
tJiuate system, say yl, ,.., y"-2, it will also be critical in any
olher coorciinate system. Thus, the definition of the critical point
is in varian l.
Question: Whal are critical points of the smooth function
"_I
k (a) =  1./ (XI)2 on the sphere 8"-2 c: T pV"-I? Furthermore, what
1=1
\'alu8s does the curvature k (IX) take at these critical point (i.e.
along the direclionsof a of corresponding to the critical points on S" -2)?
Theorem 4. The critical points of the curvature lunction k (a) on
the phere S" -2 c: T l' V" -t, are exactly the points:f=e" 1 .;;;; i .;;;; n -1
(i.e. the ends of the vectors :f=e,) when 1. , -=1= AJ lor i -=1= j. At these
points k (a) has the value, 1.,,1';;;; i";;;; n - 1, In this sense, the prin-
cipal curvatures are the extremal value. of the curvature function k ().
T! ome of the eigenvalues {AI} coincide, the theorem will read: the criti-
cal point. of the function k (a) are exactly the ends of all eigenvector.
of the form Q.
Proof. Let liS consider an arbitrary bilinear symmetric form
B (x, y) defined on the vectors x, y E R"-I (this can he, for
eXAmple, the second f1lndamental form Q (x, y) = qI)Zlyi). Constrnct
the rUnclion f (.I:) = B :x; j , x E 8"-2, and find all its critical
points. Obviously, I (;z;) = B (x, x) if Ixl = 1. That xoE 8"-2 is a
critical point for I (x) means that : I",. =0 for any vector b tan-
gent to 8"-2 at xo. where :, 1", denotes the derivative of I (x)
with respect to b, i.e. ..9- 1 = of(y', ..., 11"-2)\ .bl, where yl are
db x, iJyt I",

174 4, Smooth Mantlold. (EumpZ..)
looal coordinates in some lIeighbourhood of %0' Using an equiva-
lent definition of this derivative, we have d d b ' I ' d/( a Y(I}) I '
o t I-O
whero l' (I) is an al'bilrar' curve entirely lying on the sphere
S"-2, and 1'(0)=%0' d:t) Ir=o=b (see Fig. 4.15). It follows that
dl (X/ t » 1,=0 =  B (1' (t), Y (e» 1,=0' Since the scalar product {,)
Bxo
Figure 4,t5
is fb:ed in R"-I::::> 8"-1, we can associate bijectively with the form
B (x, y) a symmetric operator B: R"-I-+ R"-l such that B (x, y) =
(x, By), i.e. i(z) = B (x, x) = (x, Bx), where oX E sn- I is the end
of the vector x pointing from 0 to XES"-1. Hence,
ala) L. =-- :t B (y (t), Y (t)) Ito = :, (i' (e), By (t»
='( :t 1'(t), BI'(t»4-(y(t), B :t y(t»
= ( :1 V (t), By (t» + (B1' (I), :, l' (t»
= 2 ( :, (y (t), By (e» ,
Since :t i' (t) 11=0 = b, we have d f (x) /,,' = 2 (b, Bxo> = 0, i.e.
{. Bxo> = 0, and hence for any b E T"osn-. the vector Bxo is or-
thogonal to f",. S"-1 (Fig. 4.15). Thus, Bxo is collinear to Xo. i.e
}..(xo),xo = Bxo. where }.. (x-o) =I: ° is a (real) eigenvalue. Hence, the
<,ritical points of f (x) = B (x, x) on S"-2 are only those points

4.2. SUorlac,. FIrst and Second /0"11 ,'''am,..tal Form. 175
which are the ends of the unit eigenvectors of the form B. Since an
orthogonal basis can be chosen from the eigenvectors of R, A (.To) coin-
cides with the eigenvalue for Xo. The theorem is proved.
Let It = 3 and V: be embedded in R3. Then the Euler [ormula
becomes k (a) = AI cos CPl + A. cas. CP., where cos CPt + cas J cP =
I; slIppase Al  ).,., then k (a) = (AI - A.) cas. CPl + "'.. Apparently,
k (a) has a minimum for cas. <1'1 = 0: k (a) = A, and 1\ maximum
for cos. <PI = 1: k (a) = AI' If Al "': Al' then k (a) -= A (--= A. = "'.)
(see Fig. 4.16).
)(2
1
X,(x'F + X 2 ()(2)2 = k("I > 0 
)(.
1
'fi,
Figure 4.i6
Geometric interpretation of the prindpal curvatures leads to all
elegant geometric pattern, the so-called quadric of normal curva-
tures: Al cas. CPl + . ., I- An _lCOS. <!In -J = canst. The directions cor-
responding to the extremal (critical) vailles of the curvature coincide
with the principal axes of this surface. If all principal curvatures
AI' . . ., An_1 are non-negative. the quadric is an ellipsoid with the
principal semi axes 11Vf" 1 :::;;; i :::;;; n - 1, i.e. the ith principal

176 4. Smooth Man/folds (E.rampZ..)
semia"is is equal to Vrl' where rl is the corresponding curvature
radius of the [th normal section 1'1 = R" n V"-I.
Let liS analyse the case of a two-dimensional surface V 2 c:: R". It
appears that ihe Gaussian cllr\'ature K (P) determines some impor-
tant local characteristics of a two-dimensional surface. We recall
that K (P) = J...J . J... 2' and three cases are thlls possible:
(a) K>O, (b) K<O, (c) K=O, whence
{ J...I > 0 { I..I < ° { i'l> 0 { AI < 0
( a ) or . ( b ) or .
'-2>0 "-2<0' 1.. 2 <0 i..>O'
{ "'I = 0 { i'l * 0
(c) '-2*0 or 1.. 2 =0 or A'="'2=O,
II is sufficient to consider onl:r caseS: (a) }.I > 0, A 2 > 0, (b) 1.. 1 > 0.
Al < O. (c) I..t = 0, A 2 =/:: O. "'1 = I.. l = O.
Figure 4, j 7
h.
Figure 4.18
III case (a) the shape of the "urfar!! 1" near the point P is (locally)
as that in Fig. 4.17. lIere r" Iie on lhl! sirle of the tangent plane at

4.2. Surf(U$. Flrot and Second Fundamsrtlal Forms 177
P. In case (b) the shape of V2 near P (locally) is shown in Fig. 4,18.
Here v lies on both sides of TpV a . In this case the point P is called
a ,addle point (or simply a saddle). sometimes it is called a hyperbol-
Figure 4.19
Figure 4.20
Ie point. In the third case (c), A. ::;6 0, "1 = 0, the surface V2 near
P is (locally) as in Fig. 4.19, Since Al::;6 Aa, 1' has at this point
t\\'o orthogonal principal directions. e1 and 6 2 , One should not believe
thnt near the point P, at \\hich AI = A = 0, the surface 1'2 has the

178 4. Smooth ManIfold, (Ezomple,j
local structure of a plane (from the metric point of view), 'fhe fact
is that jf the second fundamental form Q valJihes at P (i.e. when
 = "'2 = 0) the local structure oC V: near P is noL qundratic. As
an example of V2, we consider the graph z = / (x, y) = Re (x +
iy}3 = :r;3 - 3xy3 (see Fig 4.20, the so-called monkey saddle). The
!lrst Cundnmental form (!! at the point (x = 0, y = 0) is gfJ = 6fJ.
Figure ,s.21
The second fundurnentaJ form Q at Lhe point (0, O) is degenerate.
since "'I = "2 = O. Here Q = (qi/), where q/J = /xixJ, i.e. Q =
6( _  =); hence Q = ° at the point (0, 0). The family of level
lines of the function I (x, y) at (0, O) is shown in Fig. 4.21,
,s.2.,s, THE GAUSSIAN AND MEAN CURVATURES
OF A TWO-DIMENSIONAL SURFACE
Now we shall fmd an explicit form of the Gaussian and
mean curvatures of a two-dimenSlOnal surface defined explicitly.
Let V2 <:: R3 be given as the graph z = f (x, y) where (x, y, z) are
Cartesian coordinates in R3. Let / x = I v = I (0. 0) = 0, then the
coordinate plane (x, y) is Langent to V 2 at the point (0, OJ. Since
 ( /x Ixy )
(!!](O,O)=(u/})=E. QI(o,o)= Ixy Iyy'
we hElve K = detQ= Ixx/yg -/. H =spurQ=fxx+l yy = }ol + A 2 .
The curvatures Hand K at points distinct from (0, 0) can be
calculated as follows. Let (!! = (; ) and Q = ( ), then

4.2. Surfa",. Fird and Stcond Fundamental Form6 179
CS- I = +(_ -;), where g=det<B. Hence,
1 ( GL-FM GM -FN )
(M-'Q=7 -FL+EM-FM+EN '
H - I'P\lt (<!I-IQ) =.!.. (GL - 2F M + EN),
g
K = det (@!-IQ) = ro:r:: .
If z -= f (x, y), then ds = (t + g) d.r + 2fxf d.r dy + (1 + f') dy2.
gince the radius vector of V given by the graph z = f (3', y) is of
I he form r (x, y) = (x, y, f (x, y», we have
f:u:=(Q, 0, !",x)' r"u=(O, 0, fxg)' fgy=(O, 0, luy)'
n= grad (z-f) _(-f",-fu,1)
I grad (.-f) I ii+f1+f'
A[;= /"U N= luu
V1+/+f& 11"1+&+{1,
L= f""
Vt+f+/ I
whence
H.,..= GL-2MF+EN (I +fW ""-2/"I/lf,,u + (I-l-Ii.)fg"
t+Ji+/£ (t+l1+f6)""
K-- fx:rfuy-fh
- (t,n+n)"
The Gaussian and JUean curvatures are scalar functions delined !It
l'1Ich point of a surface. and they are a) so in\'arinnts of the surface,
in particular, these curvatures do 1I0t depend 011 the choice of local
cnordinates,
Let us calculate the Gaussian and mean curvatures of a standardly
(.mbedded sphere 8 2 c: R3. Since any normal section of S at an arlJit-
rary point P is an equator, J... 1 = ].,2 = 1/R, where R is the sphere
radius. Hence, the curvature of any normal section is i" = 1/R
and therefore K = 1/R2, H = 2/R; in particular, the Gaussian and
mE-an cur\'atures lire ('onsta1lt.
The Gaussian aud mean curvaturcs of a two-dimensional plane are
('qual to ero.
Definition. A two-diml'nsional surface 1-'2 c: R3 is clilled the
surface of comtant curmture if its Gaussian curvature is constant.
For example, a standard sphere 8 2 and a Euclidean plane are
Illanifold.!! of constant curvature.
Defmition. A two-dimensional surface y 2 c: R3 is cal1ed the surface
(Jf pMith'e, zero or negatiL'e curL'ature if tho Gaussian cur\'ature at all
points of this surface is positive, zero or negative, respectively.

180 . Smooth Mantfold. (E"'l1mpI4')
A standard two-dimensional sphere is a manifold of positive
(constant) curvature,
Problem, Prove that the surface V2 c R3 given by the equation
:: ++ :: =1 (ellipsoid) is the surface of strictly posilive
curvature, provided the semiaxes a, b, and c are not equal lo 0
and co. Hint: the ellipsoid can be written parametrically as
x=acos6coslp, y=bcos6sin<p, z=csinO.
A Euclidean two-dimensional plane is a manifold of zero con-
stant curvalure, The graph Z=:1:2_y2 is an example of a mani-
fold of negath'e curvature; obviously, K = {;irtf2 =
(1 +4 (:;1, y'})' < O. In this example the Gaussian curvature is
a variah e function. It would be desirable, by analogy with a sur-
face of positive or zero curvature, to construct a manifold of con-
stant negative curvature, V 2 c:: R 3 . Now we 1>hall give an example of
such a surface, thereby proving the following statement: In a three-
dimen.io1!al Euclidean space there exist (locally) surfaces of contant
posith'e, zero or negative Gaussian curvature.
Let us consider a smooth curve y located in the first quadrant of
the plane (x, y) and having the following property: the length of
the tangent from the point where it touches the curve to the point
where it intersects the x-axis is constant and equal to a (Fig. 4.22).
y
A
y
lAB I '" 8 '" canst
IA,a.1 '" a
8 0
IC
B
Figure 4.22
As the point A is moving along the curve ,\" the point B slides along the
x-axis and the segment A B has the constant length, a. Such a curve
can be obtained if the points A and B are connected by an inextensi-
hIe string of length a (the initial positions of the points are Ao and
B o. see Fig. 4.22), and B is made to move along the x-axis Then A
will descrihe the curve y which touches the y-axis at Ao and has the
x-axis as an asymptote. The points A and B are assumed to move
along the plane without friction. In this case the velocity vector

4.2. Surfac... F!r&t and Second Fundamental Furms 18\
i alway$ directed along the string connecting A and B, Le the
TIlotion of IJ is uniquely defined. Let us derive a differential equation
of the curve y. From the triangle AB:r we have (see Fig. 4,22):
Illn (j1 = - y;. where y = y (x) is the graph of y, and a. sin q> = y,
whence
-y;
sin !j! = I ' Le,
I I+(y)'
ayk
--y;
}/1.. (!I;')
x,=
l' a'-lr
!I
\\ here x = x (!I) is the graph of y, Henco,
y y
x (y) = -  + V a Z _ yZ dy == _ a  dll
 a y fa'-y'
II
+
.. V a 2 _yZ
a
1/
i dy
= _a 2 J y 'If;;o=:y;
a
,1., .
Y u- - y-
r-
=-V al_y1. +  In ( a+a'-Y' ) .
- ,,- "'-y'
Thus, wo ha\'e derived the explicit expression for x = x (y). Let
11 consider the !ourface which is obtained if the curve y is rotated
ahout the x-axis (see Fig. 4.2:-\). S\lch a surface V' is called the
Figure 4.23, Beltrami surface
lleltrami .urlace or pseudosphere; the latter term will he explained
below. To find the Gaussian curvature of the Beltrami surface, we
have to calculate tnI.' Gaussian curvature of a !ourface of rotation.
Let us solve this problem in the general form.
Let us consider in R3 (x, y, z) the surface V obtained b' the rota-
lion about the x-axis of a smooth curve x = :z; (y) (the generator)
lying in the plane Oxy. A coordinate network, parallels and meridi-
ans, is obtained on VZ, and this network is orthogonal because a

182 4. Smootl. ManIfold, (E:rample8)
each point the coordinate lines intersect at right angles (see
Fig. 4.24).
Lemma 4. At each point of a surface of rotation the priru:ipal dtrec-
tions, i.e. the directions corresponding to tke principal curvatures At
and "2' may always be considered as coinciding with the directions of
the meridian and parallel through this point.
Figure 4,24
Remark, The words "may be considered" have the following mean-
tog: if 1.. 1 '* "2' the principal directions are defined uniquely and
coincide with the directions of the meridian and parallel; if Al = "2'
any direction is principal. including, as a particular case. the ortho-
gonal directions of the meridian and parallel.
Proof, As we know, the principal directions in T p V 2 are only
those orthogonal bases e l , e 2 in which the forms (g and Q are both
diagonal. Apparently, the Hr:;t fundamental form e is diagonal in
the coordinate system generated by the meridians and parallels
(see, for instance. the calculation of e for a surface of rotation).
It remains to prove therefore that the second fundamental form Q is
also diagonal in this coordinate system. Let (u, v) be the coordinates
on 1f2 which generate a coordinate network. meridians and parallels.
We should prove that
M = (r UD ' n>;;;; 0,
where r (u t v) is the radius vector of V 2 , n is the normal to '2. and
Q = ( L M ) is the second fundamental form. Consider cylindrical
M N .
coordinates (r. cp, x) in R3 and let V2 be given by the generator
r = r (x). Then the radius vector r (x. q» of VZ is of the form r (x.
<p) = (x, r (.x) cos cp. r (x) sin cp) (see Fig. 4.25). whence r",q> =
(0, -r' sin cp, r' cos cp). The normal to Y% is gi ven by n =
(7'. - cos cp, - sin q»/V 1 + (r') . Obviously, (r",q> , n> = 0, so
that Q = ( ), which is what was to be proved.

4.2. Surfaccs. Firs! and Second Fundamenlal Fo,ml t83
Lemma. 5, The Gaussian curvature K (P) of a surface of rota-
tion V2 at a point P E V2 is of the form I K I = , (1 )')l , where
r ,= r (x) is the equation of the generator of V2 tn cylindrical coor-
dinate.
i
Remark, In other words, K = ;",1.." where 11 - AI is the
r 1 +(r')'
curvature of the normal section along the parallel at the point P,
anc! "2 = k (x) is the curvature of the plane curve y = y (x) at
Jt
z
z
Figure 4.25
a point x, i.e. the curvature of the meridian. The normal section
along a parallel does not gonerally coincide with the parallel itself
(the formula for "1 will be proved below),
Proof. According to Lemma 4, the principal directions coincide
with the directions of the meridian and parallel through the point P
and therefore K (P) = "'1"'2, where "" and A 1 are the curvatures of
plane curves-the meridian and the normal section along the parallel,
respectively. Since the parallel is a circle, we shall use the corre-
sponding formula to find the curvature of the normal section. Since
the meridian coincides with the generator r = r (x), it follows from
1'"1
the Frenet formulas for plane curves that I A.a (x) I = (1 + (r')")3/' . We
now find A 1 (x) for the normal section along the parallel at the point
P = (x, r (x». Consider the parallel as a plane section of VI; then
the curvature k (e, a') of this section (along a delined by the par-
aUel) is equal to 1/r (x), since the radius of the circle (parallel)

184 4. Smooth M4nttotd8 (E:t:4mples)
is r (x). Here 6 is the angle between the normal to V and the vec-
tor 111 lying in the plane of the pnralJel (Fig. 4.26). Recall that
1
k (6, 0;) = r&; e . k (0, 0;)
(see above). Here k (0, 0;) = c 0 = cos e, k (6, 0;) is the CUlvatl1re
in question, A.J' Thus, it remains to lind ('os e, where 6 is the angle

x
Figure 4,26
between the unit vectors m and n. Since in the plane :rf)r the vec-
tor m has the coordinlltes (0. 1) and the VE'ctor n the coorcli-
nates V 1 (- r', 1), we have cos a = V 1 and
1 + (r')' i + (r')'
;.. ( x ) = t , whence
I r y1+(r')'
I K I _ I I r" I I r" I
- r V t +(r')2 (i-l-(r'J2)"/2 r (1 + (r'J2) 2 '
The lemma is proved.
Lemma 6. The Beltrami surface is a mani/old 0/ constant (strictly)
negative curvature.
Proof. Since the BeItrami surface is a surface of rotation, we can
IIse the formula of Lemma 5 to calculate the Gaussian curvature.
The function y = y (x) is the inverse of the function
./- 4 ( 4+ J / aa_y. )
x=a.(y)= - y a 2 _y2+_ln
2 a-ya'-Y'
obtained above. As was already calculated,
, -l/
Xy= !J
whence :r! __ a'a
r 2 }I a'-r' '

4.2. Surfaces. First and Second Fundamen!al Forlns 185
and consequen tly
to, -%'1 -z"x' 1
K= r{I+(r')')2 = ( ' ) J (1 ' 1 ) 2 r(I+(z'}a)2 = -/ii'-
% r T(Z7
The minus sign appears because the curve y = Y (.:I:) is convex down-
ward and therefore the eigenvalues AI> 11. 2 have opposite signs relative
to any normal n (P), Thus, K = - lIa 2 , and this completes the
proof of the lemma.
We have constructed in R3, at least locally, the surfaces of constant
positive. zero. and negative Gaussian curvature. The manifold of
constant pOSitive curvature (a sphere) is compact and closed (without
boundary); the manifold of zero curvature (a plane or a cone formed
o

= =
l'igure 4,27. Section of the Beltrami surface
by straight lines emerging from a single point, fmite or infinite, and
moving along an arbitrary smooth plane curve V in R3) is non-com-
pact and open (without boundary). The manifold of constant nega-
tive curvature just considered differs from these two examples in
that the surface of negative curvature is not a closed manifold and
l'annot be extended to inftnity. This s\lrface (see Fig, 4.23) has a bound-
ary, the circle of radius a with centre at the origin 0, anrl it can be
hown (we shall not concentrate on this point) that the surface cannot
be extended beyond this circle ill such a manner that the condition
K (P) = - cta < 0 is satisfied.
This surface is usuaJly"completed" by adjoining a surface symmet-
ric to the initial one about the plane yOz (Fig. 4.27). The resultant
lIrface has It "cusp circle" where the surface is not a smooth sub-
manifold in R 3 . This circle formed by singular points is not acciden-

186 4. Smooth Mantlold8 (Ezamp$)
tal, It seems at first sight that these singularilies could have been
aVllided if the "fllnnel" constructed above had been extended beyond
the circle of radius a, as in Fig. 4.28, (We recall that at the point AD
on the y-axis the graph of tho generator touches this axis.) Once
,slld\ an e\:tt)n.ion has been carried out, we find that the surface of
rotation tll1l obtained is not a surface of negative curvature: the
part swept Ollt by the arc A nC (see Fig, 4.28) has positive ClIn'llture,
.since the aT(' AoC i convex upward (IInlike the arc AnA).
y
A
x
Figure 4,28
Hence, it is doubtful whether the Beltrami surface can be smoothly
extended beyond the circle described by the point A 0 around the
x-axis such that the surface curvature remains negative. Thus. it is
rather diffu\ult to construct in R3 a closed, compact or non-compact,
but infinite in all directions manifold of constant negative curvature.
Hero we can easily see the difference between a surface of constant
positive curvatnre and a surface of constant negative curvature
(surfaces of variable negative curvature extending in R3 inlinitely
in all directions do exist: for example. the surface of hyperbolic
paraboloid z = x - y2 considered above; here the Gaussian curvature
is negative and vanishes atinfinily if, for example, x 2 + y2_ 00).
Indeed, a standard two-dimensional sphere has a constant positive
curvature in the metric induced by this embedding.
To get a deeper insight in the situation under discussion, we shall
find the induced Riemannian metric on the Beltrami surface embed-
<led in R3. Introduce in R3 cylindrical coordinates (x, r, q» where
x = x, y = r cos <p, z = r sin <p (i.e. the x-axis is the rolation
axis). Then tho metric induced on the surface of rotation with the
generator x = :& (r) is of the form
ds 2 = (dx (r»)2+ dr 2 + r 2 dtp2 = (1 + (.£;)2) dr 2 + r Z dcp2.
In our example x; = - Y7 (see above), i.e. ds Z = arrt +
,.2 dq>2,

4.2. Surface.. Flrsl and SUP/ld Fundamtnla! Form. 18;
Statement 1. The Riemannian metric illduced lit! tlu' Bel/rami .I"facf'
by the ambient Euclidenn metric ill a Lobachel',I..iall metric.
. Proof. By virtue of the transformation u =-!p. v = 1/r we have
v' du 2 du-I- du'
ds ,::... du 2 +-;;r- = -r-' which, apparently. proves the state-
llIent Th\l, the BeItrami sure,ICE' is isolnE!lric (IO('ally) to a Loba-

f
---
'"
J>igure 4,29
chevskian plane, i.e. we have constructed an isometric embedding of
/I domain of a Lobachevskian plane in a three-dimensional Euclid-
ean spare. A question arises: which parl of the Lobachevskian plane
IIt!mits isometric embedding in R3 (/IS the Behrami surface)? Note
!ir:;t of al! that the entirE' Beltrami !'urface Is not Isomelric to any
piece of the Lobachevskian plane. Indeed. the Beltrami surface is
Ao
8 0
Figure 4.30
homeomorphic to II disk wilh a punctured point; hence, if this
ring were homeomorphic to a domain In the Lobachevskian plane, an
infmilely distant point of the Beltrami surface would be mapped into
a finite point on the Lobachevskian plane (see Fig. 4.29). This contra-
dicts the fact that an infmitely distant point of the Beltram. surface
is inflOitely removed from the "funnel" neck.
The real situation is, however, thal the Beltrami surface cut
along any of its generators is mapped isometrically onto a domain of
the Lobachevskian plane (see Fig. 4.30). The corresponding domain

188 I. Smooth Manlfa/d. (E'zamp/es)
(in the Poincar6 model) i!> shown in Fig. 4.: J 1 as (00, AD. Bo); it lies
between two parallel ".lraight" lines (in the Lohachevskian sense)
emerging from the point 00 011 the absolule and the arc Aol3D v.hich
co
Figure 4.at
is II part, of length 2n. or the cirrIo (ill the Euclidean sense) tangenl
to tbe point 00 on the ahsolute.
Thus. the domain (00. AD, Bo) is an infmite strip lying betwecn
two parallel straighllines and bounded from one end hy thp. arc AoBo.
Figure 4,32
Let uS consider, in the Poincare model, two families of coordinate
lines that form an orthogonal network (both in the Euclidean sense
and in the sense of the Lobachevskian metric, since the two metries
differ only by a conformal factor). One famil)' is the set of straight
lines (in the Lobachevskian sense) emerging from the point 00 on
the absolute, i.e. the family of circular arcs (in the Euclidean sense)
meeting the absolute at right angles. The other fnmiJy represents

4.2. Surfaces. Flrlt and Second Fundam.ntal Forms 189
the set of circles tangent to the absolute at the point 00 (see Fig. 4,32).
This family of trajectories coincides with the set of level Jines for
the real Rnd imaginary parts of the complex-analytic function I (z) =
1 z, i.e. ReI (z) -=1.. cos cpo 1m j(z) = -1 sin (I), where z = rei'P.
r r
The eqnations of thee lines are of the form
t t 1. t
r cos q> = cons I' r sm cp = cons 8'
This coordinate network is orthogonal. Whereas one of the families
consists of straight lines in the Lobachevskian geometry, the trajec-
tories of the second family are nol such straight lines. These trajecto-
rie arc uniquely characterized by the property that aII the "perpen-
dirulars" drawn from points of a trajectory are parallel to one another
and intersect at the point 00 on the absolute (recall that the absolute
docs not bolong to the Lobachevskian plane) (see Fig. 4,32). It is
R simple matter to prove (verifyl) that any wo trajectorios of the
econd family are congruent, that is, th!!y can be transformed into
each other hy an isometry of the Lohachevskian plane (i.e. by a diffeo-
morphism preserving the Riemannian metric). Moreover, this
iometry {;an he choen in the form of the homographic function w =
as.l..b I b . I
--;- d ' w lere a, ,C, and d are appropriate comp ex numhers.
cZ-r-
Bllt we <;hall not need this fact below.
Let us consider an arbitrary trajectory of the second family
(a circle tangent to the absolute at the point 00) and a pair of points
on it, AD and BOI spaced at a distance of 2n (we assume that a = 1
and the radius or the Euclidean Poincare model is also equal to
unity). Then the strip lying between two perpendiculars (AD' 00)
and (B o ' 00) drawn at Ao and B is isometric to the Beltrami surface
cut along a meridian, and the orthogonal network of meridians anti
parallels on the Beltrami surface is transformed into an orthogonal
network of trajectories of the first and second families on the Poincare
model in the strip (00, AOI Bo) (see Fig. 4.33). On a Lobachevskian
plane (as well as 011 a Euclidean plane) there always exists reflection
about an arbitrary straight line; in particular, reflecting the strip
(00, AD 1Jo) about the straight line (00, AD), we obtain a new strip
iometric to the initial one (00, AOl Bc) and realized as a cut of the
Beltrami surface in R 3 . Reflecting then this new strip (<X), AI' Ao)
ahont the straight line (00, AI)' we obtain the strip (00, A9, AI)
with the same properties, etc. ReflecUon about a straight line on
a Lobachevskian plane is an isometry, so that a trajectory of the
second family through the points (A o ' Bo) is transformed into itself
because any isometry preserving the point 00 transforms trajectories
of the econd family into trajectories of the Same family. All seg-
menls (A R' A h -I)' 1  k < 00, have the same length, 21t. Exactly

190 .t. Smooth MaTillolds (Ezampu,)
the same procedure leads to Ihe strips (00, BIl' BkI)' 1  k < 00,
having analogous properties. Thus, the ('ntire disk D" bounded by
1\ trajectory of the second family (a circle) is deromposed into inli-
00
FIgure 4.33
nitely many strips converging at the point 00 on the absolute. We
can now construct the mApping of the cntin' disk D2 onto a Beltra-
mi funnel (without a cut) in R 3 such that each strip of the t'pe
(00, A /<, A /<-1)' (00, Ao, Bo), Ilnd (00, BI" B Il . 1 ), 1  k< 00, is mapped
.'jgure 4.34
isometrically onlo the Bcltrami surface, the disk D2 being wound
infinitely man' times onto the Beltrami surface (see Fig. 4.34).
Question: Can the entire Lobarhevskian plane, alld I\ot only the
Etrip (00, AD, no) (or any other strip isometric to the former), be
realized in R3 as a two-dimensional surface of I-onstant nE>gative cur-

4.2, Surfac.t. Fir$/ and Second Fundam.nlal Form. 191
vature? As was shown by Hil bert, the answer is no, The following
question also arises: Can we construct in R3 a complete two-dimen-
<ional surface of negative (variable, in general) curvature from zeC(>
hy a negative number? It turns out tbat such a surface does not
I'xi!>t either. The proof of this fact is not trivial and belong Ii}
.'I V. Eftmov.
This is one of the fundamental distinctions between the metric5
of positive and negative curvature. Note that the induced mEotric
011 the Beltrami surface coincides with the Lobachevskian m(J\ric.
In Ihis case, ihe Lobachevskian metric can be deflDed "abstractly",
irrespective of the embedding in a Euclidean space, and because-
(If the above coincidence this metric has a Gaussian curvature.
Although the initial definition of the Gaussian C\lrvature relied
'Ipon the embedding vn-I_ Rn and, hence, the Gaussiancurvature-
depended on the first and second fundamental forms, it appears now
Ihat this curvature seems to be independent of the second form,
i.e. it is only a function of the Riemannian metric @J: in particular.
Ihe curvature K remains unchanged under an isometry of V n - 1
in R". This assertion, which permits the Gaussian curvature to be-
expressed only in terms of the Riemannian metric, will be proved
below.
Let us now consider the mean curvature of a surface V 2 c: R".
As wa aheady noted, the mean curvllhlre is determined by the
way V 2 is embedded in Ra, i.e. by both the first and the second
fundamental forms. While studying the properties of the Gaussian
curvature, we have founll, in particular, two-dimensional surfaces of
a given constant curvature, It was a rather simple matter to give sev-
eral examples of surfaces of constant positive, zero, and negative
(:urvature. Furthermore, it is easy enough to describe all surfaces V 2
of constant c\lrvature, but we shall not prove this statement here.
For example, a two-dimensional smooth, compact, closed Riemanni-
an manifold of constant positive curvature is homeomorphic either
10 II sphere or to a proJcctive pI ane. The clescription of surfaces of
tonstant mean curvatnre is a much more complicated problem than
in the ..se of the Gaussian curvature. Let us consider a surfac
(If the mean zero cnrvaLUTO, the so-called minimal surface. Incidental-
ly, such a surface is characterized hy the property that its area is,
locally, minimal in comparison with the !Ire a of other hypersurfaces
that differ f.'om the initial one only inside (any) ban of a rather
small radius (see Fig. 4.35). The physical model for a minimal
snrface v' 2 c: R is a "soap f.1m" formed on a wire loop when it is
taken out of a \'essellilled with soap water. Generally, the same loop
can support several minimal films. Let liS consider VIe:: R3 and
derive an equation for a two-dimensional minimal surface. Sinc
GL-2MF+EN .
H = EG-FZ , the equatIOn H = 0 takes the form GL -

t92 4. Smooth Manllolds (E;r;amplt.)
2MF + EN = O. If the surface is given by the graph z = f (x, y),
then
dll 2 = (1 + I) dx 2 +2f"fll ds; dy + (1 + f:) dy1.,
L = In M 1:</1 N 1 11 /1
Y1+/H/' Y1+t.Hf3' = Y1+f+t '
and hence
(1 +f) fu- 2/"11/"'/11+ (1 + /) /1111 = O.
The very form of this partial differential equation (whose solutions
are minimal surfaces) shows that these solutions are rather complicat-
ed. (Example: a Euclidean plane R 2 c: R3 is a minimal surface.
since Q "'" 0 (verifyl).)
Figure 4.85, Volume or yn-L ;;as volume or v n - 1 ,
Here is a more complicated example of a non-compact, minimal
surface. Let us consider in R3 two orthogonal straight lines II and l2
intersecting at point O. Fix the line II and let the line l2 move along
1 1 at a constant velocity a and simultaneously rotate about II at a con-
stant angular speed (,) (helical motion). The straight line 1 2 sweeps 01lt
a two-dimensional smooth submanifold V 2 c: R 9 called a right helicoid
{see Fig. 4.36).
Exercise: prove that a right helicoid is a minimal surface. To
-simplify calculations, we introduce on the helicoid the coordinates
induced by the cylindrical coordinates in R3 with the axis 1 1 = Oz.
Note that any minimal surface V 2 c: R3 has a non-positive
'Gaussian curvature, since Al + AI = O.
Another example deals with a contour r c:: R 3 , where by r we
mean a smooth embedding of a set of non-intersecting circles in R.
Consider a non-compact minimal surface formed by the rotation
.of a smooth curve y (t), given by y = a cosh':', about the x-axis, As
a

4.2, Surface.. Flrd and Second Fundamrntal Forms 193
ig known from mathematical analysis, this curve describes the shape
of a freely sagging heavy chain fixed at two points A and B (Fig, 4.37).
J'lgure 4.36
\
\
\
y
o
x
Figure 4.37
Here we assume that the chain is lixed so that the curve y (x) does
not intersect the x-axis. The gravity point.s downwards along the
I'igure 4.38
}>'Igure 4.39
Y-I\xis. The corresponding surface of rotation is called a catl'tloid
(Fig. 4.H8). Let \lg calculate H. It follows from Lemma :5 that
H=A+A= i y"
I 2 )1)/1 + (y')' (1 + (y')')'!'
I
= ( a cosh"; ) -I ( 1 + sinh 2 .;. ) -2'
3
t ] ." ( " :t ) -7
- 2' cos I a 1 T 5111h z " = O.
Thus, 1\ cntenoid i n minimal surfnl'c. The part of the catenoid
\\'hi<oh is bounded hr the cit'('lcs formed h)' the rotation of the points

194 4. Smooth MaTltlolds (Examples)
A and B around the x-axis .ep.esenls a minimal snrfa('c spanned
by tho contour r consisting of the two boundary circles This exam-
ple shows that the p.oblem of linding a minimal surface with a given
boundary I'ontoll. does not have a nnique solulion. Indeed, besides
the solution mentioned above (Fig. 4.:iS), there C'l.ists another mini-
mallJlm with the same boundary contonr, namely, two disks spanned
by the houndar)' drcles (Fig. 4.:19). Whereas this minimal film exists
B
A
"",,.-----......
/; I,.:.( "
" 1 ...
" 1;:':'.1 "
, :' J "
, f" ,-
.......... i.:' ,-
....." \,,:..1 _.....'
--:.:1--
x
Figure 4.40
for any two arbitrary poinlS A and B, a catenoid cannot always be
spannf.'d hy Lhe bonnclary circles. Apparently, if the distance be-
tween Lho poinLs A and n is largl! enOllgh, no catenoid can he constrm.t-
ed (Fig. 4.40). This becomes especially clear when we begir, to
draw the houndary drdes apart, thereby extending the catenoid
constructed for sllffldently close points A anli n. The procedure is
shown in Fig. 4.41. Upon extension the soap film breaks.
A'
--A 8- B
--u--- --- .
Figure 4.41
Anoth"r example of the contour that admits two ohltions of the
equ,llion H = 0 is shown in Fig. /,42. lIere the two minimal fihns
.Ire homeomorphic to each other. The Fllape of the minimal films
('hallgt'. depending 011 the way the cir('le 8 1 is emuedded in R3. For
11Il' slalldard embedding of 8' in the plane (.2", y) there is onl' one
minimal him spanned hy this {'ontom, anrlthis film ('oin(irles with
the disk .r 2 + y< 1. If 8 1 makes two revolutions about the z-axis,
the solulion of thl' equation H = 0 reprcsl!nts a Mohills band (see

4.3. Surfaces. Firlt and Second Fund4mtntal Forms 195
Fig. 4.43) In the case of three revolutions, the solution is a "triple
Mobius band" (Fig. 4.44). It Can be obtained by moving a "trefoil",
composed of three segments of equal length intersecting at the same
angle 211/;{, along a circle standardly embedded in a plane in such
II way that after a complete revolution along the circle the trefoil
5J
First solulloo
Second solullon
=-
Figure 4,42
turns into itself with a rotation throllgh 2111:) (see Fig. 4.44). The
lonstruction of such a surface is similar, in certain senses, to the
(onstruclion of a right helicoid. A "triple Mobius band" is homeo-
morphic to the surface with self-intersections shown in Fig. 4.45.
lInlike the examples considered abo\'e, the minimal film shown
in Fig. 4.44 has many singular points, i.e. the point.s allY open neigh-

Figure 4.43
hOllrhood of wbich is not homeomorphic to a disk. The nE'ighhour-
hood of each such point has the structure shown in Fig. 4 46. i.E'. three
liulf-disks glued along the common diameter. It is a simple matter
to underst,md that (my singular point of a two-dimensional minimal
!ilm, which is a non-compact surface without boundary, has the
structure shown in Fig. 4.46. Indeed. if two half.disks converge at
,\ singular point, then the neighbourhood of the point is homeomor-
phic to a disk. In the case of four half-disks rOn\'crgent at a singular

196 . Smooth Manlfola. (Ezampl..)
point (see Fig. 4.47), there exists an area-reducing deformation of
the neighbourhood, The total length of the segments AB, AC, AD,

 ..-_1...."
/ ,
I \
I \
I I
I I
2 " {
'---'"
Flgu 4044, a = G  : ).
and AR exceeds the total length of A'R, A'R, A'A n , AnC, and AnD
(the decomposition of a four-fold singularity into three-fold singu-
Figure 4,45
larities), For a minLmal HIm with boundary this statement becomes
invaJid. An example is given in Fig. 4.48 where four-fold singular
points fill the segment An. Interestingly, l!n! same contour r, but

4.2, SUrf(iC'. FI"t and Second FundalrlentaZ Form. 197
made of a wire of a !illite thickness, can span another soap fIlm with
three-fold singular points located along AB (see Fig, 4.49).
Figure 4,46
Decomposillol\ ot
 seCllon C Slng  Ulerille8A_A'A'
A .. A' A"
R D R D
Figure 4.47
Figure 4,48
The so-called harmonic surfaces V 2 c: R3 arc closely related to
minimal snrfaces. Let V 1>c given as r = r (u. v), where u and v
are curvilinear coordinates on V.
Definition. The radius vector r (u. v) is called harmonic with
respect to the coordinates u, v if :: + : = 0, i.e. .1.r = O. where t.
is the Laplacian in the ('oordinales u, v.

198 4. Smooth Manifolds (E:zampl$)
The radius vector r (u, (,) which is harmonic in the coorclinale
(u, v) nec(1 lloL be necessMily harmonic in oLher coordinates u', L".
Definition. The surfare Vc: R' is calJ('cI hal'monic if iL can be
define!1 bv a harmonic radius vocLor r (u, 11) in curvilinear coordi-
nates u, t',
The raclins \'('('lor r (u, l') is :lairllo he minimal if its mean curva-
ture is identically zero. The fun<:1 ion H = A( + A1 is a scalar and, in
Tnere is no
soap film
here
Figure 4.49
particular, remain unchanged under a regnlar coordinate transforma-
tion on II plane, sO Ihat if a radins vector is minimal relative tl'
a coorclinate system, iL will also be minimll1 relative to any other
regular ('oordinate svstem. Thus, the concept of a minimallilm does
not depend on the coordinates valid on this hIm. This is not however
the case for a harmonic surfaco, and one usually speaks about a har-
monic mapping r: D (u, v)_ R3 (x, y, z), where D (u, v) is a domain
on the plane (u. v) and r (u, v) is the mapping clefining the surface
V 2 c: IP. The mapping r, which is holrmonic in some coordinates,
will not generally he harmonic in other coordi nates (give an exam-
plel). lIere is an e-.:arnpJe of II harmonic surface: let r (r. y) be deHneci
by Ihe formula r (x, .'I) - (r. y, .r z - y), where x, y, and z are Car-
tesian coordinates in I\", i e. the surface II! is given by the graph
z = x 2 - y re[erro<1 to a Cart('ian coordinate system, Appat'entl,
( a' , a' ) 0 ' I f 0.' I . L." t
(jX'i' ,- ago r .." ,I.I! tIC sur ace z '"'" X' - y- IS Incmontc uu_ I
is nol minimal [,eclltI«' If = 0 only at the point (0,0), while at other
points H =1= 0,
We recall that ('\It'vilinear coordinates (u, v) on a plane V 2 c: R'
are called conformal if in these coordinates the metric: Edu -

42. S..rfacts First and Second F..ndamental Forms 199
2Fdudu + Gdu induced 011 V' by the ambient Eudille<ln metric iJ'
diagonal, i.e, E = G and F = O.
Remark. Let lIS consider a two-dimensional smooth Hiem<lnninn
manifold sl1pplied with the Riemannian metric ds. with real-8nal.tic
coefficients E, F, and G which are considered as functions of thl'
local curvilinear coordinates u, v (this metric is not necessarily
induced by the embedding of M in R 3 ). Then for any point P E JI/
there exists a neighbourhood U = U (P) such that in U we can
introdnce coortlillates p. q (they are real-valued nnalytic functions o[
the initial coordinates) in whichds2"is of the form A (p, q) (dp. + dq),
Le, the coordinates p, q aro conformal. The proof is not difficult,
bllt requires the use of the theorem on the existence or II solution
to a special partial difrerential equation ,the Beltrami-Laplace
equation), and this theorem is beyond tile scope or onr course.
What is a relationship between a harmonic and a minimal vector
in R3? The example of a harmonic vector whil'h is not minimal
has aiready been given. Is tIle converse l.ltement valid, i.e. is nny
minimal vector II harmonic one? The an\'.'er is no (give nn ()amplel).
Yet. the following statement boLd!; tfUC
Statement 2. A minimal vector written in conformal coordinates IS
a ho.rnwnic one.
Proof. Let (u, v) be conformal coordinalC's and let f (/I, I') be the
vector describing a minimal film. We recall that E = G = {r,..
r,,> = {r lt , Tit>. F = (rlt. rv> We need to prove that r lt .. -i- rfl' = O.
Put  = r ltu + f"D' Since H = 0, we have CD - 2MF ..L HN = IJ,
i.e. L + N = 0, whence {a, n> = {r"", n> + (rrro n> = L - N = U,
wbere n is a normal to Vl. It remains to prove that (a, r , .) =
{a. r,,> = 0, since in this case the scalar products of a and the
orthogonal vectors r lt , rv will vanish, i.e. n is a ;j;efO vcrtor. Differell-
tiating the identities E = G and F = 0, we obtain (r ltu ' 1'" > -.
(r"., rv>, (r u " r lt ) = (r,.,. r.), (r lt ", r.,) - - (r", r lt .), (rill'
r.) = - (r", r v .')' whence (a, r,,> = (r".. r,) - (r. u ' r, > = O.
Similarly, (a, f D ) = O. The statement i:; proved.
Remark. We have aserted above (without proof) that conform",1
coordinates can be introduced in a neighbonrboocl of any point 011
a two-dimensional real-valued annlytic surface. It cnn he proved
that any minimal film can he derme(1 by 1\ real-valued analytic vec-
tor and, therefore. conformal coordinate can always he introduced
in n neighhOllrhood of any point on a minimnl surface (lhis fact is
not trbial).
Problems
1. Prove that if the Gaussian and mean curvatnres of
a surface in R3 are identical! V zero, the surface is II plane.
2, Let the surface S be formecl by tangent straight lines to a curve.

200 4. Smooth. Mantfold. (Ezamplel)
Express the Gaussian and mean curvatures of S in terms of the
curvature and torsion of the cnrve.
3, Demonstrate in the preceding problem that the metric on the
surface depends only on the curvature of the curve.
4.3. TRANSFORMATION GROUPS
4.3." SIMPLE EXAMPLES OF TRANSF.oRMATION GROUPS
In this section we shall discuss the basic examples of the
transformation groups of metrics, i.e. such transformations of a mani-
fold that preserve the metric. Consider a Riemannian manifold M"
wi th the metric g IJ.
Definition. The difieomorphism' f of a manifold M" onto itself is
a length-preserving mapping or an isometry if this mapping sends the
Riemannian metric g'j into itself, i.e. the following identity holds
true:
iJzl (g) ax} (g)
ghJI(y)""glJ(x(y»"'OYP '
where yl, . . ., y" are local coordinates in a neighbourhood of
a point y E M n , ;z\ . . ., :J!' are local coordinates in a neighbourhootl
of a point ;z; EM", and xl =.2; (yl, .. , yn), 1 i n, are the
functions defining (locally) the mapping I, wi th x = I (y).
This is the "coordinate" delinition of sn isometry. It is sometimes
convenient to use an invariant defmition without any reference to
local coordinates which cannot be chosen uniquely. Under a map-
ping I, the differential dt sends TyM" onto T :eM", the latter mapping
being a lineur isomorphism since f is a diffeomorphism. In each of
the tangent spaces T "M" and T xfr1" one can de fIDe scalar products
(')u and (, )", respectively, induced by the Riemannian metric,
i.e. in the local coordinates yl, . . " yn, we have (a, b)u =
g'J (y) alb!, where 8, bE TuM".
Definition, The .diffeomorphism 1 of a manifold M" onto itself is
called an isometry if (a, b>y = (dt (a), df (b» '" for any a, bE
Tu M "; x = f (y)
L('mma 1. The coordinate and invariant defillitions 01 an isometry
are equil'alent.
Proof. Let a E TyM", a = (a l , . . ., a") in the local coordinates
yl, . . ., y". Hence, dt (II) E T",M" is of the form
(d! (a»! = ()Zg) a h ,
since df' ryMn_ T"M" is givcn by the Jacobi matrix. Thus,
(df (a), dl (b)" -= glJ (x (Y» i1Y;y) i!z:y) ct'a ' > --= ghpaha P ,
,,'ldch proves the lemma.

4.9. Transformation Groups
20t
Lemma 2, The set of all isonutries 01 a Riemannian manifold M"
lorms a group (in the algebraic sense).
Proof. That the composition of isometries is an isometry follows
from the rule of differentiation of a composite function and from the-
!aw of trsnsformation of gi/ under coordinate substitution. Also,
that 1-' is an isometry stems from the fact that the Jacobi matrix
of 1-1 is the inverse of the Jacobi matrix J (I). An identity transfor-
mation should he ("hosen as the unit element of the group, The lemma
is proved.
The isometry group of a Riemanman manifold M" is usually en
(lowed with a topology in the mapping space and is dencted by Iso (Ill")
Let us consider simple examples.
1. As M' we shall take the real axis (a non-compact one-dimension-
,II manifold) with the Euclidean metric ds = dx 2 , where or is the-
coordinate on the axis. Let I be the diffeomorphism of R' onto itseH
dclined by a strictly increasing (or decreasing) function x = f (y);
Ihe condition that I is an isometry implies that ds Z = (f)2dy2 =
dy, i.e. (f)2 = 1, so that J is either I (y) = y + a or / (y) =
-y + b, where a and b are arbitrary constants, Thus, the Iso-
IJ1!:try gro.lp of a real axis is homeomorphic to a pair of real axes
(proper isometries, which preserve the axis orientation, and im-
proper isometries),
Tills example is virtually the unique one to find a complete isomet-
('j group of a manifold on an elementary level. The fact is that it is
rather difficult to prove the completeness of a given subgroup in the
isometry group. We succeeded in doing so only because the smooth
function with a constant derivative is linear. Later we shall introduce
the concept of a geodesic for a Riemannian metric and shall be
able to prove the completeness of some subgroups in the isometry
group,
2. Let us consider a Euclidean two-dimensional plane and find the
isometry group that preserves point 0, the origin. We shall seek the
iometries among linear transformations of the plane (it can be
demonstrated that any isometry of a plane is linear, but here we
shall Dot concentrate on this topic). The condition that the metric
d.x l + dy2, giJ = 6ij, is invariant can be written as the matrix
('quation E = AAT, where A: R2_ R2 is a linear transformation.
This is the definition of the orthogonal group, tlJP. group 0 (2) in
ollr case, which consists of the matrices ( cs (P sin 'P ) (proper
- Sin qJ COS q>
, ) d ( COS !P sin Ip ) ( . . n ' )
rotatIOns an. Improper rotations or re ectlons .
sm (j) -cas q>
Proper rotations form a subgrollp in 0 (2) denoted by SO (2); improp-
er rotations do not form a subgroup. The subgroup SO (2) is an
i IJvariant suhgroup in 0 (2); hence, there is valid the factor group

202 4. Sm"oth Mant/olds (E>:amplts)
o (2)1S0 (2) isomorphic to Zl (the I'econd-order cyclic group) The
group 0 () il' iI subgroup in the isometry group of a circle ondowed
with the sland ,m! Riemannian metric ds = d<p". Since 0 (2) consists
of matrices, this group becomes a topological space i( the angle <p
is associated with ('ach matrix (for proper rotations). Thus, the set
of the matrices that form 0 (2) is homeomorphic to two copies of
.a circle; in other words, 0 (2) can be provided with the structure of
II smooth one-dimensional closed (disconnected) manifold (Fig. 4.50).
$0 2 (  ) , 50(2)
GO
'--'
0(2)
l'igure 4.50
Prove that 0 (2) coincides with the isometriy group of a circle
with the metric ail = dcp2. Hin.t: use the procedure similar to that
for the group Iso (RI).
3. The motion of a Euclidean p]ane can be written as y = Ax +
b, where A EO (2) and the vector b de/ines a translation on the
plane. Apparently, all snch transformations preserve the Euclidean
metric (verify!). As is shown below. there are no isometries of a plane
other than these transformations. This group can be represented
in the matrix form
T - 1 A b
-I n 1\
(Prove that the coorrespoildence ()' = Ax + b) __ T is an isomor-
phism of groups.) Hence, just as !II the preceding example, the group
Iso (R2) can be transformed into a topological space homeomorphic
to the direct product of II pair of circles and a Euclidean plane, and
therefore this set can have the structure of a smooth three-dimellsion-
al manifold (this mallifold is non-compacl and disconnected and
consists of two connected componellts).
4. Let liS consider indefinite metrics. Defillo on R2 an indefmite
metric _d:1;2 + dy2 which transforms a two-dimensional space into
a pseudo-Euclidean plane R. The matrix of the first fundamental
form B is constant: B = ( - 1 0 ) . Hence. in order to fmd all linear
o 1 '

4.9. Transformalion Groups
203
homogeneous transformations preserving this matrix, we have to
solve the equation B = ABA T where A' R -+ R is a linear map-
ping. Representing this mapping in the form A = (; ) I we arrive
at the system of equations for a, b, c and d: a 2 - b 2 = 1, ac = bd,
d 1 _ c' = 1. This system has the solution:
( :I:: cosh.p :I:: sinh "' )
A = :I:: sinh", :I: cosh 1j> ,
ur
A=(:
1
V1-'
II
Vi-II"
II
:I:: 1 1 I _. )
1 '
:I: V j _ .
wht'rc- ..!!... =, f3 -=-- tanh '.
a
ami the following sign combinations are possible:
(: :)E11 (= =)E@j2 1 (: =)E3' c= )E@j"
( + - - +)
This is a complete set of combinations. For example,
hlongs to @jl' since substitution of -I/: for tj; transfurms
( ) into ( ) (we recall that sinh (-) = -sinh 1/=.
cosh (- 1j» = cosh 1j». Thus, (3! = @jl U @3. U @l3 U @j, (verify!)
and @Ii n @Ii = 0, provided i <P j. Indeed, suppose, for example,
that @jl n @!. =1= 0, then cosh cp = - cosh 1j> and sinh <p =
-sinh 1j>, which is impossible since cosh q> > 0 for any rp. It can
be shown in a similar way that i n @jJ = 0.
Since @! is the group of homogeneous isometries of R, transforma.
tions of this group map the set {-x' + y' = 1} U {-.x' + y' =
- 1}. i.e. a pair of pseudo-circles of a real and an imaginary
radius. into itself. In the case of a Euclidean plane R2 each rotation
was described by the angle <p of the rotation of an orthogonal frame;
an analogous parameter can be introduced for a pseudo-Euclidean
p]ane R". Let us consider an orthogonal frame e l = (1, 0), e. =
(0, 1). Then the isometry A transforms this frame as is shown in
Fig. 4.51. Also, inst.ead of an ordinary Euclidean rotation angle q>
we shall introduce the angle of hyperbolic rotation 1j> by setting
 = tanh 1j> (see above), in this case @j becomes a group of hyper-

204 4. Smooth Manllold$ (Ezample$)
bolic rotations. Recall that, the group of orthogonal transformations
of a Euclidean plane. consists of two connected components, The
group of hyperbolic rotations has a more complicated structure: it
consists of four connected components, namely the sets CM" @l2'
<!I 3 . and <!I,. i.e.
{( :), (= =). (: =). (= :)},
-00<1jJ<+00. Since we have realized @I as a group of matri-
ces, the group @I CIIn be embedded tiS a subset in a four-dimen-
sional real Euclidean space of all matrices of the form (: =)
(a, b, c. dE R) and therefore @I inherits the topology which tllrns
,
" '......
, ....-
, -
, e
,
,
,
,
e,l
\
, \
, \
, \
',\
"
,
x
Figure 4,5{
<!I into a topological space. Each of the subsets <!I" h;;;;;i4. is
pathwise connected relative to this topology. Indeed, consider, for
example, matrices of the ype of @I" Then for Any two matrices
( cosh 1jJ, sinh 1jJ, ) ( cosh 1jJ2 sinh 1jJ2 ) .
, h .1o h .1. and . h h .I, we can choose a contm-
sm '1'1 cos '1'1 SID 1/>2 cos 't'
uous path 'I' (t) which connects them in the set @I,: namely,
( cosh t sinh t )
. ht h =.y(t), 1/>,l1p2' The pathwisc connectedness of
sm cos t
the remainillg subsets CM 2 , <!I 3 . CM can be proved exactly in the
same manner. Among these four connected components only @II'
( cosh 1jJ sinh 1jJ )
i.e. sinh", cosh", ,is a subgroup (verify!). None of the remaining
components is Ii subgroup. For eXllmple, the product of two matricl'g

4.9. TraM/ormation Groups
205
oftheform (= =-) gives (: :), Le, if a, /3E <!Ig. then
(X'j3 E@!. and a'/3@!2' The identity matrix belongs to <!I" The group
@:I is II subset in the four-dimensional space of the matrices
(; :), and each <!It, 1:O;;;i:O;;;4, is homeomorphic to the real axis,
'l'his homeomorphism (say, for <!II) is realized by associating with
) , ge,
(1=g, :"-..
'93r
'\ f'
Figure '-52, g1: = e = ( ). Ir,= (- _), gs= ( _). 8. =
{ -1 0 }
o l'
( cosh", sinh $ )
each matrix sinh $ cosh 1p the angle 'Ij>, the correspondence
heing one-to-one and continuous (see Fig. 4,52), The group <!I maps
the pseudo-circle of imaginary radius _,t2 + g2 = -1 into itself.
Figure 4.53 illustrates four transformations involving <!I l , 
<!Is, and <!I,. The group <!I transforms a pseudo-ciJ:cle of real radius
in a similar way,
The group <!I is commutative (verify!), In this respect it resembles
the group 0 (2), which is also commutative. The subgroup <!II is an
invariant subgroup in @:I, since g_lqg, where q E <!I l , K E <!I, is, as
before, transformation of tYPB 1, for it is obtained from transforma-
tion of type 1 by hyperbolic rotation, Thus, tb.e factor group <!I/<!I I is
valid and its order is equal to the number of connected components
in <!I, i,e. 4. Since <!I is commutative, <!I/<!I. is also commutative,
There exist only two commutative groups of the fourth order; name-
ly, Zg ED Zg and Z" A question arises: to which group among these
two is <!I/@!l iomorphic? Let us compile the table for the multipli-
cation of gl' Kg, gs, and g, which are representatives in their con-

206 4 S ",oath M anlto/ds (E;umpl..)
nected components (see above). Calculation yields
I I I 2 I 3 I 4
I I t I 2 I 3 I 4
2 I 2 I 1 I 4 I 3
3 I 3 I 4 I t I 2
4 I 4 I 3 I 2 I I
Thus, @J/I is isomorphic to Z2 ID Z2' This is a distinction be-
tween the rotation group for a pseudo-circle and the rotation group
for an ordinary circle.
5. Let us consider the isometry group of a two-dimensional sphere
defmed as a Riemannian manifold with the metric induced by a stan-
dard embedding in R3. We first turn to the case of R n and ftnd the
group of linear homogeneous transformations A preserving the
n
Euclidean metric ds 2 =  (dXi}2. Since the malrix (giJ} is of the form
;-===1
E = (" :). we have E = AAT; the solutions of Ihis equation are
represenled by ort!lOgonal matrices which form the group 0 (n). This
group contains the subgroup SO (n) consisting of proper rotations (with
the determinant equal to + 1); the remaining (improper) rotations
do not form a subgroup, The subgroup SO (n) is an invariant subgroup
in 0 (n) and the factor group 0 (n)/SO (n) is isomorphic to Z2' Let
n = 3, then 0 (3) preserves the Euclidean metric in R' and therefore
maps the sphere 8 2 , X2 + y2 + Z2 = R2 = const, into itself. Hence,
o (3) is a subgroup in Iso (S2). FurLhermore we shaH demonstrate
that 0 (3) == [so (8 2 ). Consider 80 (3) c: 0 (3). Since SO (3) is
realized as a subset in the space of all 3 X 3 matrices with real co-
efficients, identified with R9, this subgroup is provided with an in-
duced topology and becomes a topological space.
Lemma 3. The group SO (3), as a topological space, is homeomorphic
10 a three-dimensional projective space RP3.

(3. Translnrmalitl" Grnups
207
y
c
x
x
d
Figure 4.53. (a) g,: = e = ( ). jdntity mapping; (b) C2 = (- _).
I . h . . ( 1 0 ) n .
If.nl!ctl<)11 I{' alivl! to t C or.!;.n; (e) g, = , re cetton relative to the-
o -1
( -t II )
r-axis; (<.I) c. = 0 1 ' renection relativo to thc y-axis.
PrIJof. Lot A be an clement of SO (3); then there exists in R3.
a lixcd axis i (A) slIch that the action of A in R3 is rotation II hont i (A)
through an angle (j' (A). If A "}6 E, lfTl'I1 i (A) is defined uniquely.
Consider a plane IT (A) through the origin 0 perpendicular to the
,1'I:i8 i (A); choose in n (A) an arbitrary vector 0) and let e. he th
vector ohtailled from 0, by the rotation thrOllgh the angle cp (A)
(see Fig. 4.54). Complete c.' e. to a coordinate frame (e], e., e a ), C)o
that its orientation coincide with the orientation of a fixed frame
(a" 8,. aa) in R3. Then l (A) hecomes a real axis if we can define on
it direction with the aid of e a and reckon thc value of cp (A) TIJ1I!>.
we have uniquely associated with each element A E SO (3) a point
in R3; denote thi point hy P (i, c). Apparently, P (l, 1'£) = l' (i, -1'£)
ince rotatiolls about i (A) through 1'£ and -1'£ coincide. If I 'p (A) 1<
:t, P (i, Ip) corresponds only to the rotation of A. By continuously

208 4. Smooth ManifollU (Ezample,)
varying A, we continuously change P (l, cp), the converse is also
true. Thus, we have realized a one-to-one continuous correspondence
between orthogonal transformation of 1 and p!nts of a three-
dimensional ball of radius 11 whose antIpodal pomts P (l, 11) and
Figure 4,54
p 0, -11) are glued on the boundary (i.e. on the sphere of radius n).
It remains to prove that this ball glued on boundaries is homeo-
mOl'phie to RP3. According to one of the definitions of RP", this
spaee is a sheaf of straight lines in R& through the origin 0; such
 ..:::::::.:,.......s
"'m::::.. ..::::::::"..
..:::,o..:WU::::::'::.
;0 _____f
4liI1i
R3
D3
.FIgure 4.55
11. model is equivalent to the one in which we choose S3 and identify
its antipodal points. To this end, we take a hemisphere S; and
identify the antipodal points on its boundary, Le, on the eqlttor
(Fig, 4.55). A hemisphere is diffeomorphic to a three-dimensional
.disk, and the diffeomorphism can be obtained by orthogonally pro-
jecting S onto D3 (see Fig. 4.55). Thus, Rp3 is homeomorphic to D3
with identified antipodal boundary points, The lemma is proved.
Since Rpa is pathwise connected, 0 (3) consists of two path com-
ponents.

4.3 Tan'fomallon Group.
209
6, Let us consider the isometry group of a L"bachevskian plane
with a standard Riem.annian metric. The Lobachevskian plane is
assum.ed to be defined on the upper half-plane with tho metric
dl d! . The isometry group of this metric will be sought among
(.-%)'
homographic transformations of the complex plane ::t: ' a, b, C,
elE C. Let us study how the Euclidean metric dzdi behaves IIn-
I f t . a.+/. S ' d ad-be d ( .
der t II! trans arma IOn w= C:+d ' mce w= (e%+d)" 'z veri-
fy!), we have
- I ad-be II -
dwdw= I c%+d " dzdz,
Figure 4,56
thnt. the' oriented angles are also preserved, Calculating the Iaco-
bian J. (w = ::t: ), wo find that it is positive. Let z E R2,
wE R2, w = ::: ' dw =;.. (z) dz, I. (z) = (::;:;;I ' T.R2= R2;
TwR2=R2 (see Fig, 4.56). Hence, the operation dt of tll" mapping
t: z -- w is written as df (z) = ;..,z, where 1..E C. Let ;..=u+ tv,
U, u E R, then the Iacobi matrix in the rool form is ( It V ) ,
-v u
i.e. the Iacobian is eqll/ll to 1l 2 + v 2 > 0, O"t of 1111 homographic
transformations we choose those that map the upper half-plane
into itself,
al+b
Lemma 4, The transformation w = d maps the upper half-
c:'z",
plane into ilself if and only if (a, b, c, d) = r (a', b', C', d'), where
a', b', c', d' E Rt pEe, p =F 0; i.e. all the coefficients la, b, c, d)

210 <I. Smooth Mantfolds (Ezamp$)
are proportional to the quadruple of real numbers (a', b', e', d') and
ad - be > O.
Proof. Let the quadruple of the coefficients be proportional to the
real quadruple (denoted by a, b, e, cl). Apparently, the real axis is
mapped into itself. We now demonstrate that if the point z belongs
to the upper half-plane, its image w = :t also belongs to this
half-plane, i.e. 1m ( ::: ) >0. Indeed,
(as+b) (c;+d) ac(..)+bd + ads+be;
w= I cz+d I" I ez+d J' I cz+d" '
I ad-be 0
m w = I ez+d I" . 1m z > ,
since ad - be> 0; 1m z > 0. Conversely, let w = =:: map the
upper half-plane into itself. We have to prove that there exists
a common factor p such that (a, b, e, d) are proportional to the real
quadruple (a', b', c', d'). Ifz = z, then w = W, and for an arbitrary
x E R we obtain
IU-t-b '_ 4z+b
cz+d - cz+d '
b b
For .1:=0'7= Ii =A, AER. For x - 00,
-
=  =ER, h= },d, a=J.tc,
c c
for x = f ,
a+b a+b
+d =-:-::-=pER, J!c+Ml=pc+pd, (/!-p)e=(p-A)d.
e c+d
In the case of general position, i.e. when /! - P * 0, P - A * 0
we obtain c = d,  E R, Thus, all four complex numbers a, b, e,
and d lie on a single straight line (Fig. 4.57). Multiplying these
numbers by a complex factor, we can rotate this straight line so
that it coincides with the real axis. The lemma is proved.
Lemma 5. Any transformation W = :t sr.u:h that a, b, e, d E R
and ad - be > 0 Is an isometry 0/ the Lobachevskian plane.
Proof, We have
d ad-bc d
W = (e z -! d)' Z,
which is what was required.
dwdw
(w-w)2
dz dz-
(z-:i)' '

4.3. Tran$formation Groups
211
Since ad - be: > 0, wo may take ad - be = 1.
Proposition 1. The isometry group of the Lobachevskian plane
Iso (L 2 ) contains a subgroup Isomorphic to the group SL (2, R)/Z.,
i.e. to the factor grOl.lp SL (2, R) of 2 X 2 matrices with real coefficients
y
o
Figure ,51
and determinant +1 with respect to the subgroup Z2 consisting of the
transformations E and -E.
. 62+b
Proof. Let us consider w = c:+d ' where a, b, c, d E Rand
ad - bc = 1. According to Lemma 5, these transformations are
isornetries. The set of these isometries forms a group. Indeed,
( a' ( uz+b ) + b' )j ( c' ( "z+b ) + d' ) (a'a+cb')z+(u'b+db')
cz+d cz+d (c'a+cd') z+(c'b+dd')
is again a transformation with real coefficicn ls and determinant
+ 1. Since ad - bc;/= 0, there exists an in verse ttllnsformation of
the same type, Consider the matrix group SL (2, B), i.e. the group
(; :), where a, b. c, dER and ad-bc= 1, and construct the
mapping (jJ: SL (2, B) - @I' where @t is the group of transfor-
mations w= ::t: , a, b, c, dEB, ad-bc--1. Pllt q>(; :)=
:; . Obviously, cp is a homomorphism (verify!) and, further-
more, cp is an epimorphism. Find ker4p. Clearly, cp(g)=4p(-g),
so that ker q> ::J Z2 = (E, - E). Prove that ker q> = Z2' Let q> (g) =
, . az+b 0.',-' b'
cp (g). I.e. cz+d = c'z+d' I whence
b d a c a+b ,,' +b'
V=7=J.., 7=7=/1, c+d =7+Ci" b=Xb', d=}..d',
a= j!a', c= p.c', p.a'd' + }"b'c' = !J.c'b' +}"d'a'.
(!J.- A) (a'd'-b'c') =0, V.=A, a'd' -b'c'= 1.

212 . Smooth Malllfold, (Ezample,)
Thus, v= :' =.;,.= ;, =A, i.e. g'= i.g, 1..=::1:: 1, g' = :l:g, which
is what. was required.
Problem. Prove that SL (2, R)/Za is a connected topological
space,
One should not believe that only transformations SL (2, R)/Za
are contained in the group Iso (La). Indeed, the transformation
go: z ..... -: maps the upper half-plane into itself and preserves the
y
2'
2
x
Figure 4i.58
Lobachevskian metric (reflection about the y-axis). At the same
time, go is not of the form ::t . Indeed, ::t: are conformal
(see above), i.e, they preserve oriented angles, while the transforma-
tion Ko: Z ..... -z is not conformal (see Fig. 4.58),
Thus we should also consider transformations g(z) of t.he form
w=g(z) = - a+b , where a, h, c, dER, ad-bc= 1. In other
ca+d
words, w= Q+b , where a, b, c, dER, ad-bc=-1. Let @la
c.+d
denote the set of all 8uch transformationS, The sets (!Ij and @la
are homeomorphic, since any g E @I is of the form II = got, where
! E @lit and since go and ! are isomelries, go! is an isometry as
well. Tha horr.<!omorphism is established by multiplying the set <9s
b 1!1 Al (t;I I<l ,aa+b (%:+/1 I d
y goE""t. so ""I n "":= 0, smce - +  'jiE-:--, ndee,
ca.. '\1.+6
:' + +: preserves oriented angles, while a+1J does not.
3 '\1.+6
Lemma 6, The set @I =<MI U (&:1 2 = {t} u {go!} is a group in which
@I is a subgroup and @Ja is not a subgroup,

4.a, TrallS/ormatton Group
213
Proof, Consider the set of all real.valucd 2 X 2 matrices with
the determinant :::!:: 1. Apparently, this is a group; we denote it
by L (2, R), Le, {(; ), ad - be = ::I:: 1} . This group, as a topo-
logical space, is disconnected, for it has two connected compo-
nents: L (2, R) = LI U L 2 . where
L 1 ={(; ), ad-bc=-1}. L 2 ={(; ), ad-bc=-1}.
The subgroup L 1 was denoted above by SL (1. R), Let us construct
rp: L -+@J. If
A=(: :)EL I , then Ip(A)=fE@J.. f(z)=' ::t: '
ad-bc= +1.
j[
B=(: :) ELt, then Ip(B)=gE@3t, g(z) = :;:: '
ad-bc= -1.
Clearly, Ip is a homomorphism (verify!), Furthermore, the mapping Ip
is an epimorphism. it is not one-to-one and has a kernel. To find the
kernel t it is sufficient to fmd the inverse image of the identity ele-
ment of the group @3. As in P.roposition 1. it is a simple matter to
prove that ker (<p) = Zt. where Ze = (+E; -E)i this subgroup
is the centre in L (2, R). The lemma is proved. This means that we
have also proved the fonowing statement.
Lemma 7. The group @3 = @31 U 1S t = {I} U {g} is Isomorphic
La L (2, R)/Z g , where
L(2, R)={(: ;), ad-bC=::I::1} , Ze=(E. -E).
Thus, in the isometry group of the Lobachevskian plane we
have found 1.1 subgroup consisting of two path components. namely,
@3 9i: L (2. R)/Ze. How many parameters are needed to describe the
elements of this group? Since L (2, R) = {(; ;). ad- bc = 1} ,
the element gElS is determined by three independent parameters.
Lemma 8, The group SL (2, R), as a topological space, i8 homeo-
morphic to the direct product of a circle and a Euclidean plane and can,
therefore. be provided with the structure of a smooth three-dlmen810nal
(non-compact) manifold, Correspondingly, L (2, R) is homeomorphic
t? the direct product of two copies of a circle and a Euclidean plane.
Proof. It is known from algebra that any linear homogeneous trans-
formation of a Euc1idean plane with the determinant +1 can

214 4. Smooth Manifol.u (Ezampl..)
uniquely be represented as the composition of the proper rotation
and triangular transformation (the basis orthogona1ization theorem).
Thus, any matrix g E SL (2, R) admits uniquely. a representation
as the matrix product g = ( cs <I> sin <1» _ ( : f ) . Since the
- Sin IF COS rp (%
( cosrp SinCP )
matrices. fonn a circle and the matrices
- sIn q> cos <p
(: f ) form a Euclidean plane, the lemma is proved.
The topological structure of L (2, R)/Za is more complicated. As
is shown below, the suhgroup L (2, R)/Z. is, in fact. the whole group
Iso (L 2 ). Note also that the isometry groups of a sphere and a Loba-
chevsJrian plane have the same dimension and can be endowed with
the structure' of a smooth three-dimensional manifold,
Remark on the isometry group of a pseudo-Euclidean space R.
We recall that the isometry group of a two-dimensional sphere coin-
cides with the isometry group of R3 that preserves the point 0 fixed,
[)  " C1
,PI !:, I,'I///'
!f! f; a'"
[) .\  
.'. .....
w..... ::l.."!
, d
Figure 4.59, (a) g, = e, identity transformation; (b) g., reRection
in the plane 1/<; (c) g., reflectlon at the point 0: (d) g., reRec-
tlon in the plana yz.
i.e. with the group 0 (3). Similarly; a Lobachevskian plane can be
realized in R as one of the sheets of a two-sheet hyperboloid
(a pseudosphere of imaginary radius), and any motion of the Loba-
chevskian plane will be induced by an isometry of R;. Since a pseu-
dosphere of imaginary radius consists of two connected components,
Iso L. is not the whole group Iso (R')o. where Iso (R)o denotes all
isometries of R: that leave the point 0 fixed; reDection-induced trans-

4.3. Transformation Groups
215
formations, which represent two sheets of a hyperboloid, should also
be included. Thus, the group Iso (R)o consists of four connected
components. Figure 4.59 shows four transformations, glo g2' g3'
and g" which preserve the pseudosphere S => L 2 and belong to
different connected components of the group Iso (R3)o.
7. Let us consider again the group of motions of a Euclidean plane.
The transformations y = Ax + b, A EO (2), bE R2, found above
can be written in the complex form w = az + b, where bEe,
a E C: I a I = 1, i.e. w = e;OP. z + b. This transformation group is
isomorphic to the matrix group consisting of matrices ( ),
where a = ei'l'. That the Euclidean metric is preserved directly
follows from the identity dw = adz; dwdw = I a 1 2 dzdz.
Just like in the case of a plane, it is a simple matter to verify
that the group Iso (R n ) of linear isometries of a Euclidean space R"
can be represented as the transformation group y = Ax +- b, where
the matrix A belongs to the orthogonal group 0 (n) and tl$ vector b
delines a translation. The group Iso (R n ) is isomorphic to the matrix
group cQnsb;ting of matrices of the form
=8.
o . .
Thus, the group Iso (R n ), as a topological space, is homeomorphic
to the direct product 0 (n) X R n , where 0 (n) and R n are considered
as topological spaces. This decomposi tion is not, however, a group one.
4,3.2. MATRIX TRANSFORMATION GROUPS
We have seen that all the examples of groups considered
above are topological spaces on which the structure of a smooth
manifold can be introduced in a natural way. We have used the
topology which arises on a transformation group after it is embedded
in a matrix group whose topology is defined in a usual manner:
matrices are assumed to be close if their elements are close, Thus, we
arrive at a class of smooth manifolds such that points on them can
be "multiplied", this multiplication satisfying all axioms of an
algebraic group. Let us give the following definition.

216 4. Smooth Manifold. (Ezamplt.)
Definition. A Lie group is a smooth manifold M" endowed with
two smooth mappings j: Mil X M" _ M" (multiplication) and
v: M" _ M" (construction of the inverse element), usually denowd
as j (x, y) = x 'y, v (x) = X-I, and having a marked point e EM"
which satisfIes together with j and v the relations: (1) oX '(y ,z) =
(x. y) ,z, (2) e. x = x ,e = x, x' X-I = x- 1 . X = e,
The operations f and v are usually required to be continuous, but in
all examples considered below the group operations are smooth,
that is why smooth mappings f and v have been used in the definition
of the Lie group.
Definition, The set of elements g of a Lie group @I that can be
connected by a continuous path with the unit element of @I is called
the connected component of the unit element of the group @I and is
denoted by @:Io.
Statement 1. The set l!J o is a subgroup in @I. Furthermore, @I 0 is
an invariant subgroup in @I, So there exists the factor group @I!(So'
Proof. Let gl' g. E @:Io; we need to prove that gl ,g. E (So' Accord-
ing o the delinition of @lo, there exist continuous paths VI (t)
and 12 (t) such that'l'l (0) = e, '1'1 (1) = gl and '1'2 (0) = e, '1'2 (1) =
g2' Consider the path '1': I _I!J, where 'I' (t) = '1'1 (t) ''1'. (t). Since
multiplication is continuous in @I, the path 'I' (t) is continuous and
therefore gt' g2 E @lo because y (0) = '1'1 (0) . '1'. (0) = e; 'I' (1) =
'1'1 (1) . 1'. (1) = gl 'g., Demonstrate that for any Co E @lo and any
It E @I tho element ggog-1 belongs to @:la' Since go E @lo, there
exists a continuous path 'I' (t) such that y (0) = e, 'I' (1) = Co. Con-
sider another path qJ (I) = gy (t) C- 1 , it is continuous because the
multiplication operation is continuous. At the same time, (jJ (0) =
e, q> (1) = ggog-I, j,e, ggog-1 E l!J o . The statement is proved,
We have already got acquainted with the examples of Lie groups,
For instance, the group of orthogonal matrices is a Lie group (veri-
ly!), Let liS consider the most typical examples of matrix groups,
AI! these groups are Lie groups; we shaH not, however, prove this
fact, confming ourselves to certain particular cases,
t. Full linear groups @:IL (n, R) and @L (n, C). Let us consider
(l Euclidean space R" and the set of all non-degenerate linear homo-
geneous transformations of R" into itself, i.e. the set of all non-
&ingular n X II. matrices with real coefficients. This set is denoted by
@:I L (II., R). The set @:IL (II., C) is defmed in a similar way.
Lemma 9, The sets @lL (n, R) and @lL (n, C) are LIe groups.
Proof. Let us consider, for certainty, the set @lL (n, R), That
L (II., R) forms a group (in the algebraic sense) is obvious, It
remains to prove that @:IL (n, R) can be provided with the structure
of a smooth manifold such that aB group operations ne smooth,
Clearly, l&JL (II., R) = R'" "{det g = OJ, where the Euclidean
space R,'" is identified with the space of all matrices of order n (over
the field R), Since the equation det g = 0 is polynomial, the

4./J, Tramformatlon Group.
217
set Rn', {det g = O} is open in Rn', i.e. this set is a domain in Rn'
and, therefore, a smooth manifold of dimension 110 1 . Multiplication
of matrices is a smooth operation because each element of the mat-
rix AD is a second-order polynomial in the elements of the matrices A
and B, Each element of the inverse matrix A -I is a rational function
of the elements of the marix A (the denominator of the function is-
non-zero since A is non-singular), The matrix E is the unity element
of the .group @:!L (n, R), In a similar fashion we can demonstrate that
C!JL (110, C) is a Lie group. The lemma is proved.
2. Special linear groups SL (n, R) and SL (110, C). The group
SL (n, R) is derIDed as a subset in C!JL (110, R) given by the equation.
det g = 1. Clearly, this set is a group and a topologial space. The
group SL (110, R) is, in fact, a smooth manifold, but we shall not
pt'OVQ here this assertion. The group SL (110, C) is defined as a sub-
group in @!L (110, C) which satisfies det g = 1. The dimension of
SL (n, R) is 110 1 - 1 and that of SL (n, C) is 2110 2 - 2,
3, Orthogonal groups 0 (n, R) and 0 (110, C), Let us consider R"
n
with the bilinear form (a, b) =  aib! which defines a Euclid-
i Clo 1
ean scalar product. The group a (110, R) is defined as the group of
real matrices A of order 110 that preserve this scalar product, i.e.
satisfy the relation (Aa, Ab) = (a, b) for any II, b ERn. The
group 0 (110, R) is usually denoted as 0 (n). The group 0 (110, C) is
defined in a similar way. The group 0 (110) contains a subgroup
denoted by SO (110) and called special orthogonal group: g E 80 (n) if
det g = 1-
Lemma 10. The group SO (110) is path wise connected and coinCide!;
with the unity component of 0 (n), The factor group 0 (110)/80 (110) is
isomorphic to Zit i,e. 0 (110) consists of two connected components.
Proof. It is known from algebra that for any element go E 80 (110)
there exists an orthogonal transformation g EO (n) such that
a = ggog_l is matrix of the form
cos 'Pl sin !PI
- sin 'Pt COB !Pl
o
o
. . .1
[-:::: ::::

218 4. Smoolh Mimi/aids (Ezamples)
(provided n = 2k) and of the form
CDS <P, sin <P, I
0
-sin <P1 cos CPI
I
'. .1
0 CDS <PA sin'l'h 1°
-sin CP/t cos q>h
0 1 1
(provided n = 2k + 1). As a continuous path'll (t) connecting Co
with the identity matrix E we may consider the following family of
matrices:
CDS cp,' sin q>lt
-sin <PIt coo <PIt
I 0
I...... .
, where y (t) = g-I a (t) g
a (t) =
o
Thus, SO (n) = (0 (n))o' The fact that the set of orthogonal ma-
trices with the determinant -1 is homeomorphic to SO (n) completes
the proof of the lemma.
Sometimes, it is convenient to represent 0 (n) as a subset in Rn'
defined by the !lystem of equations AA T = E, where Rn' is identi-
fied with the linear space of matrices A of order n.
Consider in Rn' the form (A, B) = spur AB T . Clearly, this sca-
lar product is Euclidean in the basis formed by the vectors e'l (each
vector in Rn' is identified with a matrix of order n, and all elements
.of the matrix are zero except for the element equal to unity and
located at the intersection of the tth row and jth column, 1  t ,
j n). If
A = . a{el} B =. ble/J'
1 1 J t, J
T '" ..
then (A, B) = spur AB = "-I ab:,
i , J
which, obviously, coincides with the scalar Euclidean product in the
orthogonal basis {e,l}' Identifying each matrix A ERn' with the vec-

4.9. Tr"".jormation Group.
219
tor A =  a{elJ' we can asociate to this matrix the Euclidean
H
length of the vector A, where II A W = spur AA T. As was already
noted. 0 (n) is realized as a subset in R'" satisfying the equation
AA T = E. Hence, for A EO (n) we have" A II = Vii. i.e. 0 (n)
is located in a standard sphere s,,'-t c:: R'" of radius Vn(Fig.4.60).
4. Unitary group U (n) nod special unitary group SU n). Let us
consider a complex space en referred to the coordinates %1, . . " z"
and provide it with the Hermitian scalar product (a, b) =
n
Re -, alp associated with the bilinear complex-valued form
i-;'1
f ",iV. Let U (n) stand for the group of all linear operators
i":[
in C" that preserve this scalar product, i.e. the group of all com-
plex-valued matrices A of order n such that (a, b) 51 (Aa, Ab) fOf
Figure 4.60
any a, bEen. This condition is equivalent to the matrix equation
AA T = E. where the bar denotes a complex conjugate quantity.
Thus, if g E U (n). then det g = ei'l>, Define SU (n) as a subgroup
in U (n) such that det g = 1.
Lemma 11, The groups U (n) and SU (n) are pathwise connected.
i.e. coincide with their unit components.
Proof. It is known from algebra that for any go E SU (n) there
exists a unitary transformation g E U (n) such that a = ggog-1 is
a diagonal matrix of the form
( ei'l>t eiq>, 0 ) ,
!Pi + . . . + CPn = 2l1t. l E Z
o e/'I>"

220 4. Smooth Manifold, (E::ampl.)
(here el<P, 1  k n, are the eigenvalues of the operator go)' As
regards the continuous path y (t) connecting go and E, it is sufficient
to consider the continuous family of matrices y (t) = gola(t) g,
where
[ 6 1 '1'1 1
a (t) = 1'Pn_lt
l
o
n-I
i(21r1-1  'P)
6 k..1
Apparently, y (0) = E and y (1) = go. That U en) is connected can
be proved in a similar way, and for I' (t) we may take the path
y (t) = g-Ia (t) g, where
( 6 i <P'1 )
'. 0
a (t) = 0 . el'Pn-,1 ,
el'Pn l
The lemma is proved.
Problem. Demonstrate that U (n) (as a topological space) is iso-
morphic to the direct product of SU (n) and 8 ' .
Problem. Let (g be a connected Li e group and let H be its discrete
normal subgroup. (The subgroup H is discrete if the unit element
of «!I has an open neighbourhood U such that it contains only one
element of H, the unit clement,) Prove that any disct'ete not'mal
subgroup H lies at the centre of «!I, Le, commutes with the whole
group «!I.
It is sometimes convenient to represent U (n) as a subset in R2n'
defined by the system of equations A:;F = E, where R'i.... e; en'
is identified with .the linear space of all complex-valued matrices A
of order n, Consider in Cn' the scalar product (A, B) = Re spur AAT
and consider in R2n' the basis consisting of matrices
I 1 1. I i I, where all the elements arc zero except for one lo-
cated in tbe kth column Bud jth row, 1k, jn, If A, BEC n ',
n
then (A, B) = Re spllr AiF = 2J a{b"{, which coincides with the
I, j=1
Het'milian scalar product in C n ' identified with R 2n '. Assuming
each matrix A to be a vector, we can associate with it the Euclidean
length of the vector A: 11 A 11 2 = He spur AA T . Hence, U (n) lies
inside the sphere S'>\'-1 'of radius Vn,

4,3. Transformation Group.
221
Now that we have demonstrated path wise connectedness of SO (n)
and SU (n), we can prove the folJowing statement,
Lemma 12. The group fSL (n, R) consists of two path components,
The group IJJL (n, C) is pathwise connected.
Proof. The group fSL (n, R) splits into the union of two subsets,
<.Yo = {g: det g > O} and fS 1 = {g: det g < O}, These subsets
are disjoint since the determinant of the matrix is a smooth function
of its coefficients, i.e. of the coordinates in Rn'. Also, fS 1 is homeo-
morphic to @jo' the homeomorphism being realized by the mapping
( 1 )
'. 0
g -+ ag, where a = 0 . 1 -1 . It remains to prove that fS D is
connecte.l. Since each element gE fSoCIIO be intorpreted AS a basis in R n ,
application of the familiar procedure of basis orthogonalization
shows that g can be represented as g = IX . <p, where tx E SO (n) and cp
is a top-triangular matrix. The matrix g can now be deformed contin-
uously into the matrix IX; to this end, it is sufficient to consider the
continuous path l' (t) = lX.rp (t), where
( 'PI I +(1_I)
<Pnt +(1-t)
'P (t) = 0 . . .
'Pnt+ (1-1)
.,,}
Connectedness of SO (n) proves the lemma for @jL (n, R). Let us
consider fSL (n, C) and construct a smooth mapping f: @jL (n, C)-+-
8 1 setting t (g) = det g. The image of eL (n, C) is a circle,
and the inverse image of the unit element on Sl is a subset of matri-
ces in fSL (n, C) such that dot (g) = 1, i.e, {-I (1) = Sf, (n, C), Since
a circle is pathwis8 connected, it suffices to prove pathwise connec-
tedness of the group SL (n, C), Applying the orthogonalization,
which transforms an arbitrary unimodular complex basis into a
Hermitian one, we pass, as before, from the connectedness of SL
(n, C) to the connectedness of SU (n); the latter group is con.
nected by Lemma 11. Thus, Lemma 12 is proved,
Remark, Connectedness of fSL (n, C) and disconnectedness of
rML (n, R) can be explained (not quite rigorously) as follows: the
group fSL (n, R) is obtained from Rn' by removing the hypersurface
det (g) = 0 which "subdivides" Rn' into two domains, one of them
being precisely the connectedness component of the unit element
of rML (n, R). In the complex case, fSL (n, C) is obtained from
Cn'  R2n' by removing the subset det g = 0 of codimension two

:.!22 4. Smooth Manifold. (E.'J:mpl..)
(from the real point of view), since the comlJlex equation det g = 0
is equivalent to two real equations, Rc (detg) = 0 and 1m (det g) = O.
It is clear intuitively Lhat a surface of "codimension two" does
not subdivide R2n. into two parts.
We have defined U (n) as a group of matrices which preserve the
n
real-valued scalar product (a, b) = Re 2j akb k . However.
k=1
besides this scalar product, there exists in C" an associated bilinear
complex-valued form (a, b) = i: akb k which, naturally,
k=1
leads to the group of matrices U (n)' that preserve this form, i.e. sat-
isfy the identity (Ba, Bb) = (a, b) for any a, b E C n , A question
arises: do U (n) and U (n)' coincide?
Lemma 13, The group U (n) and the group U (n)' coincide.
Proof, Since U (n)' pre.<!erves (a, b), it also preserves (separately)
the real and imaginary parts of (a, b), and since (a, b) = Re (a, b),
preservation of the form (a, b) implies preservation of (a, b). Thus,
U (n)' c: U (n), Conversely, let A E U (n), Le, (Aa, Ab) = (a, b)
for any a, bEen. Since this equality holds true for any a, bEen it
is also valid for a pair of vectors ia and b, i.e. Re (ia, b) =Re (A (ia) , Ab).
Since the operator A is complex, A (ia) = tAa and hence
1m (a, b) = 1m (Aa, Ab), that is, the imaginary part of the form
(a, b) is pre.<!erved. Thus, A preserves (a, b), i.e. U (n) c: U (n)'.
The lemma is proved.
In what follows we wi:l not distinguish between these two invari-
ance groups.
We now discuss the operation of "making real" which permits the
identification of en with R21\. Let Cl' . , " en be a Hermitian basis
in en, this means that any vector z E C n admits a unique decomposi-
tion z = Z I Cl -I- . . . -I- zn cn . where ZR = Xh -I- tyR, :c R , yn E R.
Consider the set of orthogonal (with respect to (a, b» vectors CI' . . .
. . ., en, lCl' ' , '. le n . Then for any z E C n there exists the decom-
" . "
position z = 2j XhCIt -I-  l(iek), where x R , yk E R. This
k=1 k=1
enables identification of en with R2 n . The mapping rp: C" -+ R2n
defined by the formula (I' (z) = (Xl, . . ., xn, yl, . . ., yn) will he
called the operator of "making c n real", What does it happen to the
Hermitian form (a, b) under (p? Clearly,
n n
(a, b) = lJ akTl< -+  (:.d< + tyA) (c A _ td k )
k=1 k=1
" n
= lJ (xkd'+!/,dk)-I-i 2j (yRcR_xAd A ),
h1 =1

4.3. Transformation Groups
223
Le,
(a, b) = Re (a, b) -+ i: (.1. 4 ,4 + y 4 d 4 ) ,
4=1
In other words, the Hermitian scalar product in C" becomes th
Euclidean scalar product in R2".
What do we mean by saying: "define in R'>' a complex structure"?
Consid,er the operation qJ: en -+ R,n, then an orthogonal basis
{ek' Ie,,}, 1  k n, arises in R2", and R2"  R" e IR n = Crt.
Thus, in R2 n there arises a Hnear operator A such that A (x) = lx,
Clearly, At. = -E and A (iek) = -e", i.e. in the orthogonal basis
(el' . . "e", lei' ' , ., Ie,,) the matrix A is of the form A = ( -),
where E is the identity matrix of order n. This operator has 2n eigen-
values Ax == . . . = An = + i, An+l = . . . = A ln = -i (verify!)
and A is also orthogonal: A E SO (2n).
Definition. We say that an orthogonal opera/or A E SO (2n) in R2''.
such that A2 = -E, de{itwS a complex strUl:ture in R2".
This definition will be justified if we can identify R'" with cn
via A. Since the operator A is orthogonal, it can be transformed, by
an orthogonal rotation of the basis, into
COS '1', sin '1'1
-sin '1'1 CDS '1'1
I
I".
o
A=
o
I
I cs q>n sin 'l'n
- SID 'l'n COB 'Pn
Since A 2 = -E, it follows that
<p,,=  +l,,;t, lkEZ.
Under another basis transformation (permutations), we can repre-
sent A in the form A = ( -). Thus, we have obtained an ortho-
gonal basis ell ' . " en' t J . ' . " t n such that A (e,,) = t", A (t,,) =
-e4' Spanning the space R"{e,,} by el' . . ., en, we obtain the
decomposition R,n = Rn{e,,} e R"{t4} such that A: R"{e,,}_
R"{t 4 }, A: R n {t 4 }_R"{e,,}. Hence, any a E 2n admits the
unique writing a = x + Ay, where x, y E R" felt}. Smce A 2 = -E,

224 4, Smoolh Manllold8 (Exampll&)
we obtain C by considering all linear combinations of the vec-
tors {e}.
How many complex structures can be defined in Ran? In other
words. how can we describe all A E SO (212.) such that A I = -E?
Since A = -A-I and A-1 = AT, then AT = -A, La, A is a com-
plex structure if and only if this operatior is skew-symmetric,
Lot us consider the transformation Ip: C n  Ran and let fil c:: Ran
be an arbitrary two-dimensional real plane. When is U a a complex
straight line, Le, the image of a complex straight line under cp?
Apparently fit is a complex straight line if and only if it is an
invariant plane for the operator A: Rail -+ Ran. The proof is left
to the reader.
What does it happen to a unitary group under the transformation Ip:
en -;- Ran? Let A E U (12.); since this operator preserves the Hermi-
tian form, cp transforms A into the operator q>A which acts in R,n
and preserves the Euclidean form because the Hermitian form is
transformed into the Euclidean one, Since cpA preserves the Euclid-
an form, it is an element of the group 0 (212.). What is the explicit
form of the embedding cp: U (12.) -+0 (2n)?
Proposition 2, The morwmorphism cp: U (12.) -;- SO (212.) that arises
.under the operation cp: en -+ R'''' can be written as
( c _B )
A=C+iB -; B C ESO(2n),
where C andB are real, Furthermore, IpU (12.) = SO (2n) n cp@!L (12., C),
Proof. Let 01' . . ., en be a basis in en, then cr: C" -+ Ran transforms
this basis into el' . . " en, ie l , .. '. iOn. whence (C + iB)e" =
Co" + B (io,,), (C + iB) (ie,,) = - Be" + C (ie,,), La, q>A =
( -). Straightforward calculation shows that det (cpA) =
1 det A II, i.e. det (lpA) > O. We now prove that cpU (12.) =
SO (212.) n IpL (12., C), Let cpA E cpU (12.), then, on the one hand,
cpA E SO (212.) (see above), and on the other hand, IpA is obtained
when the transformation is applied to a complex-valued non-degener-
ate operator, i.e. IpA E Ip@JL (n, C), whence IpU (n) c: SO (212.) n
cp@JL (n, C). Conversely, let g E SO (212.) and g E cp(!!L (n, C).
This means that g is of the form ( c -B ) . i.e. g E IpU (12.), The
B C
proposition is proved.
There exists another obvious embedding U (n) =' SO (12.) as a sub-
group of real matrices.
If A E U (12.) is the operator of multiplication by i. i.e. Az = tz,
the transformation cp sends this operator into ( -), We have

4.3. Traruformalton Groups
225
demonstrated above that the set of complex structures in R2 n coin-
cides with the set of all orthogonal skew-symmetric matrices. If A E
SO (2n) is a complex structure, there exists a matrix C E SO (2n)
such that A = CT O C-l, where
To=
0 1 I...
-1 0 0
[-
0 () 1
-1 II
The converse statement is also valid: any operator of the form
CT oC-. is a complex structure. Thus, the set of fOil complex struc-
tures in R2 n can be identified with the sot of matrices of the form
CToC-l, where C E SO (2n).
It can be proved (the proof is omitted) thal all tho matri:.: groups
considered above are Lie groups.
5. Symplectic group Sp (n) is derLnefi in terms of Lhe algehra or
quaternions Q. Recall the def'lI1ition of Q. Consider R4 rC£l!rred 10
an orthogonal basis whose vectors are denoted by 1, i, j. Ir, so thaL
any q E R4 is wriLLon as q '=ao,1 of- a1'i -:- a 2 .j - a 3 .!,-, wlwre
aD, a l , a 2 , a 3 E R. Multiplication in R4 is delined 011 the basis 1,
i, j, k and is extended thon to all vcdors in R', Th tahle is of Lhe
form
I 1 I 1 I i I k
1 I 1 I i I j I k
I I i I -1 I k I -j
i I i I -k I -I I t
k I k I j I -I j -1
Thus, we obtain a four-dimensional ,algebra over R which is associ-
ative but not commutative. This algebra is called the algebra of
quaternions Q. There exists in Q the operalion of conjugation:
q-... q = a O .! _ at i - a 2 .j - a 3 .k and he operation of finding
tbE' inverse element (for a nOli-zero clement): q-l = q/l q I 2, where

226 4. Smooth Manifold. (Ezamples)
3
I q 1 2 = q'q =  (al). The algebra Q admits the scalar real-
1=-0
valued product (g" q2} = Re (q..g2)' where (9,'92) denotes the
product in Q. The elements q E Q are caBed guaterniol., the quater-
nions with aD = 0 are called imaginary. The coordinate a<> in the
decomposition q = a O .1 + al.i + a 2 .j + a 3 .k is denoted by a O =
He (q); thus, 9 = Re (q) + 1m (q).
We now consider an n-dimensional quaternion space Q" with the
basis e " . . ., en; any vector of this space a E Qn admits a unique
representation in the form a =  tlek' where qk E Q.
1<=1
Definilion. A symplectic group Sp (n) is the set of all linear quater-
nion transformations of Q" preserving the point 0 and leaving
invariant the foIJowing scalar poduct in Qn:
n
(a, b) = Re (2J akjjk),
k1
i.e. Sp (n) = {A: Qn -+ Q", (a, b) = (Aa, Ab), a, bE Q"}.
The number n is called the quaternion dimen!lion of Q". The space Qn
can be identified canonically with cn. Take n = 1; then Q' =- Q.
Let q = a O .1 + a'.i + a.j + a 3 .k; \Ising the multiplication table,
we can write q in the form q = (a O + all) + j (a 2 _ a 3 i) =- z' + j#
where ZI = aD + a'i, z = a 2 + a 3 i are complex numbers. Performing
this operation along each quaternion coordinale in Qn, we oblain
the identification Qn  C.". As in the complex case, there e:\isis,
along with Sp (n), the invariance group of the quaternion-valued form
n
(a, b)= L1 akEk, air., Ii'EQ.
k=1
The following statement is valid: the invariance group of (a, b) coin-
cides with the group Sp (n). The proof is left to the reader as a useful
exercise.
Let liS consider the operation of "making complex", i.e. identilica-
tion of Qn with C." (not to be confused with complexifJcalion!).
How does this operation change a quaternion-valued form (a, b)?
Fut n = 1, then
a.b -+ (p+jq) (c+jd) =(p+jq). (c-dj) = (pc-/idj)
- (p dj- jqc) = (pc+ qci)'+ (- pd+ qc) j,

4.$. Tran.formatlon Groups
227
where a -+ p -I- jq, b- C .!.. /iI (see above). We have lIecl the rela-
tions jq=qj, r -: -1, and (i:1i' -.'" b.a (vel'ify!). For an arbitrary n
the form (a, b) hecomes
n n
 (pAth + qAJA) + L (r/'c h - pAd A ) j,
h=l %=1
where
a(al. ...,a n ) -+ (pl+jql, ...,p"+jf;n),
b= (h', .,., b n ) _ (C l + jJi, "" c" +i([n).
"
Apparcntly, the form (a, b)Horm =-  (rFc4 + flJA) coincides with
h=:at
o.
the Hermitian form in C 2n . and Lhe form (a, b),. = L (qV'_pAd h )
AI
is !;kew-!:Iymmetric. i.e. (a, b)sk< ..." - (b. a),.. If Lhe operator A:
Q"-..Q" pre.serve (a, b)qll"I' it will prcserve, niter Lhe IIlentUicalion
01 Q" with C 2n , two forms: (a, bhl<rm and (a. h).k.. Thal the
operator preserves (a, b)Herm moans Lhat i L hecome a unitMy
operator. \Ve have proved Lhat Sf' (n) call I,e cmhedued in
[j (2n) as a lJbgr()lIp of tho'!e e]ements that preer\'e in cn the
n
skcw-!>yrnmct ric form (8, b).ks  L (qA r " - pAd A ). "ere C" i
If'::: I
a;.IIJ11e(1 Lo he represented in Lhe form en e en, i.e. {he coordina-
1(>5 in C" lire tli\'idcd into two grolJp: a ..." (pl. .... po.. qt, ..., q").
Problem. Finrl the e\:plicit formula for the emhetl(\ing Sp (n)-
U (2n).
Proposition 3. The groups L (n, R), CMD (n, C), 8L (n, R), and
SIJ (n, C) are non-compact, and the groups U (n), SU (n), 0 (n),
80 (n), and Sp (n) are compact (a topological .pace$).
Prool. That L (n, R) and L (n, C) are non-compact follows
from the fact that these groups are unbounded Euclidean domains.
Compactness of U (n), 8 U (n), 0 (n), and SO (n) is a direct conse-
quence of the fact that these groups have been realized as bounded
subsets in the sphere. Compactness of Sp (n) is a consequence of the
fact that this group is realized as a dosed subset in U (2n). The pro-
position is proved.
6, The list of the Lie groups considered above is far from being
tomplete. There also exist hoth compact and non-compact Lie groups.
For example, among non-compact Lie groups of great importance
are the grollps 0 (n, k) and SO (n, k). 0 (n, k) is the form invariance
k n
group (in R n ): (a, b) = -  a1b i ..:.. 2: a;b J , SO (n, l..) is the
i=1 jeh+1
subgroup of unimodular matrices in 0 (n, k).

228 'fl. Smooth Mnnfjolrk (Ezampl.)
Let \IS consider some Lie groups of low dimension. We have seen
that SO (2) is homeomorphic to a circle; SO (3) is homeomorphic
to Rp3. Clearly, U (1)  SI. LeL liS study Sp (1) and SU (2)
PropoSition 4. The group. SU (2) and Sp (1) are isomorphic (in
:the algebraic sense) and are both homeomorphic to a ,phere S", The
grQup SO (a) is a factor grou.p of SU (2) with respect to the subgroupZ2
consisting of the elements (E, -E).
Proof. The group Sp (1) acts in Ql = Q as multiplication by
a "scalar", a quaternion, i.e. each A: Q_ Q, A ESp (1) is of the form
Aq = q .'ii, where q E Q and a E Q i!! a fixed q\laternion. Since
<q" q2)"uat = q\' q2 is preserved under A, we have (ql' g2)QUat =
(Aqt, Ag,)quat, i.e. q\q-;' = qlaaq. = 1 a 1 2 Ql,Q;, i.e. I a 1 2 = 1,
1 a \ = 1. Since 1 a ,. I b I = I a.b I (verify!), Sp (1) is isomorphic
(in the algebraic sense) to the group of all qllaLernions a E Q such
that I a I = 1. SlIrh qllRlernions form a sphere S3 because I a 1 2 =
(aO) + (a l )2 + (a 2 )2 + (a 3 )2. The tlnity element of the group S3
is 1 = (1, 0, 0, 0). We now demonstrate that Sp (1) is isomorphic
to SU (2). Consider the emlJedding Sp (n)_ U (2n) (see above).
Since n = 1, this embedding is of Ule form ( !!. !!. ) , i.e. the set
-q p
of such matrices form SP (1) in U (2), and furthermore pp + qq = 1.
Indeed, the mapping A: q- q'a-resu!Ls in q_ qa  (a. + jti) X
(p + /i) = (a.p - q) + (a.q + tip) j, i.e. the operator matrix in
the basis e l = (a. = 1, Ii = 0) and e. = (a. = 0,  = 1) is of the
form (: -*). which is what was required (after the substitution
q -- -g). Since each A ESp (1) is an isometry of R, it preserves the
four-dimensional volume, i.e. det (: -) = 1 and therefore! p I+
I q \2 = 1, which is what we need.
Thus Sp (1) is isomorphic to the unimodular subgroup in U (2),
and the dimension of this sllbgroup is 3. Since the dimension of
SU (2) in U (2) is also 3, Sp (1) and SU (2) are isomorphic. We
have also proved that the elements of S U (2) can be written as
(   ) , I P 1 2 + I q [2 = 1.
-g p
We now prove that there exists a homomorphism ;: SU (2)-+
SO (3) such that it is an epimorphism and has a kernel isomorphic
to Z2' Realize SU (2) as a group of unit quaternions in Q and put
f (a) = aqo.. where a E S3, I a 1 = 1, q E Q, and TIe q = 0, i.e. g are
imaginary quaternions that form R 3 e R"orthogonal to 1 E Q. Then t
is a homomorphism because f (a\.a2) = a\a 2 qaii 1 = / (a\). f (a.).

4,4. Dynamical Sy&tenu
229
The mapping f (a) transforms RS (imaginary quaterniOIlS) into itself
and is an isometry, since Re (aqlo.aq;;) -.- He (ag;qa)  UP (qlqz)'
Thus, f (a) E 80 (;) (the image f (8 3 ) lio!' in SO (,1) sint'e S3 i COI1-
neded), Find the kernel of f. If ago 15 q for ""y q E R'I (fie q - 0),
then ag - qa, I a I = 1. i.E', a = ::!:: 1. sin('e ('nmmutation of a \\'ith
all imaginary quaternions means lhat lh!' imnginary pllrt of (I is
zero, SiIH'C ker f 'r:: Zz = (E, -Ii), \\c 11.1\0 dim (S" Z!). ,!,1II1I1
dim SO (:{) -= :i implies thatf is an epimorphim, The propo"ilinn
is proved.
That SO (:3)  S3/Z l , corresponds to the r(>pre('nl<1linn of Rpl <1'<
tho quuli£!nt S3/Z l , where Zz acts 011 S3 115 IIIlIlliplication oj 1\ \'N'.
tor hy -1.
Problems
1. Prove thnt 0 (2) coincides \\'ith llll' isometr)' gronp of
A drclc (with the metric d,v 2 - dq:>2) ,
2. Provo that th(' group SI, (2, R):Zl is patli\dse l'onnct'ted,
3. Prove that Lhe group U (n), as a lopologic'nl space, is hOIlH'o-
morphic to the direct product or SU (n) and 11 /'ir('le SI
4, Let @ he a connected Lie group anellet H he 11 diJ,crNc in \'l1ri nil t
suhgroup. Pro\'e that H lies at the centr(' of @,
5. Find an explicit Corm of lite embedrling Sp (n)c U (2/1). h'
analogy with the embedding U (n)c SO (21/).
4,4, DYNAMICAL SYSTEMS
We nr,t recall the simplellt properties of the gradient of
a smooth function. We are ahead}' familiar with th£! vector fi('lci
v (x) = g-;;d f (x), where f (x) is a smooth fllfirtion on R". In clIrvi-
linear coordinates, however, the gradient is not /I vector tield, for it
is not transformed as a veclor, and therefore the gradient can be
interpreted as a field if only there exisls a Riemannian metric.
--+-
The gradient of a funclion, grad I, is relatt'd 10 the derivative
of f in the direction of a veclor a E R U by  = (a, g;d f), where
 is the directional derivative, (,) is the Euclidean prodnct. Let
f be valid on R n , then this function defines a level hypersurface (we
shall speak abont a le\'el surface for It = 3 and about a level line for
n = 2). A level hypersurface is described by II - 1 parameters
(n parameters in R" are related by one condition: namely, the equa-
tion f (x) = C = COllst), that is why the dimension of the hypersur-

230 4. Smooth Manifolds (E:camp!e.)
fare {/=c}isequal ton-1, provided {I-c) is a smooth mani-
fold.
Definition, The point Xo E {I = c} is called II critical point of the
function I if g f (xo) = 0; otherwise, Xo is called a non-slngu.lar
(ncm-crttical) point.
Example. Let the function z = x 2 - y2 be defined on R2; the level
line z = 0 is of the form x = :!: y and consists of two straigllt lines
The point 0 is the only critical point of this function.
Let Xo E {f = c} be a non-singular point. The vector a applied at
the point Xo is called tangent to the hypersurface (t = c) if there
exists a smooth curve 'I' (t) entirely belonging to {J = c} and such
that y (0) = Xo and y (0) = a.
Lemma 1, Let f (x) be a smooth function on R" and let Xo E (t = c)
.......
be a non-singular point. Then the vector grad f (.to) Is orthogonal to the
hypersurface {t = c} at the point xo. i.e. g-;;d f (xo) Is orthogonal to any
vector a tangent to {t = c} at the point Xo'
Proof. Since (a, g;;;.d f) =- : , it is sufficient to calculate the
derivative of f with respect to a, i.e.  f{i' (t» 11=0' Since
t(1'(t»=c=const and y(t)c{f=c}, we have  =0. The lemma
is proved.
If Xo E {/ = c} is a critical point, then g;;d f (xo) -= 0, and we may
aSS1,!me, formally of course, that ';;d f (x) is orthogonal to {f = c}
at any point of this hypersurface.
.......
Because the (tireclion and magnitude of grad f (x) show the direc-
tion and rate of the inerease of j, the fnnction f always grows at
a maximal rate a.long the normal to a level hypersurface
We recall that a smooth vector Held v (P) is said to be defined on
a smooth manifold Mil, if at each point P E .111" there exists a vector
v (P) E T p M" smoothly dependent on P.
In a locdl coordinate system Xl, ' . ., x'll a vector fIeld v (P) can he
given by the set of smooth functions Vi (xl, . . " xn), 1  I  n.
Continuous fields are defined in a similar way.
Definition, The \loint P" E .111" is called singular for a vector
!lelc! v (P) if v (Po) = O. The singular point Po E Mn of the field
v (P) is called holated if it has an open neighbourhood U in which
the field v (P) does not have singuillrities other than the point Po'
or great importance in physics, however, are discontinuous ftelds,
which are smooth vcry\Vhrre except at several isolated discontinui-
ty points on /If". The examples are given below.

.. Dvnamlcal Systems
231
Let v (P) be a smooth field on M n . We recall that the trajectory
y (t) is called an integral curve of the flelcl v (P) if ;, (t) = v (y (t»,
i.e. if the tangent velocity vectors to y (t) coincide with the !ield
vectors v.
Figure 4,61 Figure 4.62
Consider several examples. Let / (P) be a function 011 R! and
-.-
'Y (P) = grad f (P).
(a) f (P) = x 2 + y2, g;:;;d f = (2x, 2y). The integral curves ar<>
rays emerging from the point 0 (l"ig. 4.(1).
-+
(b) f (P) = -x! -y\ grad f = (-2x, -2y). The integral
curves arc rays converging to the point 0 (Fig. 4.(2).
-+
. (c) f (P) = -x + y2, grad f = (-2x, 2y). The integral curvcs
are hyperbolas (Fig. 4.63).
y
iJ
 rf!?'
Figure 4.63
For all the fields (a), (b), and (c) the point 0 is singular. The fu IIC-
tion / has at 0 a minimum (case (a», a maximum (case (h)), and
a saddle point (case (e)).

232 4. Smooth Manifolds (Ezamplts)
A vc(.tor lie]d V on /11" is frequentl y meant as the flow of liquid
through the manifold and it is a.,sumed that each particle of Jiquid
ha., a \ clodty represented hy a vector. The singular points of the
liquid flo\\' arc thll sinIJar point" (If the lielcl; for example, the
'illgular point in Fig. ..'1.01 is a ()lITte (tbe liquid "flows out" from the
point 0). IIIlIltlJe 'ingular point in Fig. i\ n2 is a sink. The integral
(un eK 01 the lield ,. arc 'olllelime ( alll'd !<treamlines of a Jiquid whose
110\\ i.., dl'( !'illed h thig \'el()('it) field. XIIL every real liquid flow
geuer(1te" a licit] in the :-ensc we mean The fa( tis thal the coordinates
of OUI' liel.h do Hill tlep(illd on lime; in other word, liquid nows
correspunding to slIth lields .Ire i>leady. Tjme-dependent flows are
unstl!ady, They ('an he modelled lIy a family of fjelds v (P, t) which
SlwlOthly dupend on the parameler t.
flo\\' (',1/1 we lind c\plil'iLly the integral curves of a field? The rele-
\")lIl appMatu ha heen (leveloped ililhe thcory of ordinary differ-
clltial cqllalioll. Each field v .- (.I (1'). . .., v" (I'» can hc iden-
lined wilh It y.lem of orllinary dillercntial cquatiolls d:e h ..,.
h (Xl. . . .. .t") , 1:S:;;; k n. We recall lhat u :;ystem is said 10 he
(1U1(J/lomou. if ils right-Iuind side d()e not contain the paramelcr t
explieiLly. Sleady Jiquid flows are described by autonomous systems.
Lt. 1 us ('cHlsider a particular case where the behaviour of an inte-
gral curve tall he descrihed in father simple terms.
Definition, Lel, on Mn, he given a system of differential equations
describing a !'Ield v (P). A smooth function 1 (P) on M n , which is
contant along all integral curves of the flow, is called an integral
01 the sy.stem.
Let f be an integral of the syslem. The hypersurfaces {j = c}
foliale M" as c varies Consider a fixed hypersurface 1 (P) = co.
It follows from the definition just given that if an integral curve
has a common point with {j = co}, the trajectory lies enUrely in
{f = co}. This impJies that the field v (P) is tangent to the surface
II = 1 (P)} at each poinl P (see Fig. 4.64), that is, each surface
{/ -= c} is folialed into integral curves of the now v. Thus, the order
of the initial system of equations can be reduced by one if the field y
i re!>tricted to the surface {J = c}. We recalllhat in a neighbourhood
of any point, which is non-singular for grad I, the surface {t = cj
i 11 smooth manifold of dimension n - 1. H we have two function-
ally independent integrals, f and g (i.e. almost in all points of M n
grad 1 and grad g are linearly independent), the order of the system
is reduced by two units (see Fig. 4 fi5), elc. If we have a set of n - 1
functionally independent integrals, tile flow can he integrated com-
plelel Y lie. all inlegral e,urv'cs can he desui hed as y (t) = {I) = cd n
n {In-l -= cn-d, where y (0) = 1', Ik (P) = Ck' 1:S:;;; k
n-1.
Definition. Let rJelds V,, . . ., v he given on M n . These fields ara

4,4. Dynamical Systems
233
called linearly independent if the vector VI (P), . . ., Vh (P) are lin-
earl" independent at each point P EM".
l{.!;omc lield Va; vanishes at a point, the entire syslem VI' . . ., VA
is no Inng'er inclependent.
Proposition f, U:t M" =-= (S be a Lie group. Then there always exisf
all @! II linearly independent .,mooth I'eclor fields VI' . . ., v n .
Proof. Supposc <!S is a matrix group and consiuer on (S the left
translation La: g -+ ago Clearly, L" is a diffeomorphism of (S.
}'igure 4.64. The integral curve shown Figure 4,65
csnnot exist.
ConsIder the unit element e E @. Define n linearly independenl
vectors e l , . . ., en in T.(!/) and consider the differential dLa: T,,@_
T,,@, Put VA (a) =-= dLa(e A ). Since La is a diffeomorphism, dL" is
non-degenerate, i.e. the vectors Vh (a), 1::;;;;; k n, are linearly
independent.
Below we shall give examples of such M n on which any smooth
f'Leld does have singular points (zeros). An interesting example of three
independent fields can be constructed on S3. Since S3 is homeomor-
phic to SU (2), there exist three independent fields on S3, according
to Proposition 1. Let us hnd their explicit form. Let S3 = {q E Q,
f q I = t}. Put VI (q) = iq, V2 (q) = jq, and V 3 (q) = kg. These
fields are tangent to S3 at the point q. Indeed, calculating the scalar
product, say (iq. q), we obtain (iq, q) = Re (iq.q) = Re (i I q 12) =
I 9 r lo . Re l = O.
Problem, Find the integra] curves of the above fields on a sphere S3-
in explicit form,
Among smootlJ f'lc!cls on Mil of special interest is the class of the
so-called gradient or potential 1i('lds. Discuss the concept of gradient
on M" in greater detail. Let Mil he provided with a Riemannian met-
ric amI let! be a smooth fUllction on M". Then, grad! = { :;i } is allt
element of the space dua! to T p M n . To obtain a vector, we consider-

234 4. Smooth Manlfala.. (S:x;amples)
(in a local coordinate system) the field v (P): Jc (x) = tf" (x) a£) .
Calculation shows that the set Vi, . . ., v n is transformed exactly
in the same way as the coordinates of a vector; this set will be de-
-..
noted by grad f.
Definition. The field of the form v (P) = g;;d I (P), where I is
a smooth function on M n , is called a potential field on M".
Lemma 2. A potential field does not admit closed integral curves
without singular polnts.
Prool. SupPose there exists a closed integral curve (such trajectori-
es are sometimes caJled periodic solutions of a system). Then'; (t) =
v (x), where x (t) is a solution. If f is the potential, then
df. = (;, v (x» = gfJ;'v i (x) = g!jv' (x) vi (x) = I g;;;d f jZ> O.
dr
This inequality shows that I is a strictly monotonic function which
grows in the direction of increasing t, But since the trajectory I' (t)
is closed, the point "l (t) will return to the ini tial posi tion, which
contradicts the continuity of f. The lemma is proved.
For example, the fIeld shown in Fig. 4,66 is not potential (the liq-
uid rotates round the origin).
 fgl'tili
.  x
Figure 4.66 Figure 4.67
Let us consider flows on a two-dimensional manifold. For simplic-
ity, we shall study flows locally; in this case we may consider a flow
on a Euclidean plane. Let us treat the flow as a motion of liquid of
constant density (equal to 1), We shall study how the masS of
liquid at each point changes with time. As is known from the theory
01 ordinary differential oq!otations, each flow is invariantly asso-
ciated with a one-parameter grollp of diffeomorphisms, i.e. shifts
along the integral curveS of the field. The singular points of a now are
y

11.4. Dynamical S"slems
235
lixed points of the action of this group. Let the field v have the coor.
dinates (P (x, y), Q (x, y». Let us consider the change of the mass in
an infmitesimal rectangle with sidesdx and dy (see Fig. 4.(7) (here (x, y)
are Cartesian coordinates). If dm is the mass of liquid in the rectangle
(dx, dy), then dm = dx.dy. Let .v (t) = v (y (t» be a system corre-
sponding to v. If Y (t) is a solution, t is defined along V (t) to within
a shiftj we may assume that y (0) = p, where P is a point. Set
f,. (P) .= Y (to), were Y (0) = P. The mapping ft. is a diffeomor-
phism. Let us consider P = (x, y) and apply to the rectangle (dx, dy)
the inverse mapping 1(-1>.1). where l' (t) = p, The image of (dx, dy)
under f(-I>./) is an inlinitesimal parallelogram r Applying hr, we
transform r into (dx, dy) (see Fig. 4.68). The point P = Y (t) corre.
Figure 4.68
sponds to the value t, and the inverse image of P under 1M corre-
sponds to t - 6.t. As t varies, tho mass dm (t) = dx (t). dy (t) also
varies.
Lemma 3. The change of mass in an infinitesimal rectangle (dx, dy)
can be expressed as
 (dm(t» = (  -I-  ) dxdy.
Proof. We shall assume that 16, defines a new coordinate system
in (dx, dy) (trans[ormation by II small diffeomorphism). III the
initial coordinate system, dm (t) = dx (t).dy (t), and in the new
coordinate system we have
dx (t+6t).dy (t+M)
= d (x (t) +6t.x;) ,d (y (t) + fJ.t, YI)
= (dx (t) + 6.td (xi»' (dy (t) + 6td (yi»
= (dx (t) + 6tdP (x, y»' (dy (t) + f.tdQ (x, y»
= ( dx (t) + 6t ( = dx + : dY))
X ( dy (t) + fJ.t (  dx +  dY)).

236 I. Smoolh Manifold. (E:r:amplt&)
Since in the first faclor dP (x, y) is calculated with re5pect to the
increment along the x-axis, the increment Ay in this factor vanishes
and only the term ax (t) +- At :: dx remains. Similarly, in the
second factor the increment along the y-axis, i.e. t1x, vanishes, and
only the term dy (t) + /:,t - {) {)Q dy remains. Tim!>.
'/
(dx (t) + t1t ' dx ) (dy (t) + At  dY)
=dx(t),dy(t)+6t.dx(t)dy(t) (  -I-  )
(small quanti,lies of higher orrlers are neglected). Hence,
t1 dm (t) . /j.[. (  -I-  ) dx dy,
i.e.
:t (dm (I». ( : -:  ) dx dy.
The lemma i:j proved.
Difinition. The function div v = oP :' Y) -I- oQ :' 1/) is called
the divergence I)f a flow v. The now is illcomprpssible if div (v) = O.
Lemma 3 implies that the change in mass is zero if and only if the
now is inCQmpressible. Let us consider an analogue of the expression,
obtained in Lemma :I, for a fmite domain. Let D he a bounded
domain with a piece-smooth boundary and let D (t) he a domai.l
obtained from D as a result of the shift by time t along the integral
curves of v via one-parameter group. Let the area of D (t) be equal to
V (t) (if the density of liquid is unHy, V (t) is the mass contained
in D (t».
Proposition 2. For any bounded domain D (t)
dV (I) .
(iI= J  divvd:r:dy.
D(I)
Proof. It is sufficient to subdivide D (t) into infmitesimal rectang1es
and apply Lemma 3.
Definition. The !low v = (P, Q) is called irrotational if P II = Qx'
This condition can be modirted. Let us consider an arbitrary smooth
contour C on a plane (a circle without self-intersections), C =
y (t); a.t each point of the contour there arises the number (v (y (t».
;. (t» (see Fig. 4.69). The rotor of the flow v along C is the number
 (v (i' (t», ;. (t) ) dt.
e
Lemma 4, The flow " = (P t Q) is irrotational if and only if its rotor
along any smooth contour vanishes.

t.4. Dynamical Syl/em$
237
Proof. Let D be a domain bounded by C. It follows that
j (v (')' (I)), y (t)) dt = J P (')' (t» dx (t) + Q (')' (t» dy (I)
'1(1) V(I)
2n
= ) (p : + Q :: ) dt
G
= H (P"-Q,,,)dxdy, y(O)=y(2n).
D
Here we have used the Stokes formula (see Chapter 6). The lemma
is proved.
Proposition 3. An irrotational flow v on a plane is potential,
i.e. there exists a smooth function a (XI y) such that v = grad a (x, y).
The jorm Pdx + Qdy is the exact differential oj the functiult a (x, y)
tkfined uniquely to within an additive constant.
Figure 4,69
It is sufficien t to integrate the system of equations
Q= :; under the condition : =  . Integrating the
It
first equation with respect to :c, we obtain a (x, y) =  P(z, y)d:r+
G
g (y). Differentiation with respect to y yields
Proof,
P=t
oz
oa (z, y)
ily
r ap (z, y) dx+ dg (y)
; Oy dy ,
G

238 4. Smooth Mani/olds (Eo:ampltl)
whence
'"
Q(x, y)= I' dL"+
.I (1y dll '
o
y) = Q(x, y) -Q(O, y) +, which gives g' (y) =
dll
1/
i.e. g (y) =  Q (0, y) dy+ C, whE're C =. const. Th\l,
o
'" 1/
a(x, y)=  P(z, Y)d.r+.\ Q(O, y)dy+C,
o 0
If we started integration with the equation Q = :: ' we would
1/ '"
obtain a(x, y)=  Q(x, y)dy+  P(x, O)dx + C. The function
o 0
a (x, y) is the now potelliiai. Let us describe a (x, y) in geometric
terms.
or Q(..c,
Q (0, y),
y
'"(2 ()(, y)
"'Illtl,,
(0,0)
'ri
)(
Figure 4.70
Consider two piecewise smooth paths 'I' = 1'1 U 1'2 and 1" = 'I' U '1';
(Fig. 4.70), Clearly,
'" 1/
a (x, y)=. P(x, Y)dx+) Q(O, y)dy=  (Pdx+Qdy),
o 0 y
1/ 0:
a (x, y)=  Q(x, y) dy+  P(x, 0) dx=  (Pdx+Qdy),
o 0 y'
i.e. a (x, y) clln be found by inlegrating the differential form U) =
Pdx + Qdy along either y or 1" leading from point (0, 0) 10
point (x, y).

11.4. Dynamical Syl/.ms
239
Propoition 4, Let v b, an Irrotational flow. Then it is potential and
the p0/<'"lial is of the form
a (x, y) =  Pdx+Q dy=  (0),
y y
where -y is an arbttrary piecewise smooth path from point (0, 0)
to point (x, y). The integral  w does not depend on the path.
y
Proof. We firt prove that  (P dx+ Q dy) does not depend on
y
the path (provided the initial and terminal points of the path
y
()t, y)
o
x
Figure 4,71
are fixcd). oLet -y' b; another path from (0, 0) to (XI y); consider
a = .\ w- .\ 0) =  w, where --y' is the path -y' orhmted ill
y y' YU-Y'
the opposite direction (see Fig. 4.71). Then,  (Pdx+Qdg)=
YU"-Y'
J (Pdx+Qdy)=O, because C=y U _y' is a closed contour and
c
the flow is irrotational (corollary of the Stokes formula). Thus,
a = 0 and  w =  w. Since we have proved independence of ) w
Y y' Y
of the path, it is sufficient to choose one of the paths of Fig. 4.70
to evaluate  w numerically. The proposition is provE:d.
Y
If we choose another initial point of the integration path, a (x, y)
changes by an additive constant. Let v be an irrotational , incompres-
sible flow. Then its coordinates, P and Q, satisfy
_ _.!!L _ p_ 8a Q {Ja
8z - 8y' 8y - IJz' - tii ' = ay ,

:240 4. Smoolh M ..nifolds (Ez..",plu)
whence :: + : = O. The linear differential operator of the second
order l:J. = ::. + 0:: (written in Cartesian coordinates) is called
the Laplacian. The function I (x, y), such that M = 0, is called
harmonic.
We have proved that the potential of an irrotational, incompres-
sible flow is a harmonic function. The potential a (x, y) is usually
-considel'ed together with another potential, b (x, y), called a con-
jugate potential or tht" potential of a conjugate flow. To define
this poten tial, we consider the system : =- - Q. :: = P. The func-
tion b(x, y}, which is a solution of this system, is called a con-
jugate potential. Let us prove the existence of a conjugate potcn-
  iJb  8b - on
tial. Put P= -Q, Q=P. Then, -ax=P, 7;=Q, provided=
- = ' : = :: . This is the system that was integrated above
to fwd a (x, y). Th\ls, b (x, y) does exist and plays the role of a (x, y)
for the flow v = (P, Q) = (- Q, P). The flow; = (P. Q) is called
conjugate to the flow v = (P, Q), The converse is also true: the poten-
tial a (x, y) is conjugate to b (x, y), i.e. the potential dOllble-conjug-
te to a (x, y) coincides with a (x, y). The flows v and'; are ortho-
gonal: (v, ';;) = - PQ + QP = O.
Let us consider the plane (x, y) as a plane of complex variable z =
.x + iy and consider also a complex-valued function I (x, y) =
JL (x, y) + ib (x, y), where a and b are the potential and the con-
jugate potential of the incompressiblo flow v = (P, Q). Instead
of ,  we will simply write gv' gr' Since a r = P, a v = Q, b x =
= - Q, and bll = P, we have ax = b v ' a v = -hr. The function
t (x, y) = a + ib is called complex-analytic, and the equations
for a and b are called the Cauchy-Riemann equations (conditions).
The functions a and bare called the real and imaginary parts of t,
respectively: a = Re I, b = 1m I. Let us recall some properties of
a complex-analytic function.
- 1 - -I -
Let z=x+iy, z=x-iy, then x=2'(z+z), y=---z---(z-z).
Thus, any function g (x, y) = u (x, y) + iv (x, y), which can be
expanded in a convergent series of x and y, can be written as g (Xt
y) = g (Ztz). By the rule of differentiation of a composite function,
a 8z (} iJy (} 1 ( 8 . 8 )
a;=8ZiiX+8%"8V=2' a;;-8Y

4.4. Dynamical Sy,'em<
24t
_ 0 1 ( 0 8 ) A
Similarly, a. =2" ar + t 8Y' mong the set of functions
g (z, z) we will choose those that depend only on z. Analyticallr,
this is written as  g (z, z).... O. Such functions are called com-
a,
plex-analytic, thl'y can only be expanded in a convergent power
. f . hI S' og 0 + . 0' .
senes.o one vana e, z. mce 0; = ,gz lKv = ,I.e. u" + IV", +
i(u v +ivv)=O, and the condition =O is equivalent to the
iJz
Cauchy-Riemann equations, u" = vv' U v = - v".
Thus. we have proved
Throrem 1. Any irrotattonal, incompressible flow v = (P, Q) can
be represented in the form v = grad a (x, y) and the conjugate flow
'; = (i>, Q) is represented in the lorm  = grad b (x. y), where / (.r, y)"""""
a (x, y) + ib (x, y) is a complex-analytic lunction uniquely defi'led 10
within an addltive constant. The converse Is also true: ill (z) I' a com-
plex-analytic lunction, the flows v = grad Re I (z) and ;- = grad 1m I (z)
are irrotational and incompressible, and they are conjugate to each
other.
The integral curves of v and ;- are orthogonal. The function I =
a + ib is called the complex potential of a flow. We now turn to
studying singular points of a vector field on a plane.
Let v be an irrotational. incompressible flow. How can we find
zeros of v and ;? From the Cauchy-Riemann eq\lations we have
I; (z) =+ (1..- ifv) = a,,-ia v = b v + ibo:.
Lemma 5, The points at which I; (z) vanishes coincide with zeros 01
the flow v (the same points are zeros of ;),
The prool directly follows from f; (z) = a,. - ia v = by + ib
Thus, to find zeros of the flows v, ;, it suffices to solve the equation
f' (z) = O. Until now, we have considered only smooth fields. All
the considerations can, nevertheless, be repeated for 1\ now with
isolated discontinuity points which are not, of course, solutions of
the equation f' (z) = O.
Here is a question of practical importance: let v be a' incomprg-
ible, irroLational now, i.e. v = grad Re I (z), where I (z) is an
analytic function, how can we find integral curves of this fIeld? There
i no need to solve explicitly the corresponding system of dHrerentifil
eq 11ft Hons.
Proposition ;), Let I = a + ib be a complex analylic lunctiol/,
v '"" grad a,; = grad b. Then b is an Integral lor the field v, alld a Is

!!42 d. Smooth Manilold. (E;r;amp' J
an integral for -:;. i.e. the integral curves of v are level lines of the fu.llc-
tion b, and the integral curnlS of'; are leL'elli12es of a.
}o'igure ,7Z
Proof. It
Figure . 73
is sufficient to find d':. and db
iv /iV'
For example,
db ---..
d'V={v, grad b>=a",b",+abu=O. Similarly, d ",,0, that is, a
dv

.
.

:::;::;:;-
-
l>igure ,i.(j, Thu potcntial Flgure, 75
<P (..., y) is a many-valuud fUnc-
tion.
and b /:fe constant along the corre!\ponding integral curves. The
proposition is proved.
Examples, Lel f (z) "'" Zk, k;;;;' 2, f' = J.:zk-l. Clearl', f' (z) = 0
onl' at the point 0, f = r" (ros k<r - i sin k(f'), i.e. a = r lt cos k%"

4,4. Dynamical Slld.m,
243
b = r< sin kq>. Figure 4.72 shows the integral curves of grad a
(for k = 4). The origin is a singular point obtained as a result of the
Figure 4.76. Dipole.
confluence of several singularities of lower order (see below). Let
f = z-k, k  1, f = r- h (cos kip - i sin kq». Figure 4.73 shows the
Figure r.,77
,
'RA
Figure 4.78. k = 6.
integral curves of grad a (for k = 4). Let f (z) = In z. The integral
r.Urves of v and; for this function are shown in Fig. 4.74. This is

244 4. Smooth ManlJoltk (EzQMplts)
a logarilhmic singularity. Let us consider the J oukowski transforma-
tion, f (.1:) = Z + 1/.1:. The integral curves (for one flow) are presented
in Fig, 4.75. Problem: Construct trajectories of the conjugate flow.
Figure 4,79
1 .
Let f (z) =  (In (z + (X) - In (z - (X)), (X is rea1. The pattern of
8 flow is shown in Fig. 4,76 (construct the pattern of the conjugate
flow). This case is an illustration of the confluence of singlliarities,
Figure 4.80
For (X oF 0 the !low pattern (see Fig. 4.76) coincides with that for
a dipole (two charges at singular points). As (X- 0. f (z)_ t; (z) =
1/z anel the dipole field becomes the field of a now with a flrst-
order pole (see Fig. 4.77).
"
Let liS consider a polynomial Pit (z) = II (z - a,) with simple
j1
real roots, The corresponding flow is shown in Fig. 4.78. Let all the
roots (x, vanish (the polynomial is transformed into p (z) = ?!'
with only one zero of order k). Deformation of the flow is illustrated
in Fig. 4.79.

4.4, DynamIcal Sy.loms
245
Problem, Construct the deformation of the funclion :- to the

funclion (.If (z - '.t/)) _t and draw the pallern of now deform a-
11
tion.
Figure 4,81
Thus far, we have considered flows on a plane. Thee examples can
he transferred lo a sphere S. The stereographic projection deflDes
a correspondence between the points of S2 and the points of an
D 2
J'igure 4.82
extended complex plane (i.e. the complex plane with the adjoined
infmitely distant point),
The flow,' = grad Re z on R2 has only one singular point. infInity,
this is a first-order pole (when considered on 8 2 ) (see Fig. 4.80).
Figure 4.81 shows the qualitative behaviour of grad Re (z -; 1/;),
rad Re In z, and grad 1m In z.

2.46 1#. Smoolh' Manifold. (E:z:ampl..)
Problem. Construct the pattern of the trajectories for the fol-
lowing flows:
-I). I ( a:+b ) t ..It
(a+tj-')' n c:+d ,.+Inz, :o"+lnz.
The attempts at constructing on 8 2 a smooth field without singular-
ities have failed; In particular, the flows considered above do have
singularities on 8 2 . This is no accident. Let us explain intuitively
why any smooth field on 8 2 has a Singularity. Suppose a field without
singularities is constructed on 8 2 . Represent 8 2 as the union of two
disks: 8 2 = D, U Dz' where D, is the disk of small radius e with
centre at the north pole N, and D 2 is the complement to D, (see
c
Figure 4.83
Fig, 4.82). Since the field is smooth, we may assume that it is con-
stant in D, and its vectors at points A, B, C, D have directions as is
shown in the figure. Introduce on D, and D 2 Cartesian coordinates
(xl, yl) and (x 2 , y2), respectively. Therl in D 2 the field Y (in these coor-
dinates) has the pattern shown in Fig. 4.83. Such a pattern can be
explained as follows. Stretch D, along the meridians so that it occu-
pies almost the entire sphere except for D 2 of small radius. Then we
obtain the pattern of Fig. 4.83. Since the radius of Dz is small, it is
clear intuitively that this domain should contain a singularity,
otherwise, the trajectory pattern would coincide with that in Dl"
4.5. CLASSIFICATION
OF TWO-DIMENSIONAL SURFACES
This section deals with the structure of two-dimensional
manifolds, We shall start with general definitions, then consider
particular examples, and in conclusion give the formal proof of
the classification theorem.

4.5. CllU$lficallon of Two-Dlmen8lonal Surfae$
24;
4,5.1. MANIFOLDS W1TH BOUNDARY
Let us extend the concept of a smooth manifold by in-
cluding into it subsets in R" defined by systems of equations and
inequalities. Let R c:: R" stand for the half-space given by :en;;;;>.t 0,
(xl, . , ., x") E R n and let n:- 1 denote the boundary x" = 0 of R.
If f: R" -+ Rl is a continuous function. then at all points. including
the bondary ones, we shall define differentiability of f. If Xo E R
is an. interior point, i.e. :en > 0, the definition is customary. If
"0 E R-l, i.e. :J!' = O. we say that f is differentiable at the point Xo
if the following expansion is valid:
n
f(x)=f(xo)+ al(xi-xH o (x-x o )'
,..,1
where lim °1(:Jt-) - 0 for X-X o , x"O. Then al= 1 (xo) for
x- b
i=1.2....n-1anda,,=lim  (/(x, .... X-I, x+h)-f(x,
h-+O
. . ., x», The latter limit may be called the partial derivative
:" of the function f at the point x o - Smooth functions of class
cr I r = 1, 2, ..., OCI, are defined in a similar way.
Definition, A metric space M is called a .mooth manifold with boun-
dary if there exists an atlas {U..} and coordinate homeomorphisms
<Po.: U"'_ Vo. c RZ, where Vo. is an open set in R, such that the
functions of coordinate transformation
qWP,.': Vd=<i'o. (Ua. n U)_ v,... = <P"(Uo. n U,,)
are smooth functions of class C', r = 1, 2, . . 'f 00. The linear
coordinates (xl, ..,:J!') in R induce local coordinates in the chart
U..: x (P) = :r!' (<po.P). In any local coordinate system w have
then x (P);;;;>.t O. A point P E M is called an interior point if  (P) > 0
and it is called a boundary point, if x (P) = O.
If P is an interior point, then x" (P) > 0 in any local coordinate
system (x., . . ., x"), and if P is a boundary point, then in any local
coordinate system (xl",., x") x"(P) = O. Indeed, suppose that
for a point P there exist two coordinate systems (xt, . . ., x n ) and
(yt, ; . ., y") such that x" (P) > 0, y" (P) = O. The coordinates
(x t , . . ., x") deline a homeomorphism of some neighbourhood U eM
into the domain Vc RIO, and tho coordinates (y1, , ,. yn) map
homeomorphicaUy the same lIeighbourhood U into the domain
V' c RZ. Consider the transition function as a smooth homeo-
morphism cp: V-V'. 11' = cplt (Xl, . . ., x") satisfying the conditions:
(a) y" = Ip" (x t , .... xn);;;;>.tO and (b) y" = cp" (x, ..., x) = 0
for a point Xo = (X, . . " x) E V. These conditions mean that at
the point Xo the funcUon Ip" reaches a minimum. Since Xo is an inte-

248 4. Smooth Manifolds (E:r:amples)
rior point for the domain V eRn, the gradient of q>n vanishes at
"0' i.e.
8<pn , 1
q;i"(x., .... x)=O, i= , ..., n,
Hence, the determinant of the Jacobi matrix II ::: II at the point
x is zero, which contradicts the condition that the transition func-
tion from one local coordinate system to another is smooth. Thus,
for a manifold with boundary there exists an atlas {U a,} with local
coordinates (x, . . " .r) such that in any chart we have the strict
inequality x > ° at the interior points and the equality .r = 0
at the boundary points. If the chart U a does not contain boundary
points, the condition x > 0 may be rejected without loss of general-
ity. The set oM of boundary points is a smooth manifold of dimen-
sion n - 1. Indeed, the intersection Wet = aM n U" can be chosen
as a chart and .r. . . " X-I E U", as coordinates. The coordinate
homeomorphisms q>" (see above) map W" homeomorphically onto
V", n Rt, and the transition functions remain smooth, for they
are restrictions of the functions of coordinate transformation to
the manifold M.
If 8M = 0, we arrh'e at the earlier concept of a manifold, hence-
forth, this manifold will be called closed.
Examples. 1. Consider a ball D" defined in R" by the inequal-
71
ity  (xl)1, the space D n is a manifold with the boun-
i1
dary sn-I.
2. In general, let f: R" _ R A be a vector function satisfying the
conditions of the implicit function theorem, La, at all points P E
M = {f (P) = o} the differential df has the maximal rank. Con-
sider the ystem of equalities and inequalities: f (,xl, ' . ., x") =
0, . . ., fA-l (Xl. . , " x") = 0, and f" (x t , . . ., ,1;"» 0 and let M'
be the set of solutions of this system. If at each point P E M' the
rank of II :: II, 1  t  k - 1, 1  s  n, is equal to k - 1, then
M' is a manifold with the boundary aM' = M, Indeed. by the
implicit function theorem for the system f1 = 0, . . " f"-1 = 0
there exists an atlas at points P EM' ....... M. Consider a point P E M.
Let N' be the set of solutions of the system f1 = 0, . . '1 fA-l = 0
in some neighbourhood of P, Then II< may be considered as a function
on N'. Clearly. the differential of fl< on N' does not vanish, Hence,
according to Theorem 1 of Chapter 3 (Sec, 3.5), N', is diffeomorphic
to the direct product of the interval (-e, e) and the manifold N
which is the intersection of M and the neighbourhood of the point P,
i.e. N' = N X (-e, e). Thus, the inequality In> 0 defines the

4 5. Cla..i/ica!lolt of Two-D imen$iol<4/ Surfaces
2411
chart N j X [0, e) in which the set (xl, . . ., xnk) on the manifold N
/lnd the function I k  0 can be chosen as coordinates.
3. In the preceding example one can see that the Cartesian product
of the manifold M with the boundary 8M and the closed manifold N
is Ihe manifold M X N with the boundary aM X N.
4,5,2. ORIENTABLE MANIFOLDS
Definition. A manifold (with boundary) M is called
orientable if there exists an atlas {U".} such that the Jacobian of the
transformation from one local coordinate system to another is positive.
We also say that the atlas (U..} for which all Jacobians of the transi-
tion functions are positive defines an orientation on M; such an
atlas is called oriented. Two atlases {U..} and {U} define the same
orientation if their union {U..) U {U} is an oriented atlas.
Any atlas on an orientable manifold can be made oriented by tlllt
transformation of local coordinates in each chart. Indeed, let
{U.., (x:', . . ., .x» be an arbitrary atlas consisting of connected charts.
Since the manifold M is odentable, there exists on M an oriented
aLIas {Vp, (y, . . " yU)}. Let us consider an arbitrary point P E U...
5(",:. . . ", z)
/lnd let P E V p . If iJ( , H) (P) > 0, we retain in U.. the
lip. . . " up
8(3":' ,..., .r)
coordinates (x:'. . . ., x), If, on the contrary, iJ( 1 ") < 0.
!lp ,..., 
we shall take (-x1., .r, . . ., x::') as new local coordinates, Thus, we
have made the Jacobian of the transition from (x:'. ..., x) to
(y, ' , ., y) positive at least at one point P of each chart U... Since
U.. are connected, the J aeabians are positive at any point of U...
It is convenient to say that an individual chart defines a local
orientation on a manifold M; also, if the Jacobian of the transition
from the coordinates in one chart to the coordinates in the other
chart is positive, the two local orientations on M are said to be
consistent. If an orientation is defined on M by an oriented adas, the
local orientation of another connected chart can be consistent or
inconsistent with the orientation of JI{, In the former case we say
that the local orientation coincides with the orientation of M, and
jn the latter case we say that the local orientation is opposite to the
orientation of M.
Proposition 1, On a connected oriented man/lold there exist exactly two
distinct orientations, any chart defining a local orientation that coincideS'
lL'ith one of the orientations of M.
Proof. Let {U.., x:', . . ., x} and IV p. yA, . . ., yB} be two oriented
atlases, If the local orientation of one chart, VB, coincides with the
orientation of the atlas {U "'}, then both atlases define the same orien-
tation on M. Indeed, it is sufficient to determine the sign of the

250 4. Smooth Manl/olds (Ezamplts)
iJ(z:.,...,' z)
Jacobian iJ(' n) at any point P E u"" n V.... Let Po E
v y ' . . " v...
U "'. n Vp be an arbitrary point. Connect P and Po by a conti-
nuous curve q>: 10, 1]..... M, q> (0) = Po, q> (1) = P. Consider the
infimum to of those numbers t E 10, 11 for which the sign of the
Jacobian at the point q> (t) is negative for a certain choice of the
charts containing cp (t). Then, on the one hand, to =1= 0, i.e. cp (to) ;/=
PD' On the other hand, the point q> (to) and the point q> (t), at
which the Jacobian is negative, lie in the same chart. The choice of
the charts U"" 3 q> (to), q> (t), and V p :3 q> (to)' !fJ (t), is insignificant,
since the Jacobian is negative at the point q> (t) for any two charts U'"
and V.,. that contain q> (t). Since the 1acobian is a continuous, non-
zero function, its sign is negative at the point q> (to) as well as at the
point II' (I') for t' < to. Hence, to is not the infimum. The contradic-
tion shows that both atlases define the same orientation on M, In
a similar way one can verify that all charts with a local orientation
opposite to that of the atlas {U a} form an oriented atlas. The prop-
osition is proved.
There is another concept of orientation of M. We say that the
basis (e1, . . ., en) E R" defines the orientation of a Euclidean space.
Two bases are assumed to have the same orientation if the determi-
nant of the transition matrix is positive. Then R" acquires exactly
two distinct orientations. To lerine the orientation of M means to
define the orientation of the tangent space T pM at each point P E M,
Let q>: 10, 11 __ M be an arbitrary path. For any basis we can con-
struct at the initial point (e 1 ,..., en) E T <;>(0) M It continuous
family of bases (el (t), . . ., C n (t») E Tq(t) M which begins witlt
(e1O . , ., en)' All the more, any two families defme the same orienta-
tion in T(I) M. If M is orientable, the orientation at the terminal
point q> (1) does not depend on the choice of the path q>. Thus, to define
an orientation on M means to defllle the orientation of the tangent
spaces T pM in such a way that the orientations are consistent along
any continuous path connecting the initial and terminal points.
Theorem 1, If a manifold with boundary M is orientable, the boun-
dary 8M is also an orientable manifold.
Proof, Let {U a (x)., . . ., .:r;)} be an atlas on M. The last coordinate
.x in any chart U a is a non-negative funcion, M is orientable
by hypohe5is. then by replacing, if necessary, in each chart U a
the coordinate x). by -.z;, we may assume that the transformation
!acobian is positive at each intersection U "" = U a n U. As
charts on 8M we take, as before, the intersections W,," = U ex n {)ll/I,
and as local coordinates we take the first (n - 1) coordinates. Let
us demonstrate that det I l i.J Y 7 II " > 0 at an arbitrary point
iJy I.J"'l
PEW ex n W. Indeed, in t e ambient manifold M the Jacobi

4.5. Classification 0/ Two-Dimensional Sur/ace.
251
matrix of the coordinate transformation at the point P is of the form
().r 'J '1-1 B.rJl
, r,/.
ax iJx "xb
fJ(x,,) _ ox:X  n., II..
u:,.
8 (:l(I) - 8xp-' 8..;-' 8xU-'
az 8.r-L {} ..
z"-
az {}zH azg
Since x . xK -= 0 in the intersection W" n W, we have ,)z:;' - ()
lJxA t::::I t
-' . II (J",) {} (y,.) iI.r
1"",,!n-1, so thdt dct1(T=dct""if()'-:-;;->O. Hence,
. /I ' J 11 y (lZ
'x *U, Since .r:1O in thc interscction U" n Un, we have
"Z{l
x: (P) > O. Thus, dpt il cJ « Y"' » (P) > O. Thl' theorem is proved.
IXII !I(I
Defmition. Let M be an oriented manifold with boundary and let
(U"'(x:.... " .x)}, xO, be an atlas deiining the orientation
on M. Tho orientation on OM defincd by the atlas W" = U '" n aM,
(y;', ' . ., y _1) = (x:', . . ., x;l _1), is called an orientation consistent
with the orientation on M.
Examples
4. Consider again a ball D c R". To dcfine an orienta-
lion on Dt, it is suffIcient to fix at a point P a basis of the tangent
space. If P is chosen on dD2 = St, t.he basis can be so chosen that
the hrst vector el is tangent to the boundary and the second vector e.
points to the centre of the ball (see Fig. 4.84). Then the orientation of
the boundary defined by e l is consistent with the orientation of D2.
The figure shows another direction of the orientation of a two-
dimensional ffianifo]d as thc direction of the rotation of the tangent
plane along the smaller arc from c 1 to Ct.
3. Consider a three-dimensional ball D3 and its bouf1dary 8',
Figure 4.85 shows how the orientations of D3 and S' can be made
consistent using the basis (el' C1, C3) and the right-hand screw rule.
G. This is an example of a non-orientable manifold. Consider
a Mobius band in the form of a square with two parallel sides identi-
fied in opposite directions (Fig. 4.86), Fix a basis (e 1 , c.) at the
point A and construct a continuous family of bases along the curve
ABA. At the end of the curve the hasis changes the orief1tation,

252 i. SmQoth Manifold. (Ez;ample.)
i.e. the Mobius band is non.orientable. The boundary is, however, an
'Jrientablo manifold.
Let us now turn to the description of two-dimensional manifolds,
The simplest one is S2, which is diffeomorphic to Cpl, We have
t'lgure 4.84
already considered the torus T2 = 8 1 X SI which also admits the fol-
lowing representation. Let @J = Z (a) Ea Z (b) be an Abelian group
whose generators a and b are defined on R2 by the shifts a (x, y) =
(x + i, y) and b (x, y) = (x, y + 1). Considering the factor
I
I
I
I
cD
I
I
I
I
I
I
I
t
, . .
. ......
Figure 4.85
space R2/Z E9 Z, we obtain T2, The square O x 1, O y 1 is
called the fundamental domain which acts as an elementary "brick":
applying the elements of @J to this domain, we can "pave" the entire
plane (see Fig. 4.87). The functions f (x, y) on R2 (x, y) hich are

.5. CllZ8Slficallon 0/ TWD-Dlmm,lonal Sur/acts
253
invariant relative to the transformations Z (a) e Z (6) are smooth
functions on T2.
Let us consider on RZ (:1;, y) a class of smooth functions I which
are invariant relative to another group @j (a, b) whose generators
<Ire defined on RZ by b (:I;, y) = (:I;, Y + 1), a (x, y) = (1 - :1;, Y + 1).
r -.- :: }W :t::i::t::.:.:::
Ie..... '.............'.'............,..
T_.: it il_ j .,
figure 4.86
(Find relations between a and b!) The fundamental domain is shown
in Fig. 4.88. Arrows indicate identification of the sides in accordance
with the action of the group, The surface which arises after factor-
ization is shown in Fig, 4.88; it is called a KleEn bottle, A torus and
a Klein bottle are smooth manifolds; a torus js an orientable mani-
fold and a Klein bottle is a non-orientable maniCold (verify!),
y 1
(0, 1) a
b iif1i b
0 a (1. 0) x
_.
a
Figure 4.87
These two examples suggest a simple way oC constructing other
two-dimensional manifolds. Consider gluing on the boundary of
a square shown in Fig, 4.89. Since a square is homeomorphic to
a disk, we obtain a projective plane. Let us depict this plane in R3
making appropriate gluings, Figure 4.90 illustrates the process of
gluing leading to a model oC RPl in R3. We obtain an object which
i not an immersed submaniiold, two singular points, A alld 8,
ueing an obstacle. The figure also shows sections of the model hy the

254 i Smoolh M ant/aids (EzampltJ)
planes orthogonal to the singular segment. Although we have failed
to immerse RJd in R3 by this method, an immersion does exist
(this topic is discussed below).
y 1
Figure 4.88
Lemma 1, HI" is homeomorphic tCl gluing along the cClmmCln bClundary
of a di$k and a Mobius band.
Figure 4.91 illustrates the prool of the lemma.
. i#
8 ..,..............'..8
II!fi«w.t*
b
Figure 4.89
An equivalent formulation of Lemma 1 is as follows; elimination
of a disk from RP2 gives a Mobius band. The situation is shown in
Fig. 4.92 where the remaining Mobius band occupied such a posi-

4.5. Cla8sificatioll "f Tit'o.DlmtI/8111I/al SI/rfart8
251>
lion in R3 that its bonndary circle lies in a two-dimensional plane
(but at the expense of two singnJllf poinls, A and B); snch II posi-
t ion of the Mobius band 2 in RJ is cAHad II cross cap.
1 . .:;:;:t:i?::r::
a'" . . '. ......', 8
NHtrtfr&
/)
Figure 4.HU. E\'oluliIJ" IIr Ih., $1'1'Ii()n.
Let liS COMidl'1' l'l and np from allolhel' point of view. Appl' to
llw1' manifold!' II hon1l'0morphi:\1\\ sho\\'n in Fig. 4.93, We may assu.
II)() lhat .\ lorus is oLJtnined frol\\ S JIY eliminating two disks and
gluing instead or LllI!m .\ handle (see Fig. 4.93) homeomorphic t()
8 1 ;-.: DI. To obtain 1\11 RpJ, olle shollid eliminate from S2 one disk
.Inti glue II l\'lObills 1,1\11(( (see Fig. 4.H3). SlIl'Il an operation is calJed
"gluing a Mobius iiI m", To model this op!)rntion in RU, H is conve-
1I1em to IIse a cross cap (see above) Let 115 apply th!) above opera-
lions., gluing a handle and a Mobilis !ilm, t\\'o times. What snrfaces
do we obtain ill this case? Consider an octagon (Fig. 4.!J4) whose
<ides are marked with lellers G, b, C, and d in stich a way that moving

"256 . Smooth Manifolds (Ezamp/e8)
<clockwise round the octagon, we obtain the word W =
aba- 1 b- 1 cdc- 1 d- 1 . Perform gluings, as shown in the figure. The two-
dimensional manifold thus obtained is called a pret:i:elj it is homeo-
_ _8 .1V a
-II'.' -,
b b D b
5 1 1tmt
. I''''''''''''''''''';'N
D .miIJlt
. 4 11 :::::itim
u .................
D :::{f;::%?::
Figure 4,91
morphic to a sphere with. two handles: 8 2 + 2r (r conventionally
denotes a handle).
Lemma 2. A Klein bottle is homeomorphic to a sp1u!re glued with
two MabiWi films.
A
D2
Figure 4.92
The proof is illustrated in Fig. 4.95. Gluing two Mobius bands
along their common boundary is equivalent to gluing these bands
into /I sphere with two holes (gluing a cylinder with two l\!ob,us
fiJms).

4.5. Classificalion oj Two-Dimensional Surfacs
257
Let us find an immersion of RP2 in R3. Consider the immersion of
a Mobius band in R3 as a half of a Klein bottle (see Fig. 4.96), The
fact is that cutting the bottle, as is shown in the figure, we obtain
two Mobius bands (see Lemma 2). According to Lemma 1, in order
to obtain Rp2, it is sufficient to glue the Mobius band with a disk.
Let us try to realize this gluing in R3 so as to obtain the immersioll
CyUnder = 5' x 0'
(0' Is a segment)
Figure 4.93
RP2 -10- RS. Place the boundary of the Mobius band in the plane of
the figure (see Fig. 4.97) and move this plane parallel towards the
reader (as indicated by arrows). Simultaneously, we carry out in
thi pIano a smooth deformation of the boundary curve of the Mobius
band, completing the initial immersion of the Mobius band with the
plane swept out by the deformed circle. At the last moment, the
circle hecomes standardly embedded, and we glue it with R disk to
obtain the immersion RP2 __ R in question. Figure 4.97 illus.
trates the structure of the set of self-intersection points.
We have obtained two ways of constructing two-dimensional man.
Holds (Fig. 4.98):
(1) k handles should be glued to 8 2 ,
(2) $ Mobius films should be glued to 8 2 .

2.';8 4. 81>1oolh Manifold. (Ezalnpl.)
Thus, we get two series of manifolds: M:=R = 8 2 + kr and
111. = 8 2 + SJ.t2 where M are orientable and M are non-orients-
ble. On MJ there always exist smooth paths lyingon the Mobius film,
Le. modifying the orientation of the frames moving along these paths
(see above). A priori, there al30 may exist manifolds of "mixed type",
Le, manifolds obtained from 8 2 by gluing k handles and s Mobius
films. This operation does not however lead to new manifolds. Let us
b
Figure 4,94
consider 8 2 + r T 11-. Fix one base of handle r and begin to move the
second base towards the Mobius film, theu move tho base along the
axis of the Mobius HIm and leave the !Hm, coming back to the ini-
tial position. Tho handlA will find itself in a new position (see Fig.
4.99); this position is called "handle turned inside out" and differs
from the ordinary position by the way the bases are glued to 8 2 ,
But (see Fig. 4.100) a sphere with a "handle turned inside out" is
homeomorphic to a Klein bottle, i.e. (see Lemma 2) to a sphere with
two Mobius films. This means that in the presence of a Mobius film
a handle turns into two Mobius films; in particular, 8 2 + r + f.l s;;
8 2 + 311-, whence S2 + kr + SJJ. s;; S' + (2k + s) 11, i.e. mani-

4.5, Cla8$Ificatioll of Two-Dimensional Sur/Gee.
259
folds of "mixed type'" are homeomorphic to manifolds of the "non-
orientable series".
As was shown, ail M call he embedded smoothly in R3. We now
prove that manifolds of the non-orientable series can be immersed
. fi:};; a.  ,::,,:::::.:::::::::J s' .   b ,..:.:':: a'
a .,"..,.,.,".,".,..,.,,.. a.'.'.:.;'. .'.... s ::1
%:}:;,:;,::.:.::::::,:::: ,:;:::t!f::::/:{:;:' a" fB
'" ;::iKq : & "": '" .'"
.'", "..'......'..,. .. b
...:.......:.................l..... 8
."Igure 4.95
rnoothly in R3. This statement ha already been proved for RP2 =
.')2 +- 11-. Represent i\l as in Fig. 4.101. Clearly, ll.n is lhe gluing
of I!cv()ral copies of RP2. Imm.:rsing each copy independently in R3,
we obtain the immersion of M in R3.
Figure 4,96
112 + 1'2
[t appears that the two series mentioned above include all com-
pact, smooth, closed two-dimensional manifolds.
Theorem 2 (clasHication theorem). Any smooth, compact, connec-
ted, closed two-dimensional manifold is homeomorphic either to a sphere
8 2 with k handles or to a sphere 8 2 with s MiJbius films.

260 t. Smooth Mati/fold. (Ezampl..)
It is convenient to prove the theorem in several stages. We say
that M2 (of the type stated in Theorem 2) is provided with a trian-
gulation (or is triangulated) if on this manifold there are fixed Ii.
(j(]j(JJ(J)@
 O. :A
Figure 4,9i
nitely many points PI' . . ., P "connected (in some order) by fll\itcl'
many segments ot smooth curves y" in such a war that:
(t) each segment y, starts at a vertex P" and terminates at a ver-
tex P, Po: '* p, and this segment contains no 1\ other than
Pa. and P,
(2) the set of all these segments subdivides 1\12 into finitely many
closed triangles wilh their vertices at points belonging to {P a.}.
(3) any two triangles 6. 1 and l!.2 due to this subdivision either do
not intersect. or intersect at one common vertex, or intersect along
one common side (i.e. along one of the segments Ii)'
These conditions forbid situations sho\\'n, for instance, in Fig. 4.102.
An example of a triangu1ation is presented in Fig, 4.103; this is
the triangulation of RP2 consisting of 24 triangles and 13 vertices.
Problen\, Conlruct triangulations of a torus and a Klein !Joltle.
The same J/2 admits infinitely many triangulations.

4.5. Classljicalioll of Two-Dtmenslollai Sllrfaees
261
Lemma 3. Any two-dimensio1wl smooth, compact, connected,
(l().ed manifold 1"1 2 admits a finite triangulation.
This lemma wiJI be proved in Chapter 5.
L('t \IS consider the triangulation of M2 (of the type stated in Lem-
ma ). l\lark (,I\rh side of each triangle with a letter and assume that
rigure . 98
different distiuct sides are marked with different letters. Provide
('ach side with an arrow indicating the orientation of this side. Di-
r('ctions of the arrows may be chosen arbitrarily. Cut M2 along all
the segments (sides of the triangles), Than M2 splits into a finite set
or triangles each side of which is marked with a letter and provided
with an arrow. We assume that while cutting 111 2 along any segment,
we assign to both borders of the C\lt the letter that marks the seg-
ment (sel' Fig. 4.104).
Our aim is to glue again all these triangles so /.s to obtain a poly-
gon. We begin the procedure with choosing a triangle (it is denoted
hy l)' Consider any side of this triangle; it is endowed with a letter
nnd an orientation. Since each letter appears in the entire set of the
liides exactly two times (becBusa JltJ2 is closed, i.e. it does not have B
houndary and each cut has two borders), there can be found another
triangle (we denote it by l!.2) s\lch that one of its sides is marked
with the same letter. This triangle is distinct from a l , otherwise the
ubdivision would not be a triangulation (i.e. one of the cases of
Fig. 4.102 would be realized). The triangles l!.l and . can be glued
along this common side in slich a way that the arrows point in the
ame direction. We thus obtain It pJane figure whose boundary sides
ar(' marked with letters and arrows. Then we choose another letter;
ngain, there can be found a Irinngle 8 3 sur.h that one of its sides is
marked with the same letter. Glue l!., and so on and so forth. The
procedure ends when all the triangles are glued. We do use all the
J riangles, because if at a certain stage none of the bounadry letters

262 4. SmlJOth M allifolds (t':r:ampZes)
of the domain thus ohtained appeared among the leHers of the re-
maining (not gl ued) triangles, then by gluing all pairwise letters we
would come back to the initial :11' (gluing eliminates ClltS) But hy
assumption, this M will be disronnt'cted, which contral.hcts the
Figure 4,99
condition of the lemma. Thlls, we IIsa all the triangle.. and obtain II
plane polygon W. That W is plane follows from tha faCl that each
time we glue a planc triangle to a plane domain only along ona side.
'fhe polygon W is not de/ined uniqllcJy (evan for a fixed triangula-
tion ; it has the following properties: (i) W is plane, (2) the boundary
oW consists of an even number of sides, each heing endowed with a
let ler and an arrow, each letter appears in the boundary two timas,
Let us fix an orientation on Wand go round the boundary (start-
ing with an arbitrary vertex P), writing successivt'ly aU the let-
ters. If the direction of motion coincides with the arrow, the power
of the corrasponding letter is +1, otharwise, it is -1. When we
come back to the point P we obtain the word W = aila' . , . a,
" k

4.5. Claulficalloll 01 Two-DimlI$iollal Surlace$
26;j
where 8", =::f::1. This word uniquely define,> the polygon W (sec
Fig. 4.105).
With each M2 (in a certain triangulation) we have associated (not
nniquely) a word W. The word W may be considered as a code of .1/ J ,
rz
Figure 4.100
.
5' + 2 ,t
ami this encoding is not unique: infinitely many .:odes correspond to
a single JlP. Our aim is 1.0 prove that any M1. (of the above type) is
homeomorphic either to .r or to M. To this end, we reconstruct
Figure 4.101
..
.
tha polygon via certain operations appropriate to certain homeomor-
phisms of the initial M2 (thase operations will not preserve the tri-
angulation, triangulatioll ceases to he effective as soon as we associ-
ate with M2 its code, the word W). If all the necessary gluings on
OW ara made, we obtain the initial manifold,
Lemma 4, The word W can be reconstructed (by a homeomorphism
oj M2) $0 that all the vertices of the polygon Ware glued into a single
point,
Proof. Let..lls divide the vertices of W into the classes of equivalent
verices. Two vertices arc assumed to he equivalent if they are glued

2r.4 4, Smooth Manifold. (l:.':tampl..)
into a single point under identihcations on OW. If W has only one
class of equivalent verUces, the lemma is proved. Let W contain at
least two classes of equivalent vertices, {P) and {Q). We may as-
sume that there axists a side It, such that its beginning belongs to
i'igurc 4.102
{I'} and tha end to {Q}. Reconstruct W by a homeomorphism of Af2
(ee Fig. .1.106). We recall that each letter (in this case c) appears in
IV exactly two times. As a result of the reconstruction we obtain
IV' wilh the following changes in {PI and {Q}. {P} has lost one ver-
4
Figure 4.103
4
tex, and {Q} has gained one vertex. Choosing another vertex in
{P} we can also eliminate it by a certain class (not necessarily by
{Q}). Thus, repeating the procedure, we can reduce tha number of
vertices in {P} until it contains only one vertax. The last stage:
siD.e {PI has only one vertex, the adjacent sides are of the form
a and a _1 (see Fig. 4.107). The lemma is proved.

4.5. Classlfical/on of Tluo.Dtmen.tonal Surface.
265
Lemma 5. Let W = A baa -lcB. Then there exists a homeomorphism
of ,.,,/Z which transjorms W into the word W' = A bcB.
Figure 4.104
Prooj, Elimination of aa- 1 is performed exactly in the same way
as the elimination of the last vertex in Lemma 4. The lemma is
Figure 4.105. W = ala.aj"'ai'a3a.o:;'a,'o.a.
proved. Here A and B denote those parts of W that remain unchan-
ged under the reconstruction.
Lemma 6. If W = BaAaC. there exists a homeomorphism of M2
.uch that W turns into W' = BA -laaC.
The proof of the lemma is illustrated in Fig. 4.108.
Lemma 7, If W = AaRbPa-1Qb-1B, there exists a homeomor-
phism oj M2 which transforms W into W' = AQPaba- 1 b- 1 RB.
The proof is iJlustrated in Fig. 4109.
Lemma 8, If W (subjected to all the preceding operations) contains
a pair of letters a and a-I, i.e. W=AaBa-IC, B<I=0. there exists a
pair b, b- 1 such that W = AaDbQa- 1 Rb- 1 T. where B = DbQ,
c = Rb- 1 T, i.e. for any pair a, a-I (B '* 0) there exists a coupled
pair b. b- I .

266 4. Smooth M4nlfo!d (EzlJmpls)
Proof. Assume the converse: let for any b E B the corresponding
letter be (e = :1:1) appears in the same word B. Glue W along a.
The result is shown in Fig. 4.110. Thus, all the vertices of W lire
IP; fO!
:P:
Figure 4.106
. }j@:, a..ig:. . .fifu1:fIJ\
":$::::::.:::::::,:;:::::,: ::::::!:i::::::!:::!!:::}:::'
''::':::::Wi$:'':: ':::=fffjffffjW)::'
Figure ,101
A
Figure 4.108
-divided into at least two classes of non-equivalent vertices. Since
the operation of Lemma 4 has been applied to W. we come to contra-
diction. It remains to prove that e = -1. If e were equal to +1, we

4.5. CIQS$lficalion 0/ Two-J)immsional Su.r/aces
267
would obtain contradiction with the form of W implied by Lemma
6. The lemma is proved.
Remark. These lemmas have a feature that each lemma preserves
lh(' property of W proved by the preceding lemmas.
e
Q
'<t
en
Q::
Thus, W is represented as the product of tho expressions aba'b-'
(commutators) and cc (squares), This follows from the fact that the ex-
pressions -a-a- are reduced to -aa- and the expressions
fP]
a
\\\\\\\\\\\\\\\\1\\\\\\1\\\\\\\11\1111
:QI
Figure 4.110
-a-a- I - to -aba-Ib- t -, Il remains to analyse the case where
the word includes hoth commutators and squares.
Lemma 9. /f W = Aaba-tb-1BccQ, there exists a homeomorphism
which transforms W into W' = Mp2q 2 d 2 N = Aabd- 1 B- 1 bad- I Q.
The proof is illustrated in Fig. 4.111. This operation turns W
into the word W = A abel-IB -Ibad-'Q. It remains to "collect to-

268 4. Smooth Manifolds (Exampl.s)
gether" the three squares, and this can be done by Lemma 6. The
lemma is proved.
This lemma is a formal proof of the assertion that manifolds of
the "mixed type" are homeomorphic to non-orientable manifolds of
A
.Figure 4,111
the second series (see above). Combining the lemmas proved above,
we obtain
Lemma 10, Let W be one of the codes of Mt. Then there exist. a lw-
meonlorphism of M which turns W into one of the following words:
(1) W = aa-l,
(2) W = alblalb;l... aBbaglb81.
(3) W = CICICC1' "CkC'"
Which manifolds correspond to these three types? Clearly, in case
(1) .1:1 2 is homeomorphic to a sphere (Fig. 4.112). In case (2) lift
Figure 4.112. WQQ-\ is not a triangulation.
is represented as 8 2 with g handles. The number g is called the ge-
nus of a surface. Indeed, for g = 1 the word W = albla;'b' defmes
II lorus (sl'e <l1Io\'e); for g = 2 the word VI' albla;'b;'a2b2a;'b-
defines a pretzel (see above); and so on by induction. In case (3)
M2 is homeomorphic to a 8phere with k Mobius films. Indeed, if

45. Cln..ificalion of TLDo.Dlm<Rslonal Surfacos
269
k = 1, W = cle, defines RP2 (see above); if k = 2, W = C 1 C 1 C 2 C 2
defines a Klein bottle, etc. Thus, Theorem 2 is proved.
There exist some other convenient representations of M2.
Thr.orem 3, Any smooth, compact. connected, closed two-dimensional
manifold Af'/. can be represented in the form W = ala2...aNa'a'...
...a?1'-s a\. where e = -1 'I and only if M2 = M} (is orientable) ,
1,'lgure .1I3
in this case N is even; alld e = +1 (N is arbitrary) if and only if
1'vJ2 = M'", (is mm-orientable).
Prool. Note that this representation of /lf2 is called a canonical
symmetric form. That W is reduced (by the elementary operations
described above) to words (1) or (2) 10lIows from the previous discus-
sion. Consider, for certainty, e = -1. We need to prove that the
representation described in tPis theorem is possible lor any M.
For what follows see Fig. 4.113. Since N = 2g is even, W is reduced

270 4. Smooth Mantfolds (Ezample.)
to W = albla;'b;'..,asbBai;tb"81, where g is arbitrary, which is
what was required. The case e = +1 is treated similarly (verify!).
The theorem is proved.
Here we do not have at our disposal the apparatus for proving that
M; are not homeomorphic to one another for gl =1= ga, that M are
not homeomorphic for 111 :;6 l1a. and also that Mi and MB are not
homeomorphic. We report these facts without proof.
4.6. RIEMANNIAN SURFACES
OF ALGEBRAIC FUNCTIONS
First, we recall simple properties of complex-valued
functions which we shall need below. Ll't en be referred to the coor-
dinates ZI, . . .. zn; if y (t) = (ZI (t), .... zn (t» is a smooth
curve, then
l(y)= J  A :A dt= .. r' ( t r + ( ':,t )2 dt,
kc::l1 a r 1cct1
wher£'! dz""=d:eA+idlt", For n= 1 we have
fJ a: fJ iJ; iJ iJ 0
0.;;= h'8Z+Q? bZ"="T:" + fl, '
=.!-+ iJi ..2--=t ( .!-_ !- )
iJy iJy fJ: iJy iJr. {JI. {Jz '
{J 1 ( {J .{J } {J ' ( a .iJ )
"8Z='2 BZ-tOji' 8; ='2 ,,&+t8i/
_ a: _ 0
af. - 8: - .
Let us identify en {zA} with Ran {xl<, II}; then, as was already
noted, any polynomialf (ZI, ,.., zn) CBII be written in the variables
{X'''. y"} as g (:e 1 , yl, ' .., xn, yn). Conversely, any polynomial
g (Xl, yl, . , " Xn, y") can be written in the variables {zA,?} as
t (zl, Zi, . .., Z", ZR),
Lemma t. The polynomial f (zl, -;;1, . . ., Z", i') does not depend on
z.. it and only if iJfJ{)Za ;a O.
Proof. The direct statement is obvious. Verify that if "" .... 0,
{J:
then f (Zl, %1, .." zn, i') does 1I0t contain the variable za,
Assume the converse and represent f as a polynomial in the powers of
zcz: f= CI) (za)J> + . ." where the coefficients CI), . .., do not de-

4.6. Rlemt1.nnlan Sur/ac., of Algebraic Funcllons
271
- - M
pend on za.. Let p be the maximal power of za., then "'"=C = P X
01
OJ (z«)P-l + . .., which contradicts the condition, The lemma is
proved.
(w) '" c'
w
z
o (Z)-C I
Figure 4,U4
We remind that the function f (zl, Zi, . , " zn, z")
analytic if ajla 5S 0, 1  cr. n. For n = 1 we have
C 1 (z) = RZ (:e, z), f (x, y) = g (z, 'Z) = u (x, y) = Iv (x, y).
then
is called
RZ (x, y) =
If o = 0,
a.
;z (u+iu)+i :/1 (u+Iu)=O,
( : - :: ) + t( : + :: ) = 0,  = :: '  = - * '
A_ __ o1 - - 
u""=u,,..= -u lIlI ' U"',,+ U IIY =0, ... +
. iJz' og" o. as .
We shaH need a complex analogue of the implicit function theorem.
The proof of the complex version of the theorem can be found in a
course on the theory of functions of a complex variable.
Proposition 1, Let J (z, w) be a complex-analytic function on
C"(z, w), Consider the equation {(z, w) = 0 in the uariable w, and let the
relation : :;6 0 be satisfied at a point Po E {f = OJ. Then there
exists an open neighbourhood U (P 0) in C2 such that in this neighbour-
hood the junction w = g (z) has the following properties: (1) g (z)
is complex-analytic, (2) w = g (z) is a solution of the equatwn
{ (z, w) = 0 in U (P 0)' i.e, t (z, g (z» = 0 in U (P 0), and this solution
is unique in U (P 0)'

272 4. Smooth Manifolds (Ezamples)
Geometrically, it is convenient to consider the solution w =
g (z) as a "graph" (see Fig. 4.114).
Definition. Let t (z, w) be a polynomial in the variables z, w
in CD and let the equation f (z, w) = 0 be solvable for w in an open
neighbourhood U (Po) (for example, let :  0). Then the function
UJ = g (z), which is a solution of this equation, is called algebraic,
and the set of points (z, w) such that f (z, w) = 0 (i.e. the zero' level
C'{w)
w.
t
Wo
C'lz)
o
zo %0
Figure 4,115
surface of t (z, w» is called the Riemannian surface for the algebraic
function w = g (z) at those points where w = g (z) is defined.
Note that if   0 at a point Po = (zo, w o ), then in U (Po)
there exists a solution z = <p (w), i.e. the equation t = 0 can be solved
for z. In this case <p (w) satisfies conditions (1) and (2) of Propo
sition 1. Thus, the condition of local solvability of the equation
t (z, w) = 0 is of the form grad t = (, : )  0, where grad f
is the "complex gradient" of the function f.
Let us write the polynomial t (z, w) in a general form. representing
it. for example, as a power series of z: t = ao (w) z" + al (w) Z,,-l +
. " where the coeffIcients ak (w). 0";;; k";;; n. are polynomials
of w. It is worth noting that Riemannian surfaces of algebraic func-
tions are non-compact and "go to infinity" in C 2 . (We assume that f
is not identically zero.) Indeed. fix a point W o E CJ (w); this leads
to the equation a o (wo)z" + a J (w o ) Z',-1 + . . . = 0 for z. According
to the familiar algebraic theorem this equation al ways has a complex
root, i.e. there exists a point Zo such that t (zo. w o ) = O. Since W o
can be made to tend to infinity along Cl (w), the surface I (z. w) = 0
also goes to infinity (see Fig. 4.115). Since we deal with a poly-
nomial, we can replace inhomogeneous coordinates in CP' by homo-
geneous coordinates Xl. x2, r, setting z = :r;1!X', w = x2/x'. Then

4.6. Rmanntan Surfaces of A tgebratc Fu"clton,
273
f (;I, W) = ap WP;l0 is reduced to n homogeneous polynomial
ap9(.:tI)O(.x2)P()'-\PH), where s is the maximal degree of the mono-
I11ials wPzo, deg (wPzO) = p + q. 'fhus, we can "compactify" a Rie-
I11annian surface by transferring it from C 2 into Cpa, Since the equa-
tion g (Xl, x\ z3) = 0 is a polynoI11ial on cpa, this level surface is
compact in CP3. We shall not go into details of this compactification,
for the procedure is not trivial and we shall not need it below.
Let us discuss the structure of a Rienlannian surface from the to-
pological point of view. From the real point of view, the equation
f (z, w) = 0 splits into two equations: Re f = 0 and 1m f = 0 in
R4, i.e. at the points of "general position" the set f = 0 is a two-di-
mensional (real) surface.
Theorem 1. Suppose the polynomial f (;I, w) is of the form f =
WO - P n (;I) and suppose the polynomial P n (z) does not have mul-
tiple roots. Then the equation f (z, w) = 0 defines a smooth two-dimen-
.ional (over R) submanifold in C2 (z, w).
Proof. Let the point Po belong to {f = O} and have the coordi-
nates (zo, w o ), where w * 0; then f I = qWS-1 * 0 at Po,
uUJ Po
La, by the implicit function theorem (see Proposition 1), in a neigh-
bourhood of Po the surface {f = O} is a smooth two-dimensional
submanifold, the graph of the algebraic function w = g (#). In our
case the solution w = g (z) is of the form w = V' P" (z) . It only
remains to consider the points Po = (zo, 0). We assert that
grad! (Po) = (- dP;z(!o) , 0) * O. Iff.Pn(;I)= 0, we have P,,(zo) = 0
and :z Pn (zo) = 0 at (zo, 0), whence Zo is a multiple root of Pn (z),
which contradicts the hypothesis, Thus grad f (P 0) * 0, and in an
open neighbourhood U of the point (zo' 0) the surface {f = O} is
given by the smooth graph w = g (z), where g (z) is an analytic
function. The theorem is proved.
Let us now examine the main properties of algebraic functions.
(t) An algebraic function is usually many-valued, Le, for some
argument z the function w = g (z) has several values: for example,
w = V... where q and n are relatively prime. The simplest case is
'the function w = Vz which has exactly two values w = :I:: V io
for each Zo * O. The function z = ct (w) can also be many-valued.
The linear function w = az + b is single-valued. The many-valued-
nes:! of w = g (z) can be interpreted as follows. Consider the ortho-
gonal projection n: C 2 -+ CI (z), n (z, w) = (z, 0); the complex straight
line H (zo) parallel to CI (w) is the inverse image of any point
Zo E CI (z). The Riemannian surface r = {f = O}, considered as
the graph of w = g (z), is also projected onto C 1 (z). the inverse im-
age of any point Zo E C 1 (z) under n: r -+ C1 (z) being represented by

274 4, Smooth M..nlfolth (E..mple$)
all the values of the function w = g (z) at the point Zo (Fig, 4.116).
Since the function w = g (z) is algebraic (i.e. j is a polynomial),
the entire plane C 1 (z) ie the image of the surface r under the projec-
tion 1\:: r -+ Cl (w), i.e. for any Zo E Cl (z) there (',ists at least one
solution of the polynomial equation I (zo, w) = O.
C'(w)
!
C'(w)
I
I
I r.
zIP)
C'lz
C'(r)
o
o
}'Igure 4,116
Figure 4.111
Let us introduce a new variable by writing w = g (z) in the form
w = p (P), there PEr is a variable point on the surface r (see Fig,
4.117). Apparently, the function IV = P (P) is single-vahled because
for each value of the argument PEr this function has exactly one
value, Thus, we have succeeded in making the algebraic function
single-valued (in the new variables), but at the expense of consider-
able complication of the domain of the argument: in the former case
the argument z varied in Cl (2:), and upon substitution the new ar-
gument P varies on the two-dimensional manifold r, Hence, the
Riemannian surface r = {j = O} of an algebraic function w =
g (z) is the domain where the function is single-valued, This prop-
erty is sometiml!s used to give another equivalent definition of a
Riemannian surface.
(2) Since for each Zo E C I (z) there exist, in general, many values
Wi = g (zo) (their number is equal to the number of the roots of the
polynomial equation I (zo, w) = 0)), one can define in an open neigh-
bourhood of each Zo Eel (z) a set of continuous (even smooth) func-
tions IV = {<PI (z)}, 1  i  k, where k is the degree of the poly-
nomial ao (z)!J' + a l (z) W"-l + . . . + a" (::) = f (z, w) in the
variable w, Each of these functions describes the variation of a cer-
tain root of the equation f (z, w) = 0 !IS z changes, The function
CPr (z), 1  i  k, can be continued to all values of the variable z,
and at cPl'tRin points these continued functions can coincide or
even interchange. The functions <PI (z) are called branches of the al-
gebraic junction IV = g (z). Each branch describes the behaviour of a

1.6. Riemannian Surface, of A Illebra!c Fun.llom
275
certain root of the equation 1 (z, 10) = 0; in this equation z is consid-
ered as a parameter and 10 as the quantity to be sought, The values
of the function 10 = g (z) that are the roots of the equation
f (zo, 10) = ao (zo) wi< + a1 (zo) W h - 1 + . . . + 4h (zo) = 0 "hang" over
each Zo E C1 (z). It follows from the implicit function theorem that
each branch defines (almost at all ponts) a smooth. even complex-
analytic, submanifold in C 2 , since CPI (z) (locally) a complex-analyt-
ic function.
(3) Let liS associate wit.h each point Zo E C 1 (z) a number k (zo)
equal to the numbcr of distinct roots of the equation f (zo, 10) = O.
C'(W)
J' It.
zit Zo
Branching
point
h(z;) < k
Figure 4,118
k(zo) = k
C'IZ)
Apparently, k (zo)  k; H all the roots of I(zo. 10) = 0 are simple.
Ihen k(zo)=k; if the equation has mllitiple roots, thenk(zo).<k.
'fIle Dumber k (:to) is equal to the numbor of distinct values of
It> = g (z) at the point zoo The points :to, where k (:to) < k, also
ha ve the property that a t them certain branches (j>/ (z) of the
function IV = g (z) merge, thereby reducing the numbor of distinct
vn]1Ies {(PI (z)) over Zo (see Fig, 4.118). Suppose f (z, IV) = w q -
P n (z), and the polynomial P n (z) does not have multiple roots,
According to Theorem 1. the surface r =: {f = 0) is a complex-
analytic submanifold in Ct. The points Zo E C (z) for which
k (:to) < k are called branching points of an algebraic functIon 10 =
g (z) (this term is discussed below). In the example 10 = 'V Pn (z) ,
thE' branching points (in the finite part of C 1 (z), i.e. in the plane C1 (z)
not completed with "infinity") are represented by the lootS of the
polynomial P n (z); and if Zl. . . .. Zn are the roots of P n (z) (they are
all simple), then k (za) < ". = q = deg.. (w Q - P n (z)), since
k (za) = 1, 1  C(  11" The "infinity" point is not considered for
the time being, Thus, all branching points of the function 10 =
'If P n (z) are isolated,

276 4. Smooth Man!fold. (E:r;amp!..;
(4) The branching points (in the finite part of C 1 (z») have the prop-
erty that several branches of the function w = g (z) merge at these
points. Since a branching point is isolated, it may be considered as
the centre of a sufficiently small disk DZ which does not contain oth-
er branching points, so that k (z) = k for z E D2, Z =1= zoo Consider a
circle in D" with centre at Zo and move along this circle round zoo
Generally, the branches of the function g (z) interchange, so that
making an appropriate number of revolutions round %0. we can pass
from one branch to another.
r = [w 2 = z(
Figure 4, t19
Here is an example, Let I = w 2 - Pn (z), where Pn (z) does not
have multiple roots; let n = 1, i.e. 1= W Z - z, Then w = g (z) =
V;' Obviously, k = 2, w = {CPl (z), !pz (z)}, where CPl (zo) =
+Vzo and <l'a (zo) = -jfzo. If z = r,ei'l', then CPl (z) =
11 r ei'l'/z, CPz (z) =; - y r-e i 'l'/2; 11'1 (z) =1= <J1. (z) for z =1= OJ 11'1 (0) =
<J>. (0) = O. The point 0 E C 1 (z) is the only branching point in the
finite part of the plane, Both blanches are smooth functions. If one
moves round the point 0 (r is constant, 0 <p  2n), the branches
£PI (z) and <PZ (z) interchange:
'''' 1('1'+2,,> t'l'
cpdz)=Yre 2 _ y're--r-;.- -Vre 2 =<pz(z).
This interchange is schematically shown in Fig. 4.119. The
Riemannian surface itsolf does not contain singular points, since r
is a smooth submanifold in C2.
Let us analyse the global topological properties of a Riemannian
surface. We shall confine ourselves to the case f (z, w) = w 2 - Pn (z),
where Pn does not have multiple roots. The surface r is defined in C'

4.6. Riemannian Surface. 01 Algebraic Function.
277
as the graph w = g (2) = V P n (2). Before studying the behaviour of
rat infinilJ', let us explain what we mean by "infinity". There are
several ways of defining this concept. This is the so-called com-
pactification problem; it is related to a more general question: how
can a non-compact, open manifold be transformed into a compact man-
ifold by "adjoining a boundary"? This problaffi does not, in general,
have a universal solution, since the concept of compactiflcation de-
pends on the pU1'pose of a given problem: namely, an open manifold
can be compactified in different ways (if this manifold admits com-
pactification).
Let us consider a straight line Cl (.z) and adjoin to it an infinite
point, i.e. adjoin this point to the plaua R2 (x, y)j in this case CI U 00
is transformed into S2 which will be called a completed complex
straight line. There is a canonical way of identifying a completed
complex straight line with a one-dimensional space Cpl. Choose for
(',pI the model: (AZ I , /...Z2) , where'" * 0, and consider the mapping
),.1 ,1 a c
It: (".zI, Az2) -+ 1..z2 = z;; E s' = CI (z) U 00. If b = d = P, we
may assume that a = 1, C = 1, i.e. b = d = p-I and (1, p_l) =
(1, p_l), i.e at "finite" points of the extended plane R' U 00 =
S' the mapping h is a homeomorphism; adjoining of the point :x>
preserves this prop('rty of h, Thus, we may sometimes assume that
Cpl  8'. Recall that the extended plane can also be identified
with 8 by stereographic projection,
Let us now consider the direct product 8 X S'; a complex struc-
ture can be introduced in this four-dimensional manifold, since
S'  CpI, i,e. 8 2 is a complex manifold. Fix on the spheres 8 2 (z)
,lIId 8 2 (w) infinitely distant points 00, and QO"" and assume that
8 2 (z) = CI (z) U 00 ." S' (w) ,== CI (w) U 00"" Thl'n the point
(00" oow) is fixed in 8 2 (z) X 8 2 (w), such that two copies of the
sphere, 00, X 8 2 (w) and S2 (.z) X 00'0' pass through this point. If
\\'0 eliminate this wedge of two spheres (gl\led at (00" 00",)), we
obtain R4 identified with C 2 (z, w) (see Fig, 4,120). Clearly,
(8 2 (z) X S2(w»'[(8 2 (z) X oou,) U (00. X S2(W»)
= (8 2 (.z)'oo,) X (82 (w),oo".) = CI (z) X CI (w) = C2 (.z, w).
This i!; a particular cIIse of the formula (X x y),I(X X Yo) U
(Xl X Y)] e; (X,x o ) X (y'Yn), wherE> Xo E X,!/o E Y, Converse-
ly, C 2 (z, w) can he transformed from a non<ompact. open manifold
inlo n compacl manifold hy adjoining to the former a houndary, i.e.
the wedge of two spheres (S x 00",) V (00. x 8;') glued at the
point (00" OO'D) (by the wedge of two topological spaces X and Y we mean
the space obtained from X and Y by identifying two points XD E
X, Yo E Y; the wedge of X and Y is denoted by X V Y). This is
one of the way'" of the compaclincation of C 2 . Another way is to

278 g, Smuolh Manlfold3 (Ezamplel)
adjoin to C a point, "infinity", thereby transforming C into a sphere
S', These two ways of compactilication differ significantly; one can
prove that S2 X 52 and S' are not homeomorphic (the proof is be-
yond the scope of this book),
<;'(Z) )( 00"
Ox S'(w)
oo
"", x S1(W)
S'(z} x 0
(11 cow)
00, X S'(W)
00 1
111
, , . :@jij@@ijji
c (z) xC (w) ,..'......,.....,...
}:g;::{:::l:
Figure 4.120. (8 2 X 8')'[(5' (:) X co".) U (00, X S' (w)) s Co (z, wI.
There is one more method to compa('tify C" and transform it into a
compact manifold. We are already familiar (in part) with this meth-
od; it lies in introducing projective coordinates. Let us consider
CP2 referred to homogeneous coordinates (.-r t , :r", 3), tlen CP" is

4.6, Riemannian Surfaces of Algtbratc Functions
279
co\'ered with three chal'Ls homeomorphic to C2:
A1={A-(x l , x 2 , x 3 ), xl*O},
A={A-(Xl, x 2 , x 3 ), x2*O}, A 3 ={i..(x', x 2 , x 3 ), x 3 *O},
Let
eta" A3......C2(Z' w), «3(A-(X I , %2, x 3 »= ( :: , ;: ),
%1 %1
Z=7' w=7'
then a,l is a homeomorphism between the chart A  lind C 2 . The homeo-
morphisms a 2 : A 2 ..... C2 (u, L') lIud a,: Al _ C2 (0:, )
21 Z3 %2 
u==zr. V=-;r-, (Z=, =%1'
IIrc constructed in a similar fashion (see Fig. 4,121). Set
CP:.", S -= {1. (0, %2, ;t.-3)}, CP = S: = {1 (Xl, 0, .z3)}.
CP= S:={1(x t , x 2 , O)},
then Cpz = A I U S, Cpz = Az US;, Cp2 = A3 US:. where the com-
pact two-dimensional manifolds Sit S:, and S: are homeomorphic
. . .l;t. . i;;!::i;'" 0 . )1.l;iii:::,
,0.... .... .... 00' ................ :.:. ''':'"0 '0 '. '0 .....
.Jj}ftfJY' ':j11!!Jlmmi: '. .:,;:..:,......
A ,( ..,;.. . A 2(U. ;..... A.{", II) Cpz = A, U Az U As
Figure ,U21
to 8 2 , Thll5, C clln be compactifled b' Eldjoining a sphere Sl, which
tu\nsrorms C into CPl; this operation can be realized in two ways
(see above). Let liS consider all the three methorls of fompactilication
of Cl and observe what happens to 1\ Riemannian surface r embedded
in C. Since r is lion-compact Rnd goes to infmity, it IIlso undergoes
compactilicalion, therehy hecoming a compact topologital space r.
Demonstrate that r is a smooth manifold.
Theorem 2, Let C be compact/fied to S X S= by adjolning to C
the wedge of two spheres S V S = (S (z) X 00",) V (00, X S': (w».
The/! the Riemannian surface rof the algebraic function IV = V P" (z )
(the polynomial Pn has simple roots) Is compactified to a compact,
"nwoth, closed two-dimensional manifold f

280 4. Smooth Manifold. (Ezarnpl..)
Proof. We first consider that part. or r which is rOlltainerl in C 2 =
($2 X 5 2 )',(8 2 V SO). Arcording to Theorem 1, this part of :r
is 1\ smooth mnnifold. Consider I' n ($2 V 52). We assert. that this
is a single poinl. i.e. f tOllrhci' the set of "infinite poinls" (8 2 (z X co",) U
(co, X 52 (w» only at onc point. Indeed. the form of w = V p"(z)
implies Iflallhe image uf "infinity" iHepresenled solely by "iutIJlity",
i.e. f ('on1aills only one infinile poi"t (00,.00",) (Fig. 4.122). Let
$2 X $2
"
Q
Figure <Ii.122
us prove that the structure of II smooth manifold can he introduced
in II neighbonrhood of tlie point (00., 00",) on r. Deline on 8 2 X S2
lhe coordinate.!> z and w which run 8 2 (z) Rnd S2 (w). Introduce new
variables u = 1/z anrl v = 1iw. Siure lhe Icansformalion z  u =
I z i)< rugular on 8 2 (z), this lrll11sfOl,ntation is also regular on
S (z) ,: S! (w), so that 00. ->- 0 1 (' S! (z). 00". ->- Ow E S2 (w).
The c>ql1alioll
w 2 =P n (z)= n (Z-ah),
"=1
where 01 =1= OJ for i =1=;, is reduced to
r
"
,ir- IT (  -0,,) =0,
ltc:t
"
un n
7- (1- a h u )=O,
h=1
that is
v 2 =
un
n
II (1-a"u)
"I

4.6. Riemannian Surface.. of A 19.bratc FuncHo1l8 281
We now study the structure of zeros of the function feu, v)=
vz_un( fl (1- a"u»)-l in a neighbourhood of the point ° (u=O,
"...1
V = 0). After the chang!' V Z = t, u = u the equation takes the form
t = n un . Calculation yields grad 7 (t, u) 10 = (1, 0) for
Il (1- a h u )
"...\
1/> 1 and grad 7 (t, u) 10= (1, 1) for II = 1, i.e. grad 7*0 at.
{he point 0, and by the implicit function theorem the surface under
<tnrly (in the variables (t, u» is a smooth submanifold in C2 (in the
neighbourhood of 0). With the inverse change, u = '}IT !lnd u =- u,
the graph of the preceding manifold undergoes square-root transfor-
mation. Siuce the equation v - t=>O defines near the point ()
a smooth submanifold, the initial graph is also n smooth submanilold.
The theorem is pr.)ved.
Under other complicLiticalions (adjoining a point and adjoining
a sphere), the corresponding !;ets r' anrl r" are also smootll
tlhl\1anirolds in the compactilied C. We shall not prove this
tatement in a gencral case, but consider several examples,
Under the compactiflcation C Z _C 2 U oo=s, the surface 1"
embedded in C2 is also compactifled by a single point, bu under
C Z -.. CZ U sz = Cp2 the situation is more complicated, Since r t.::
A 3 = {i..(x', Jf1-, XI), XI=;60}, we have to find, under the compacti-
ficiltion cz__Cpz, the intersection of r" with the adjoined sphere
ZI z'
SZ=Cpl=S:, Let n>2. The change z=%3"' w=-;;- transforms
n
the equation 11 (::-a,,)=w2 into (X2)2(:z:3)"-Z_1! (x l -a"x3)=O
11.1=1 ItImlI
(verifyl). Since S: is defined in Cp2 by the equation :z;3=O, ths-
intersection r" n S; call be found by setting :z:3 = ° in the equa-
lion of the surface, which gives the solution (Xl = 0, :£2 is arbit.
rary, :z;3=0). This means that f" n S; consists of a single point
wi th the coordinates u."  = 0, v = :: = 0 in the chart A 2 =
(J..(:z;I, :z;z, x3), x2=;60} (Fig. 4.123), To study the local structure
n
of the set of solutions to the equation (x 2 )Z (:z:3)n-z_ I! (x l -
Ami
ahx3)=O in a neighbourhood of the point (Xl =:z;3=0, x Z ="-), ons-
h  
sould make the .:hange u=7' v=7' which leads to the
equation q=v n - z _ II (u-aAv) =0. For n>2 the gradient of th&
1i!:1

282 4. Smoolh Manlfold$ (E,"ampl$)
polynomial q is zero at the point (0, 0), and it is not obvious
therefore that the surface is a smooth submanifold at this point,
Though a new change can be made, which vlucidates this fact,
we shall not concentrate on the topic and consider only the cases
n=1 and n=2.
. ( r2 ) 2 ",'
For n=1 we have w--z=O, z:;- --;0=0, (X 2 )2_X I X 3 =O,
Since r c: A3' one has to fmd Y;" n S;, i.e. to assume x 3 = 0,
Figure 4,123
which gives: Xl is arbitrary, x 2 =O, z3=0. Thus, the intersection
of Y;" with infinitely distant sphere S is a single point wit1\ the
x 2 x:J
coordinates (;1;= 7 = 0,  = "7" = 0 in the chart AI = {A (Xl, x 2 , x 3 ),
XI:;= OJ. To examine tbe neighbourhoorl of this point, it is COli-
%,2 x3
venient to make the change 1% = "7'  = -;>, which gives
{(X 2 )2-x l z3=01_ [( :: ) 2 _ :: = OJ =[CX2_ = 0]. Apparent1,
this equation defines a smooth submanifolrl in a neighbourhood
of the point 0: = f3 = 0, and this proves that the compactified SUI'-
face fN is smooth at an infinitely distant point. For n = 2 we
have wZ-(z-al)(z-az)=O where Q.:;=a l ,
( -=:.. } 2_ ( _a ) ( -a ) =0
,%3 .;t:' t xJ '2 ,
(X2)2_(XI- a 1 x 3 ) (x l - U 2 X 3 ) = O.
As for n=1, we obtain: if x 3 =O, then XZ=::!::XI, i.e. the inter-
section Y;" n S: consists of two points TI = (Xl = A, x 2 = A, x 3 = 0)

4.6. RI.",a""lan Surfac<I of A Ig,bralC Functlonr
283
and T. = (Xl = A, ;t2 = -A, x 3 = 0). In tbe chart Al = (J.. (;tl, ;r3, ;t3),
xl=;o!=O} we have T,=(ex=1. =O), T.=(a.= -1. =O) Since-
;1'
;tl '* 0, we may pass to the chart A, via tho change ex = 7 '
".' . 1 Id
 = 7' whic 1 yie s
cx 2 -(1- al) (1- a2) = 0,
grad [a2-(1-al) (1-a2)]T1T.
= (2a., a, (1- a 2 (3) + a. (1 - a,)h,T. "" (:I: 2, a l + a 2 ) =;o!= 0
III the point,; T 1 and T 2' so that the surface F- is 1\ smooth submani-
fold near TI and Tl' For /l = t, 2 we have proved that the compncti-
lill\lion r -+ F which arises lindeI' the compartiflcation C __ CP',
gives a smooth manifold. This is 111::;0 true for n > 2, but Ihe proof
of this fact is heyond the scope o( the book.
In whllt (oHows we shall asstlme that an inllnitely distant point is
adjoint to the surface rc:: C', so that r becomes a smooth, compact
manifold. Let us study the branching points o( the fUHction w =
V P'fI (z) which is now denned on the entire sphere S' = CPl.
Proposition 2. Let w' - P'fI (z) -= 0, where P", (z) ha. ollly 1I1mpie
roots aI' . . ., a". Then all these roots art' the branching polnt. of
w = V Pn (z) located in a finite part of the plane, on the .phere S2
there eJ'i,\'ls, be.ide. the.e points, another branching points, infinity,
prol'ided thf'dRgree of P" i odd. If the degree of Pn is el'en, infinity i.
not a branching point. There are no other brallching point. for thf'
function w = V P", (z) ,
Proof. Consider the case w" - z = O. The function w = :I: V;;
h,\s two branches: !PI = VZ- and !P2 = -Vi: Consider a circle of
Hnite radius with the centre at ° and go rO\lnd this point. The bran.
ches !PI and q>2 are interchanged, so that the fUllction cannot be made
single-,'al\led and smooth on a sphere. Repeat the prored\lre, taking
the poi nt 00 E S as the ('entre of the circle. On S2 all the point!l are
. 1 J . I f . a2+b . . I
eqUI\"!I ent re alive to tie trau:, ormation gronp c; + d ; In particli ar,
00 can be transformed into any "rmite" point_ A circle with centre
al oo'can also be considered as a circle with centre at O. so that cir-
cumvention round 00 also interchanges the branches (see Fig. 4.124).
We \lOW tnrn to a general case. Let ah be an arbitrary root of
P" (z). Consider a small circle 8 1 wilh centre at ak which contains
no other roots of the polynomial and represent the point z E SI
in tho (orm z = ak + re irp , where r is the radius of the circle. We

284
4. Smooth Mo.nlfoldB (EZo.mp!t8
have
w(z)= 11 il (z-a p )= V fT (z-ap)'V z-a"
p1 1>"'''
I icp
=../ lJ (z-a p ).rT.e 2
V /,,,,",,
(see Fig. 4.125), Furthermore,
I (CPo I i(cp,+2n)
w(zo)=rJ',e-r.. /11 (z-ap)-+r"f e-r-... /11 (z-a p )
V 1>"'"" V 1>"'""
under circumvention round a,,_ Sinre tllis circumvention causes the
argument of (z - al<) to change by 2n, Vz=a;; changes the sign.
At the same time, circumvention round a" does 1I0t increase the argll-
S2
Figure 4,124
ments of the numbers z - ap, p '* ', by 2n, and these arguments
take the initial values (see Fig. 4.126). i.e. all radicals V Z - Qp
also take the iDiial values and thE' transition from one branch
o
Figure 4,125
to another occurs only due to the radical V z -a" . Thus, {a,,}
are branching points. Consider the point 00 and make the
change u = 1/z, then 00 goes over to 0 and w = g (z) takes the

1.6. RIemannian Sur/ac.. of AlgebraIc Funcllon.
285
n
fonn /n. n y 1 - ak u , If n is even, circumvention round the
1 U h=1
point 0 returns the function to the initilll value, so that 0 is
not a branching point. If n = 2p + 1 is odd, W =  .;_ X
u I' u
n
II V 1- ahu IInd 0 is II branching point. The proposiUon is
h",1
proved.
As we arleady know, the surface f is the domain of single-valued.
nass of an algebraic function. Represent r as the gluing of several
a"
an
Figure 4.126
"heets," each being the graph of a single-valued algebraic runc-
tion. 'Ve start with 1111 example.
Let liS consider the function IV = :f:1IZ" which is clefind on S
and has two branching points, 0 and 00. Connect the points 0 and
00 by II smooth curve y without self-intersections and make a cut
along y from 0 to 00. Then two subsets, f l and f 2 , arise on S: f]
is the graph of the branch CPI = 11:; delined on S"y and r 2 is the
graph of the branch 11'2 = - VZ" defined on 8 2 " V, Since y connects
the branching points (and there afe 110 other branching points), cir-
cllnwentions rOllnd these points lire prohibited, i.e. we cannot pass
frol!! !PI to <r2' remaining on S" 1'. This impLies Lhat each of Ihe
sht>ets r 1 and r. ig homeomorphic to S" y, and Lhis homeomorphism
is l'ell\jzed by Lhe functions <PI anel CPi (r, is the graph of <PI) (see
Fig. 4.127). The projertioll n: C....... C' (z) along CI (w) maps homeo-
morphicaJly each of the shcels r l nlld r 2 onto S,,,y, Thus, r is
glued of two pieces, II<\mely. the sheets r, and r 2' How can we realize

286 4, Smooth. Mont/olu (Ezample8)
this gluing? Define on S an orientation 8"1d mark one horder with
" + " and the other with "_". These marks appear on the borders of
both sheets, f 1 and f 2 . Since these sheet.s are homeomorphic to
8 2 , "i reconstruction of r requires that two copies of S2, 'V should
be glued with an allowance for the position of the marks "+" and
co_". Since the branches interchange in the circumvention round the
branching points, the border "i of the sheet f 1 should be glued to the

....
Figure 4.t27
border V; ('f the sheet r 2; tbe horders "i- t and '\' of f 1 and r 2 are glued
similarly (see Fig, 4.128). Apparent y, such a gluing produces a
sphere. Thus, we have proved the following statement.
Lemma 2. The compact/tied Riemannian surface r of an algebraic
function w = :I: yz /s homeomorphIc to a sphere 8 2 ,
Let us consider the Riemannian surface of the function w =
Y(z - a). (z - b), where a:l= b. According to Proposition 2.
this function has two branching points: z = a and z =- b; thOllgh
co is not a branching point, it is, nevertheless, a singular point of
tbe compnctified surface r, for at this point the two sheets touch each
other and the values of both branches, Cj)! and '92' at infinity coin-
cide (and are equal to infinity). Tl1us. 00 is a singular point of the
immersion off in C 2 U (S2 V S)=SXS2, but it is not a singular
point of the surface F itself, Let us study the surface r, assuming
that the two sheets do not tOllch at 00, i.e. we as if "unglue" the
point 00, thereby doubling it and providing each sheet with it
"own" point co.

4.6. RImannlan Sv.rlaus 01 AlgebraIc Functlolls
287
Lemma 3. The compactitied Riemannian surfac e r (with "ungl ued
infinity") of the algebraic function w = :I: 11 (z - a) (z - b) i:r
homeomorphic to a tu.o-dimensional :rphere.
Proof. The branching points are a and b, so that one has to make a
Cllt y from a to b Ilnd repeat the reasoning of Lemma 2. The lemma is
proved.
As was already noted. the immersion 8 2 = r -+ 8 2 X 8 2 glues
two points, and therefore the true location of r in 8 2 X 8 2 is as in
Figure U28
Fig. 4.129. Since we have proved that at the point 00 the surface r
is a smooth manifold, the singular point 00 is the tangency of the
two smooth sheets, rather than a conical point (see Fig. 4,130), Let
us consider the general case.
Theorem 3. Let f (z. w) = w 2 - Pn (z), where tke polynomial
Pn (z) does not have multiple roots, Then the compact/fied
Riemannian surface r ("w ith unglued infinity") of the algebraic func-
tion w=:l:V Pn(Z) ='/ fI (Z-ah) is h01TU1omorphfc to a sphere
V 10...1

288 I. Smooth ManIfolds (Ezampl'$)
with [ n;1 ] handles ([ J de1UJtes the whole pari), i.e. 10 a mani-
iold of type M} of genus g= [ n;1 ], The immersion 0/ this sur-
. 1=1'.
, ,.."",,.
two sheals
Figure 4.t 29
face in the comptu:Ufied space C 2 U (S2 V S2) =8 2 X SZ is an em-
bedding if the degree /I is odd and an immersion gluing two potnts
00 (on distinct sheets f, and r 2 ) if the degree n is even.
1'a-..- .',
 [' .-..".
00 ,
g encY ,0,' two shea Is
: ....011
,
I
I
I '"
...
f,
11'1....
...
Figure 4.130
proof. Let n = 2p + 1; according to Proposition 2, all the
roots au ,... an and the point 00 are branching points of w=

4.8. Riemannian Sur/ac4a 0/ Algebraic Fundlof18
21\9
::i: V P n (2) = '1. f fr (Z -4,,); this set can be subdivided into pairs,
t' A=1
say (ai' a 2 ), ..., (a", a"+I)' where a n + t == 00. Connecting the points
(2 -I and a 2 " in each pair by a smooth segment '1'". 1  k  p + 1.
we obtain a sphere 8 2 with p + 1 cuts VI' . . ., '1'1'+1' Then each of
Ille br:mches, qi1 IInd fI>" is a single-valuod function on 8 2 '('1'1 U ..
. U y., H)' Indeed, we have prohibited circumventions round an
'Id;! m,'mber of branching points; with our scheme of cuts, we can only
..  r . .
Figure -i. t3t
;:0 round an even number of branching points. Suppose we have gone
l',,"nd the pairs (aj,. al,+1> ' . . ., (ai cz ' a...+1)' What does it happen
to the branches !Pl and CPt? Since the argument increases by 2n only
fot' those points that have been circumvented, the signs of the radi.
cals change an even number of times. Because we deal with an even
lIumber of points, we need not necessarily distinguish between the
concepts "inside the contour" and "outside the contour", for the num-
ber of singularities outside the contour is also even (furthermore,
on a sphere the concepts "outside" and "inside" are dual) (see Fig.
,t13t). In the circumvention along i' we have
..
«p= II V(z-a'j) (Z-4'1+1)' II ¥ z-a"
1=1 "....(i 1 , . . ., f..+1J
tt
- II (- y z- aL t ) (- V t+1 )' II ¥ z-alt =!p,
1",1 "....('1. "., fc.+n
If we choose another way of dividing the branching points into
pairs. the domain obtained by cutting along the new paths will be ho-
meomorphic to the domain considered above. Thus. the procedure
does not depend on the method of grouping the branching points into
pairs and on the way of connecting points in each pair by a smooth
curve. Mark the borders of each cut with U + " and" - It; then each
of the sheets, r 1 and 1'2' is homeomorphic to 8 2 " (Y1 U . . . U YP+1)'

290 4. Smooth Manl!oLds (E"mpu.)
Hence, to reconstruct f, one has to glue these two sheel.:ltogether by
attaching the border yj (1) on f l to the border W (2) on r 2 (Fig. 4.132).
We obtain M, i.e. sphere 8 1 with g = p = [ n;t ] handles.
Let n = 2p. According to Proposition 2, only the roots a l . . . .,
a 2p are branching points; at infinity there exists a singular point,
a rl .....,.........
( " ,. '"''
--
... ...,...."
FIgure 4,132
3
,.... C. ""
{. '"
,'-"-,....., - ."'''
namely, the tangency of two sheets when f is immersed in
8 2 X 8 2 . Again, dividing the roots into pairs, connecting the points
in each pair by smooth segments, marking the borders of the cuts,
and cutting r along the inverse images of these segments, we obtain
two sheets. The point 00 is "unglued" and each sheet is provided with
its own point 00. Finally, r is reconstructed by making gluing in
the inverse order (see Fig, 4.133). The theorem is proved.
Thus, two-dimensional surfaces arise quite naturally: as Rieman-
nian surfaces of certain algebraic functions.

4.6. RIemannIan Surfact$ of Algebrafc Fundfona
291
Statement 1. Each two-dimensional s1TWoth, compact, connected,
closed manifold of genus g (i.e. of type M) can be endowed with the
.tructure of a complex-analytic manifold.
Proof. Let liS consider r for the function w = :I: V Pn (z) : where
P" has simple roots. Since r c: C 2 , where it is deftned by the graph
It' = g (z) or by z = ((w), we have two orthogonal projections
r.
..
Figure 4.t33
1(1: (z, w) -+ CI (z) and 1(2: (z, w) -+ CI (w). At each point (z, w) E r
at least one of the projections is valid, since grad (w-P" (z» *0
at each point (z, w) E r (see Theorem 1). Thus, we obtain the
c.overing of r with local disks. It remains to find the transition func-
lions. They can be only of two types: either w = g (z) or z =
IJ) (w). Since the unction g (z) or (J) (w) is complex-analytic (see
Proposition 1), the statement is proved, because the continuation of
the complex structure to the point 00 of the surface {'holds inside
the complex-analytic manifold 52 X 8. = Cpl X CPl.
It is a simple matter to demonstrate that a manifold M cannot be
endowed with the structure of a complex-analytic manifold (veri-
fy!).
Statement 2, Each two-dimensional smooth, compact, connected,
closed manilol 01 genus g (i.e. of type M) can be provided with a con-
formal Riemannian metric.

292 4. Smooth. MtJniio/d& (EztJmDles)
Proof. The metric is called conformal if there exist local coordi-
nateS in which this metric is of the form gl/=a..13;: r t liS realie
M; in (extended) C2 as the Riemannian su!fa,e of CII! function
\II
e :::t\%)::,
.......................
:tjJ\\J
Circumvention
Figure 4.134
w = ::I: V Pro (z). Consider on M; the coordinate z which runs C1 (z).
This complex coordinate is valid in the entire M; except at the
point 0 near which we should introduce the coordinate w related to
z by the complex-analytic transformation w = g (z). Introduce in

4.8. Riemannian Srjates oj AlgebraIc FnctlolU
293
C' = R4 the metric
4
dS2=dzii+dwdw=  (dx k )2
k=1
and consider its restriction to r c: C2, We have
ds 2 (M;) = dzdz + dg (z) dg (z)
= dzdz + g' (z) i' (z) dzdz = (1 + Ig; (:) 12) dzdz
or ds2(M)=(1+u+v)(d.x2+dy2). where z=x+iy, g=u+iv.
Here
g' (z) = :. g (z) =  ( :z - i :1/ ) (u+ iv)
=+(U",+ iv",- iU II + VII) = u..,+ iv..,= IJ II + tv",.
because u'" = VII and "1/ = -v",. The statement is proved.
If we make the complex-analytic change z = rt (s), the conformal
metric derived above remains conformal. since
ds2+a. (z) d:di'= cz (1] (,)) [11' aWdgdf=p m dS,
In conclusion, a few words about the way r is embedded in C2
in a neighbourhood of a branching point. Though we have proved
that at such a point r is a smooth manifold. it appears difficult to
depict a branching point in R3, because the space R3 has "low dimen-
sion" as compared to R4 = ca. Figure 4.134 is an attempt to illus-
trate schematically circumvention round a branching point.

Chapter 5
Tensor Analysis
and Riemannian Geometry
5.t. THE GENERAL CONCEPT
OF A TENSOR FIELD ON A MANIFOLD
Let us consider a smooth manifold M". Let P E M" anel
let :c t , . , " x" be a local regular coordinate system in a neighbour-
hood of p, There also exist infinitely many other regular coordinate
systems in the neighbourhood of p, Our task is to study the objects
which BrB invariant under different coordinate transformations. The
new coordinates will be denoted by a prime: .xl', . . ., :r. 1I ',
(1) Let a E T1> (AI") be a tangent vector to M" at a point p, Pre-
viously, we have given sllveral definitions of a tangent vector; SOIDe
of them defined this voctor irrespective of the choice of local coordi-
nates, Le, as an object invariant under regular coordinate transfor-
mations (viz. definition via the sheaf of tangent curves), However,
if we want to associate with a tangent vector its coordinates, Le, to
express this vector analytically, we need to introduce a coordinate
system. If we choose another system, the coordinates will change.
Let us find relations between the coordinates of a vector in differ-
ent systems, Since
{a I' a ,/). /)zY a
a = a a;I = a iJ:x:{' , we have a{ a:x: I ' = a' iJzl a",!' and since
{'
{  } form a basis inTp(Mn),a{=al. Thus, we have
{Jz iJz'
found how the coordinates of a vector change under system transfor-
mation: the coordinates ai, . . ., an are transformed by the Jacobi
matrix () =J. We have obtained this condition from the require-
ment that a vector, as a geometric object, be invariant under coordi-
nate transformation.
On the contrary, we can define a vector as an object given in each
coordinate system Xl, . . .,:e n by a set of numbers ai, . . ., an which
are transformed while passing from the system :c l . . . ., x n to another

5.1. Tho Concept oj II T.nlor Field On II Mllnlfold 295
I' ", di h I ., IIlti' ! A I
system x ,.... X accor ngto t e awa\=a, pparenty,
the object thus defined is invariant.
(2) Let / (x) be a smooth function on M"; consider grad / =
{ ..!!L. } . This set of numbers (functions of a point P) is valid in
i):t l
each coordinate system. Let us find the transformation law, Accord-
ing to the rule of differentiation of a composite function,
of of ozl ozi
61' = o:rl' =0;1 IIz!' = I Ozi' ·
where
grad/={SI' ".. 6n}.
o! ( II",! )
I = -, ----r-- = (/T)-'.
Ozl o:t
i.e. the set (61' ' . ., S,,) is transformed in a different way than the
set of the components of a vector: namely. grad / is transformed by
the matrix (/-l)T, where I is the Jacobi matrix  . This law can
also be deduced through the requirement that a geometric object,
the derivative of a function I with respect to a vector 8 E T p (M"),
be invariant under coordinate transformation, Indeed. this deriva-
tive does not depend on the choice of local coordinates, so that
d d a f = lim J(v (e» -/(v (0»
8-0 e
where
y(t)EM n . y(O)=P. y(O)=8ET,.(M n ),
 =  dzi (I) I = aJ .!!..- = ai' -!.L..
dB iJ",1 dt 1=0 iJzi iJzi' .
Thus.
i' oj I o",i' It! .
a IIzi' = a 8;! ozl' , I.e.
, i oz!' " I !: 0:1."
aISI.=a -I'=a61' 1=61'-'
Ozi O",i
iJ:t i
61' = ----r-- I'
8",
which is what was required.
(3) Let us consider T p (Mn) and the adjoint space Tt. (M") of real-
valued linear functionals l (8), where 8 E T p (M"). We now find the
transformation law for the coordinates of these functionals under co-
ordinate substitution in some neighbourhood of a pointj introduce
in T (Mn) a basis e 1 , " " e'fl consisting of the functionals satisfying
the condition eft (e,,) = 8, where el' ., " en is a basis in T p (M"),

295 S, T1UIOr AMlllsU and RiemannIan GI)_'ry
Since 1 (a) is a real number and since the scalar does not depend on
the choice of coordinates, 1 (a)(:<) = l' (a')(:<,), whence
a=afeh l(a)=(l,ej)(aelt)=llta=l/!.,alt'=l,., :' a".
i}z'
Since a is arbitrary, we have from these equalities lit = a;A l,."
i.e. 1,.. = az:, 1 M hence, the coordinates of the functional are
az
transformed exactly in the same way as the coordinates of grad /,
The elements of rJ, (M n ) are called covectors; thus, grad / is a co-
vector.
We have introduced in T* the adjoint basis consisting of covectors
e 1 , . . ., en I so that T can be identified with r* by setting q>: T-
n
T*; q> (a) = 1, where a = alte/!.: 1 = 1: altel'. Hence, we assume
"I
that 1 mapped by the isomorphism cp into a has, relative to e 1 , . . .
. . "' en, the same coordinates as a relative to el' ,..t en' This lin-
ear isomorphism is not invariant under the coordinate substitu-
tion IT. _ (:r.'), since a vector and a covector obey different transfor-
mation laws: a vector is transformed by the matrix J and a covector
by the matrix (J-l)1'. Thus, the identification of T with r* valid
in the system (x) becomes invalid under an arbitrary coordinate su}).
stitutioo (x) _ (x'), There are, however, transformations which pre-
serva the correspondence cp: r _ T*, for these transformations
J = (J-l)T, 1.e, they are defined at a given point P by an orthogonal
Jacobi matrix. A general transformation has a non-singular Jacobi
matrix, rather than an orthogonal one. Nevertheless, T and T*
can be identified by a linear isomorphism which works for any coor-
dinate substitution, This topic is discussed below.
(4) Let C be a linear homogeneous operator which turns T p M" into
itself and let the same letter stand for the matrix of the operator writ-
ten in the basis 810 . . ., en' Makinf, the. substitution (x) _ (x.').
b . C ' JCJ l' {' iJz iJz1 j Th f .
we 0 tam = -, I.e. Cj' =  azi' c$' e trans ormation
law for the operator matrix can be deduced from the requirement
that the relation b = C (8) (a is an arbitrary vector in TpM") be
invariant under coordinate transformation, Indeed, all the objects
in this relation admit invariant (i.e. independent of the coordi-
nates) definition wbich, apparently, leads to the desirable transforma-
tion law for C, The proof of this statement is left to the reader as an
exercise.
(5) Let us consider Ii bilinear symmetric form B on T p (M").
B (a, b) = blja'b J , It is known from algebra that under a coordinate
substitution the matrix B is transformed into a matrix B' such that

5.1. Th Concept oj It Tertlor FIeld on a Manl/old 297
S ' - ax! axl .
JbJT, I.e. bi'r = a"Y ii%P bl). As before, this transforma-
tion lnw for B can be obtained from the requirement that the scalar
B (a, b), a, bET p bE) invariant under an arbitrary coordinate sub-
stitu i"n. Indeed,
B (8, b) = b!jaib J ;E; b.'i.ai'b i ' = blJ a:; ; a 1 b 1 i.e.
az i ' ax;' /)zl /)zi
btJ = bi'i' --;;-- i -;;- ; , or bi'}' = ---." - ; ' bu'
uZ ux 8,,' 8"
We have considered simple examples of transformation laws;
their distinction was indicated by different indexation of the object
components: ai, lJ' c}, b o , The superscripts are called contravariant
inrlices and the subscrips covariant indices. It is convenient to as-
sociale with each object a pair (p, q). where q is the number of co-
variant indices and p is the number of contravariant indices. Thus.
the set \If coordinates for a vectol' 8 E TpM" is of the type (t, 0).
for a covecor of the: type (0, 1 J. foe an operator of the type (t, f)
and for a bilinear form on vectors of the type (0, 2). Besides this
form, there exists the form B (l, in) on the covectors l, in E T;'M".
Cl\Ic}ation (by analogy.wit!1 B (8, b) shows that if B (l, m) =
{J I' (j l'
bHl,ms, then b"i. = (1;. :z; b H (verifyl), Le. B' = (J-l)TB (J-l).
Th objects just mentioned admit an invariant, coordinate-indepen-
dent. delinition in a particular coordinate system, and the transfor-
Ination law for the components of these objects directly follows from
the im-ariance of this definition. Thus, if in each coordinate system
thte i given a set of numbers (functions) transformed (under coor-
dinate substitution) according to the above laws, we may say that
th.e:!:! l\('t8 define an invariant object which can be studied through
its coordinates in different coordinate systems. This circumstance
underlit!!I the general definition of a tensor field on a manifold as an
invl!rint geometric object which d09.!l not depend on the choice of
the 10l'al coordinate system. A particular expression of the compo-
nents of this object does depend on the coordinate system, .
Defmition. By a tensor of type (p, q) of rank p + q (corresponding-
ly, Q tensor field of type (p, q) of rank p + q) we nuan an object defined
in. each coordtlUlte system (z) = (,1;1, . . " ,1;n) by the set of numbers
Tj::':1 (correspondingly, by the set of sTMoth functlo1i8 T:::: (;1;»
transformed under the coordinate substitution (x) - (x') according to
the law
.' .' i'
T'. .., :!' = ax. t
J"  . . J q B%h
{)Zij, a,,;'
azip a,.J;
iJ lq
 T' ... p
az jq 1. .. . 1q'

298 5. Tensor A naly,i, and Rlmannlan Geomet-;
This definition satisfies the requirement that an object can be de-
fined invariantly, i.e. irrespective of its expression in a particular
coordinate sY5h'm. Jndeed, it is known from algebra that a tensor
(tensor field) can be defined as II PQlylinear mapping
T:Tpx ... xTpxTf,x xT1>-R.
. P
defined by the relation
T ( l I ) Til j p ;1 J . l l t
aft....t 8 q, 1"." p = ;...jqQI ...a q it... 'pt
where 8k =akeik' II< = Ikelh. {ah} are the coordinates of the vec-
tor 8ltE T p (Mh) (the kth factor), {11t} are the coordinates or the
covector lit E T1> (M n ) (the kth factor). The functions T;::::: are the
coefficients of the mapping T. It is known from algebra that a
multilinear mapping can always be written in this form. If the
bases e 1 . ,." en and e'f .. .,e n are fixed. we have a useful formula
for T 11"'p
il...].
T il...lp T( I l )
ia...J q = ell' ..., es q . e 1. ., '. e P .
Indeed.
T(ei!. ....ei. ell. ..,.elp)
q
T C1:1'''C1: p ( ) Pl ( ) p ( I ) ( j )
:-.. Pl...P.' eit " . .' eJ q e 1 C1:1' .., . e p C1: p
T Cll...Q.pPI "Pq Slit "j P T II.. ip
= Pl...Pq Vi! ' " . ' Uiq'Vat ' .,. . v"'p ca ir...i..
Furthermore, T can be defined in invariant terms, since all the
objects appearing in the formula T: (T p)q X (T1»P -+ R, as well as
the multilinearity property, are invariant, A particular expression
fo.' the coefficients of TJ:::::: depends on the choice of the coordi-
nates.
Although the invariant form of a multilinear mapping is more
convenient, we shall almost always consider the "coordinate form"
because it appears quite useful in particular calculations, Let us
introd uce. for convenience, a special notation, the so-called mul ti-
indices: (i) = (i l . ' . .. ip) where the entire set of indices is re-
p1aced by a single one. Then, the tensor law is written as
(I') 8z(l') iJz(J) (i) 8",(/')1 azl
T(J') = 8%<i) 8z(J') Tw. where 8,\:(1)' =a;r.-
8x 1P
iJzlp .

5.1. Th. Concept of a T.ruor Field on a Manifold 299
We now turn to studying simple properties of t!Jnsor fields.
Lemma 1. Let (z) _ :c') be a regular substitution, Then the
. (1') oz(i-) axIl) (I) .
relauon T(j') = a%iiJ azU') T(j) (the tensor law) Implies that
(I) az(l) az O ) (I')
T(j) =   w) TO')'
OZ <lZ
. uz l u:t;h
The proof of the lemma follows from the Identity -..- --.- =
OZR 0'"
! 0 i'
17 = 6j which means that I. I-I = E I where I = ( a:1 ) is the
Jacobi matrix.
Lemma 2. Given a tensor field written in the system (x), and let the
coordinate substitutions (.:r) - (z) _ (y) and (x) _ (v) _ (y) be valid,
i.e, we pass from (x) to (y) in two ways: via (z) and via (u). Then the
expression for the field T in (y) does not depend on the way (x) is trans-
formed into (y),
The proof follows from the formula for differentiation of a com-
posite function
aJ/1 al/' 8J q oJ/' ova.
;;;;< = -a;ri' az'" = avO. az'"
Let us recall some properties of tensors known from aglebra,
(a) Tensors of rank p + q and type (p, q) (they are considered at a
point) form a linear space H such that dim H = 1Ip+o' The proof
follows from the representation of a tensor as a multilinear mapping:
the linear combination of such mappings is also multilinear.
(b) In the space H of all tensors of rank p + q and type (p, q)
there exists an additive basis; the elements of this basis (multilin-
ear mappings) are denoted by
ell  . ..  el  eh  ., ,  eiq.
p
The number of these elements is equal to n Mp (verify I) . Each
mapping
elt  . . .  el @ eh @ , ,. I8J eiq
p
is given by the formulas
et.. (1) = II". where 1 E T; (M"), e j " (a) = a;", a E T pM",
(eiJ 0 . . .  ej @ eit 0 ,. . 0 e;q) (a lt ' . ., 8'l11, . , ., IP)
P
= ell (ll) . ." . Pip (l")' eh (a,). .... ejq (a)
= It . ... ' If. ,a{1 . .., . a iq ),
p q
If T: (T p M,,)q x (T;' M")P _ R is an arbitrary multilinear map-

300 5. Ten.sor A nalVsi. and Riemannian Geometrv
ping of type (p, q), then
T ( 1 1 1 1' ) T lt...Ip it jq Z I 1 "
8 1 , . . . I 8 q , t. .. ., =- jl.. jqal ... a q il."" ip
T 1t... i p ( .0. ' i ) Z l t P )
= it...i", e;1'G1 ... <8> ell' i8I ell <8> ..' <8>e q (a 1'" .,a q , ".., .
i,e.
T , T 11" ijJ J J
= 11''''q (ell <8> '" <8> eip <8> e 1 i8I ... <8> e q).
Thus, the mappings (eil 181 ' . . i8I el i8I eit 181 . .. 181 eiq) form an
additive basis in the space of aU ttnsors of type (P. q). With the
aid of multi-indices, this decomposition of an arbitrary tensOr T
in the basis tensors can be written as T = rUJe(l) i8I e(j), The trans-
formation law for the basis tensorS Bit @ , .. i8I el @ til i8I ... 181 eiq
under coordinate substitution is of the form P
" oz(l) oz<i') .
eW) i8I ell ) = -wi --m- e(l) 181 el')
oz 0:%
(verifyl). All these properties and definitions are transferred to the
case of smooth tensor fields on a manifold; the difference is that here
we should consider tensor fields as linear combinations of basis fields
of type (p, q) with variable coefficients, so that the space of tensor
fields becomes an infinite-dimensional space.
A particular case: any tensor field of type (1, 0) admits (locally)
the representation T = Ti (x) e,(x); similarly, aoy field of type
(0, 1) admits the representation 1 = lle/(x).
5,2, SIMPLE TENSOR FIELDS
5.2.1, EXAMPLES
(1) Let a vector field T (x) be defined on M", Then T (x)
is a tensor field of type (1,0); the space of all smooth fields of type
(1, 0) is identified with the infinite-dimensional space of all smooth
vtICtor fields on M",
(2) Let a smooth covector field be defined on M", i.e. at each point
x E M" there is fixed a linear functional l (x) E T;M" smoothly
dependent on the point. This field is an element of the infinite-di-
mensional space of tensor fields of type (0, 1). This space contains a
linear infinite-dimensional subspace of the fields grad I (x), where I
is a smooth function on M".
(3) Let the, Riemannian metric giJ (y) be given on M". Since
0%' ozl
8"J' = oz/' 0:,J' g'J' the set of all gt! (x) defines a tensor field of
type (0, 2), i,e. a smooth field of bilinear forms. This tensor is
called a metrlc te1l80r.

52. Simple Tensor Field,
30t
(4) A complex-analytic manifold A{2n is characterized by the
property that in each T :rJl,p.n there is defined the operator 1 of "mll!-
tipJir.stion by imaginary unity". J2= -t:. Since this operator smooth-
ly depend!! on the point, we obtain a tl'nsor field of type (1, 1).
(5) [ner!la tensor. Let us consider in R3 J1rigid body with one f\)ced
point O. the body can rotate about the point O. Suppose the body con-
sists of a finite number N of material points rigidly connected with
<3nl' another (their mutual position doe not change during the mo-
lion), We shall only deal with orthogonal coordinate systems in R3,
Let ml' m" . . ., m N stand for the masses of the pointR and let
xf.., . . ., xlN) (i = 1,2, 3) denote their coordinates at a fixed
moment t. Let us consider the matrix
N N
aiJ = _  m..xxt!l.) + ()IJ  mOl I X(,,) 1 2 .
act cr._I
We obtain a tensor of rank 2 and tpe (0, 2), the matrix aU being sym-
metric. This symmetric tensor a'l is caUed the inertia tensor of a
rigid body (various bodies have distinct inertia tensors). The role
of thill tensor may be i1lustrated by the following example. Let us
consider a straight line with the unit direction vector I = (lI, l2, P)
through the point 0 and evaluate the sum '2./}a iJ zil j = H (1). Since
a iJ and II are tensors, H (1) is an invariant of orthogonal coordinate
transformation, Le, a scalar. Calculation yields
H (1) =  uaUlll}
N N
'" "'" i i"'" ; . 1i i'l' "'"
= - £.J m",  x(o.)l  %(..)l' + 6 'll'  m", I x(",) I
"'''1 i ; J 1
N N
= -  m", «X(a)' 1})2 +  11121 X(",) }2
1 «I
N
=  m", (l X(",Jl2- «X(",), 1})2).
",,,,,\
This expres..ion can be interpreted as follows: the factor I x" I  -
(x " ,1}2 is the squared distance from the point of mass m"
to the axis 1. Thus, we have obtained the familiar expression for the
moment of inertia of a body about an 3xis. The eigenvectors of the
matrix a i ; are paranel to the so-called principal axes of inertia of a
rigid body.
(6) Strain tensor, Let us consider a continuous medium which occu-
pies a certain volume in R n referred to Cartesian coordinates Xl, . , .
.. ., :tn. We identify the points of this volume with the points of

:102 5. TenlOr A naly&l" and R I.mannlan Geomelrll
the medium and consider small deformation of the medium, viz..
due to external forces. Suppose we are given displacements of the
points, i.e. smooth functions u;(x l , . . " x"), t::;:;; i::;:;; n. which
depend on the point and enable the coordinates of the displaced
point P to be calculated in terms of the coordinates of the initial
point P by the formula ';i=xl + u i (Xl, ..., x"). We shall consider
small displacements, i.e. thl) functiOlls ul(x) are assumed to be rath-
er small. To formulate Ihe problem correl.tly, one should deal
Figure 5.1
with infmitesimal displacements. which is equivalent to defming a
smooth vector field in a domain. though the theory of continuous me-
dia sometimes handles finite deformations. We shan use the formal-
ism suggested above. Thus, suppose a point P = (xl, . . 'f x")
shifts to the point P = {xi + u i (Xl, . . .. ,1""». How does the length
of a smooth curve change under medium deformation? Consider two
nearby points P (Xl, . . ., x") and P' (Xl'. ..., x n '). Let (6l) =
 (xi' - xl) =  (6x;) be the length of tbe Euclidean seg-
i==1 i-I
ment conneeLing P .and P'; fmd the length (6l')2 of the segment con-
necting the images P and 15' of the points P and P' after their dis-
placement as a result of medium deformation (Fig. 5.1). We have
"
(6l')2 =  (xi' + u i (z') -%1- u l {xW
1=1
n n
= l} [{Xi'_xl)+(U i (:z;')_u;{:z;»)J2=l} (6Xi+6u l )2
1=1 ;=1
n n n
= l} (6:z;i)2 + 2  6xi 6u I +  (U;)2
ic:ll ;-1 i=1
n n
= (6l)2+ 2  Ax l 6u; + l} {6U i )2.
iS ;c::a:l

5.2. Slmpl. T.ruor FI.ld8
303
! au' -"
Sillce au =A;r, we have
n n I
(61')2- (al)Z = 2  ax! au l + h (au')2 = 2   l!..1:la:c"
a:r;
'=1 Icol 1,1&
+   aul a:i' axP
ad' a:r;P
(, R. P
=  ( au! + ) axla:c"
az lt az l
1<11
+  au! aul lJ.:J!'ax p .
I /I ad' ax'P
, . P
'rhus,
( dl' ) 2- ( dl ) 2=" ( a,,! + ) dxid:J!'+ " Sui !!:!-.d:J!'dxP
LJ iJzII azl LJ azll a:ep .
1<1< 1,11,1'
If the strains u l (x) are small, we can assume
(dl')2- (dl)2   ( :;: + ::: ) dx l d:J!'.
1<"
11,,1 au ll I __"
Set !Jil< (x) = II%/r. +a;r, then (dl')2- (dl 2 )  !JII< dx dz-. The set.
of functions tJl" forms a tensor of rank 2 of type (0. 2), called
the strain tensor.
(7) Stress tensor, Let us consider an elastic body subject to a defor-
mation which produces stresses in the body. To explain this term,
we resort to the conventional model developed in the classical elas-
ticity thpory. Let du be an elementary area through a point P
in the body and let n (P) be the normal to du at P. We shall as-
sume that do is oriented and the normalis also provided with an orien-
tatioll (Fig. 5.2). In some neighbourhood of P the element cro di-
vides thE; body into two parts: one is located on the positive side of
do and the other on the negative side. The existence of stresses in an
(elastic) body suggests that there exists a force F exerted by the mat-
ter on one side of the element cro upon the matter of the other side:
this force is applied at the point P. To describe stresses in a body
usually meaDS to.find the force (due to a stress) exerted on an arbitrary
oriented infmitesimal element (the shape of the element may b&
arbitrary). It is assumed that for a given normal n (P) the forc&
F acting on do is proportional to its area which is also denoted by
du. Thus, we obtain a correspondence between nand F: with each
normal n (P) at P there is associated a vector F (n), The character
of this dependence is established in classical elasticity theory pro-

304 5. 'l'ensor A nalysis and Riemannian Geometry
ceeding from mechanical considerations. We shall not go into deails.
but only report the final result: it appears (in a good approximation
to the real situation) that the dependence F (n) may be considered
as linear, i.e. at each point P there is defined a linear operator
n._Q{n): Then.F(n) can be written as F(n) = Qp (n) da, where
F' = Q}nJda; {n'} are the coordinates of n (P). Thus. we obtain a
tensor field of type (1, 1): at each point P there exists a Hnear opera-
tor Qp smoothly dependent on the point. This field defines th", stress
f'jgure 5.2
tensor. Stresses depend on the deformation of a medium, and the
c;haracter of this dependenc.e wiH be discussed below.
It foHows from mechanical considerations (which are omitted here)
that the stress tensor Qp must be symmetric (i.e. the matrix of the
-operator Qp is symmetric). The stress tensor cannot only be defined
(and calculated) in a deformed elastic body. For example, this tensor
does exist in an ideal1iquid, It is assumed that in such a liquid there
are no viscous forces, so that the force due to a stress F (n) exert-
ed on du can only be directed normal to da, i.e. it coincides with
the force of normal pressure on da. Since F (n) = Qp(n) du, the
<lperator Qp is diagonal, i.e.
Q; = q (P) Bj; pi = Q1n; da= q (P) B;n; da = q (P) R i (la,
pi =q (P) n i (P) 11<1,
where q (P) is a scalar function called pressure at the point P. Hence,
the pressure q (P) does not depend on the direction n, but depends
<lnly on the point P (the pressure may, of course, change from point
to point). If in a medium Qp=q(P).E, such a meium is said. to
obey Pascal's law. This law does not hold in every medium; for
example, in a viscous liquid the stress tensor is of a more compli-
c;ated form, since there exist, besides normal pressure, forces due
to friction.
There is a relation between the strain and stress tensors; an anaiy-
sis of this relation is one of the most important topics of elasticity

5.2. Simpl. Tense. Fi.lds
305
theory. In tho first approximation, the stress tensor depends linear-
ly on the strain tensor for small deformations, i.e. the following rela-
tion holds true: Q = a}I'IJh where the functions a}1 form a ten
sor of rank 4 (of type (3. 1». In R3 this tensor has 81 components.
The linear relation between the stress and strain tensors is known as
Hooke's law. Suppose the medium is homogeneous and isotropic,
i,e. the form of the dependence Q = Q (TJ), Q = ex}kITJ1 remains
unchanged under orthogonal transformations in R3. that is, the
tensor ex}kl is invariant relative to SO (3). It can be demonstrated
(the proof is omitted) that this leads to the following form of the
dependence Q = Q ('1]), Q} = QIJ = 1.lTlu + i.. spur 1)' Blj. where
f.I. and i.. are the so-called Lame constants (they depend on the prop-
erties of the medium). The function 1:'1]/1 = spur 1] has a simple
meaning (here we do not distinguish between the upper and lower
indices, for we deal with R3). Consider the displacement of particles
in a medium under small deformations; the functions ut, which de-
scribe this displacement. may be looked upon as the components of
the vector field defining the strain (this field is denoted by v (P».
'" ( aui aul ) '" aul
Then spur 1J = LJ azr + a;r = 2 LJ hi = 2 div v, i.e. spur 1) is
i I
the divergence of the displacement field. For liquids, f.I. is called
the viscosity coeffichmt.
We now turn to another class of tensor fields: the fields which can
be constructed from fixed ftelds via algebraic operations thereby ex-
tending the given set of tensors.
5,2,2. ALGEBRAIC OPERATIONS OVER TENSORS
(1) Given two tensor fields of the same type and rank,
T:"::and P::..;; then we can construct II new tensor field, defining
. f C I....lp T I,...lp P I) .Ip U h d fi .
It as a set of unctions ;... iq = j,...i q + h ..J q ' sing tee lll-
tion of tensor, one can easily prove that C;::':j: is also a tensor
(verify!).
(2) If T:::::: is II tensor field and f (x) is a smooth function on
Mil, then the set f(x).T}:.:: is also II tensor field on M n , The
proof is straightforward.
(3) Permutation of indices of the same type. Given a tensor field
Tio.. .iq (for simplicity, we consider the field of type (0, q». Let us
construct II new field according to the formula PI,...!.. ,i"...iq =
T!,.. !".. i.... i q ' i.e. the operation is reduced to the permutation
(renumbering) of the tensor components. Evidently, this operation is

306 5 Tensor A naillsis and R I,mannlan G.ometry
a tensor one (verify!). This permutation interchanges two indices,
i. and ia.. Apparently, we might consider any other permutation.
Remark, Permutation of indices of different type (i.e. covariant
and contravariant) is not generally a tensor operation, because while
the upper indices are transformed by the matrix J. the lower indices
by the matrix (J -1)T (see above), and these matrices are distinct.
Example, Let us consider a field of type (1,1), i.e. the field of oper-
ators C in T p (Mn); suppose the identity q = C{ is valid in a co-
ordinate system (x), i.e, C is "symmetric", Here we have interchanged
the upper and lower indices. In terms of the matrix C, this condition
reads: C = CT, Suppose the same relation Cr = C{: is valid in any
other coordinate system. i.e. C' = (C')7'. Let A he the Jacobi ma-
trix of the transformation (x) --+ (x'); then C' = ACA -1. ACA _1 =
(ACA -I)T, (ATA) C = C (ATA), i.e. BC = CB, where B =
ATA. Thus, our condition is equivalent to the condition that the
matrices Band C are commutative, which is not always the case,
If the transformation (x) _ (x') had an orthogonal Jacobi matrix
A, 1. e, AT A = E. the permutation of indices i and j would he a ten-
sor operation.
(4) Contraction. Let Tt:p be a tensor field, Fix two indices
of different type, covariant J. and contravariant ia.. and construct
th f II . f . P II...i"_lil1.+l...lp "" T l1 ... 1 11.-1' 1..",1,1"+1...1,,
e 0 owmg unctIons: 11...i._ 1 i ,..,"';q = 4 i,. i,_,. i.",i. i...,.. i q '
This is a tensor operation (ve1'ifyl). it tranfol'ms a tensor of type
(p,q) into a tensor of type (p-1, '1-1). If p=q, the inilial
field Tt:p admits complete contractionj this means that all the
npper indie8 are contracted with all the lowE'r indices to give
a scalar function spur T which is an invariant of T, i.e. it doE's
not change under coordinate transformations.
(5) Tensor product. Given two tenSor fields of the general fonn,
T is. .il' d p ..,....I1.. W d C I,. 11'11., "a..
it .iq an 81...8 1 ' e can construct a new fie] : it ..iqll....II, =
T l1 ..ip P a,...Ct. 0 . I h . ld ' f ( )
i,...i . 11,...11,' bVIOUS y, t IS tensor fie IS 0 type p+s, q+l ,
it is qdenoted by C=T(i!}P. Tensor product is not, in general,
commutative: r (i!) P=I= P (i!) T.
(6) Raising and lowering oj indices. Consider a tensor field au of
type (0, 2); let this field be non-degenerate, i.e. the matrix A =
(au) is non-singular at each point. Then there exists the invere
matrix A _I; its coefficients are denoted by a Ii. Thus, aiahJ= 6j.
Given an arbitrary tensor field T':::, we can construct new fields
p'....ip. _ a . TIa...p and CI,.:.lp = a"hTt.:.ip.
11.i!''''q - "" }I"" J q '2 "'q "12.. J q '

5.2 Simple Tenlor Field.
307
The former 0pHation is called index lowertng and the lattet index
raising. These 3re tensor operations, for thoy are a combination
of two tensor operations: n1ulUplication and contraction of tensors.
Since the tensor field aiJ is non-degenerate, the operations of rais-
ing and lowl'ring of indices are invetsa. Indeed,
C';,.. ;1' Til..i{' &' T'lOo.i p T I , ..fp
o.ho. C j2 "iq = aha. aa . .i.. ']q = i, aJ._.i q = },i, .iq'
As al} one generally takes the metric tensor gl} on a Riemannian
manifold. For example, if M n = R n and glJ=6i}o the transformation
Jaw is the same both for upper and lower indices (in the system
where gli = &/i)' Thus, in a Cartesian system there is no difference
between upper and lower indices, and they can be raised and low-
ered arbitrarily; for instance, all the indices may be regarded as
lower. The operations of raising and lowering of indices permit, in
general, canonical identification of T pM n and Tf,M". Indeed, the
elements of T pM n are represented by vectors (tensors of type (1, 0)
and the elements of TM" by covectors (tensors of type (0, 1».
Let us construct linear mappings
A: T -- r"', A(a)=t,
whl're aE T= TpM". sE r"'= rf,M n , 61 =lJ/",aa,
B: T'" -- T, B(T)=b, bl=gio.T)a.'
Then,
«B 0 A) a)i = gio. (Aa)" = gio.Co.Jai  l\}a J = 0.1,
i.e..
BoA: T-+ T, BaA=1T
(identity mapping), Similarly,
«A 0 B) S)/=g,,,, <m)""= glo.r%= &{S/=S/,
i.e. A ,B: T'" ..... r*, AB = iT.' This means that A and B are iso-
morphisms. Furthermore, this identification of T and r'" is invariant
nnder coordinate transformation because only tensor operations were
used, In particular, the following two diagrams are commutative:
r. -- r.
t (J,)T t
A I I A
T-->-T
J
T. -- r-
I (J-'IT I
B t  s
T--T
J
(7) Symmetrization. Let us first consider a pair of indices of
t'be same type and, using the tensor field T, , --1..-' construct.

308 5 Tensor A nalysl. arnl Riemannian Geometry
a new field P., I.. J ,= T. J ./.. (by interchanging these indices).
We define general symmetrization as follows: P/1...i q = T(h. .J q ) =
:1 l] T a<J....i q ), where summation (\xtends over all permutations of
(a)
thf indices il' . . " iq. Symmetrization is a tensor operation (veri-
fy.) .
Definition. A field Tilt:::; is called symmetric if it does not
change under the permutation ,of any two indices of the same
type.
Symmetrization of a symmetric field resulls in the same field,
Example: the motric tensor glJ is symmetric,
(8) Alternatton, Let us consider a pair of neighbouring indices
of the same type and, usinr the tensor field T...l}. construct
a new field p. 0... =.} (- T...J! . + T.../J...). Alternation is defined
as follows;
p. i -T i i } =....!- " ( _1 ) 'I'(Cl)T ( ' ' )
J, .. q - I... q ql L.J " J1... J q .
(a)
Here tp (0') is the parity of the permutation (J (the notation (_1)°
is sometimes used). Alternation is a tensor operation (verify!).
Definition. A tensor field T:':) is caUed skew-symmetric if its
components change the sign under te transposition of any two neigh-
bouring indices of the same type.
Lemma 1. A lternation does not r:hange a skew-symmetric tensor
TII' ..I (for simplicity, we consider tensors with covariunt indices).
A lternation makes a symmetric tensor /Janish.
The prool directly follows from the definition of a symmetric and
a skew-symmetric tensor.
Here we end the discussion of the main algebraic operations over
tensors and turn to the concept of a symmetric and a skew-symmetric
operator. The example presented above shows that the attempt at
defining, say,  symmetric operator C = (cj) by the formula C = CT
(which must hold in aU coordinate systems) fails, lor this definition
is not a tensor one, and the relation C = CT becomes invalid under
coordinate transformations, The concept of symmetry and skew sym-
metry of an operator is valid if only the Riemannian metric exists
on the manifold (or, in a more general form, if there exists a non-
degenate tensor field of type (0, 2) or (2, 0». Let us consider M"
with the metric CIJ which generates in each T p M" the scalar product
(a, b)p = glJ (P)aib i , s, bE Tp M".

5.2. Simpl ren$or FItd$
309
Definition. An operator l' (at one point P) or an operator fteld of
type (1, 1) is called symmetric (or skew-symmetric) if the identity
(Ta, b) """ (a, Tb) holds for any a, bET pM" at this point (or at
all points for the operator Held); in the skew-symmetric e,ase the cor-
responding identity is (Ta, b) E; -(a, Tb), a, b E 1'1' A1".
Explanation (say, Cor a symmetric operator). (a, Tb) => (Ta, b>,
gi)a! (Tb)i = giJTbk = (gl}rn aib h = Tikaib h , where Tilt = gIJT(
Furthermore, (Ta, b) = gl} (1'a)ib j = gl}TkaAb i = gJATla i b " "",
TAla'bh, where TAl = g'AT{. Since TIA(lib h eE TAlaib A , we have
T IA = T hi' Thus, after index lowering in a tensor T of type (1, 1)
the symmetry condition (imposed on a tensor of type (0, 2» takes the
usual form, i.e. riA does 1I0t change under index transposition. That
riA is skew-symmetric after index lowering (in the skew-symmetric
case) can be demonstrated in a similar way.
5,2.3, SKEW.SYMMETRIC TENSORS
Here we shall consider covarian t skew-symmetric ten-
sors l' i,... (h (or tensor fields).
Lemma 2. A skew-symmetric tensor T;, .' n of maximal rank n on
Mn is completely defined by only one component T 12. . . n (the so-called
principal component); the other components differ from the principal
component by the factor (-1)", i.e.
Til...in = (-1)"1' 12 n (i.. . .., i,,) = a (1,2, . . ., n).
Proof. Let us fix a coordinate system Xl, . . ., x", then in this
system 1'i, in = (-1)01'12... n (see the definition of a skew-symmet-
ric tensor). After the transformation (x) __ (x') we have
o:z:i, azJn
l' -1'. .
I' n' =----p- . .. iJ ¥n' 'I" ' n
", {Jz ¥
. II .}n
- - { )' ( - 1) " 1" ". )T (d J) T
£J a:r' {;:rn' 12 n = el . It..n'
o
Thus, to describe II skew-symmetric tensor of maximal rank it is
sufficient to know its principal component (in one coordinate sys-
tem because the principal components in other coordinate systems
are obtained via multiplication by the transformation Jacobian).
The lemma is vroved.
Skew-symmetric tensors admit an important operation called
exterior multiplication.. Let TI" ..ih and P J, ..iq be two skew-symmetric

310 5. Tensor A nalllsl, and Riemannian Geomelrll
tensors. Define a new skew-symmetric tensor (denoted as R =
TAP) by the relation
R i1 . tkh' .J k = T[il...i Ph. iql = Itqr  (  (.:... 1)" Ta(l,. .1Ph...jq) ) .
a 0
This multiplication is a bilinear operation; the ranks of the tensors
are added. Let us interpret the operation in terms of exterior differ-
ential forms. First consider a skew-symmetric tensor T;, . i k given
at one point on M". Then TI,.. .Ik defines a skew-symmetric multiJinear
mapping T: (T.x .., X T.)(k times) -+oR, where T. = Tl'M".
Fix a coordinate system Xl, . . .. x" and consider these coordinates
as smooth functions in a neighbourhood of a point P. Then the
differentials tJx1, . . ., dr' of these functions are valid. The differ-
ential dj of any smooth function j (x) is an element of the space ad--
joint to T., i.e. a linear functional on T.. Indeed, if a E T., then
.2L = df (,\, (I)) I
da dl I=n'
where l' (0) = P,  (0) = a, i.e.
df of d:r k I of h h df at k
'd8 = a;:ii""""di" 1=0 =  a , T us, dj" = lizk a,
. ' f t I h dzl liz! I< .,i I< I..>JL I< d
I.e. I =X. t en d1= Ii#. a =Uka --=a, '=a t,
Hence, d:I;1< may be looked upon as functionals on T., i.e. ele-
ments of T.. Assuming dt small, we may consider it as a proportion-
ality coefficient, which is omitted below, If e., . . ., en E T. is a .
basis in T.. then for basis in T* we take dx 1 , . . ., dx" assuming
that d:r!' (e,,) = 6:, i.e. d:r!' (a) = aI<, Let A (d:I;t, ' . '. dx") be
an exterior algebra with the generators dx 1 , ' . ., dx" satisfying the
relations dx l A d:J + dx! A dx' = 0, i =P j, The operation A is bi-
linear, and the algebra A is generated (in the additive sense) by the
monomials d:I;l, . . , d:I;"
dx.' /). dx i , i < j; , . .; dXll A ... A dxi/!,
i\<...<il<' dx'A... Adx n ,
As is known from algebra, these monomials do not obey linear rela-
tions with constant coefficients; the multiplicative generators of the
algebra A (d:I;I. . . .. dxA) are represented by d:I;1, . . '. d:I;h, i.e.
by the basis in T., The algebra A can be decomposed into the direct
n
sum of linear subspaces AI<: A = $ AI<, where AR, 1  k n, arc
h=O
generated by the monomials d:I;!1 A . . , A d:I;il<, i l < , . . < i h ;

5.2. SImple Tensor Fields
311
arId AU  R is generated by nnily a]t'mcnt 1. The climension of lhe
algebra A is equal to 2".
Let us consider a nuw algebra A (Mn) whose e]amenls are lincar
combinAtions 00<") = Ti, I" dXi, 1\ .. . 1\ dx'" and all possible linear
n
combinations L 00("), where Ti, ..i" (x) is II skew-s'm11lelric tensor
"",0
field of rank k, and the indices it... i" are arranged in increasing
order. Multiplication in A (M") is defined helow,
We have defmed the element CI)(") in a given coordinale system
Xl, . . ., xn, O k:;;;;' n. What does it happen to this objerl under
the transformation (x) __ (x')?
Lemma 3, The elements 0)(") = T (,.. i" (x) dxi, 1\ . . . !\ dXi" al'e
defined inv(trtanlly, i.e. under the transformalion (x) _ (x') lI'e
have
T, ,dx'; /\ . .. !\ dX(h  Ti , . i. dXil !\. . !\ dXi".
1,,,.1,, n
Prool. It is sufficient to make the substitution (x) __ (x') and IIse
lhe transformation laws for T".. .i" (x) and d.7;1.
Definition, The elements 00<"1 of the algelJra I\. (M") are called
exterior differential forms.
The algebra I\. (M") is infinite-dimensional. Each exterior form
uniquely defines the co!lection of components TI,.. .f" of a skew-sym-
metric tensor; the converse is also true: any skew-symmetrir: tensOl'
(a tensor Held, to be more precise) uniquely defines an exterior form.
The apparatus of exterior rorms is one of the apparatuses to descrihe
skew-symmetric fields,
Remark. While defining an exterior rorm, we might consider lin-
ear combinations AI,. 'i dx i , !\ ' . , /\ dx i ", where AI.. ..fA is an
arbitrary tensor field (not necessarily skew-symmetric) and summa-
lion extends over all (not ordered) colleclions i l . ' , " i h . Since the
generators dx ' , . . .. dx" are skew-symmetric with respect to exte-
rior multiplication, we have
/I i, ..i" dXi, 1\ . ., !\ dx!" = AIi' ..ihl d:t"!\ .., !\ dXI1 i l < ." <ik'
i.e. any combination Ai,...." dxll !\ .. . !\ dx'k genE-rates an exle-
rior difierenlial form.
Let us derIDe multiplication in A (M"). If ,,)(k) = T 1 \. .1" dril !\ '"
.,. !\ dxj (i. < ... < i,,) anrl 00(')= Ph. J.dx il !\ ...1\ dx's
(i l < ,., <j.) are exterior forms, their product w("")=b)(h)Au)(') is

312 5. Tmsor A ..alysis a.d Riema....lan Geomdry
defined as the form
C!)(R.')= (T il .. i A dxil /\ ... 1\ dx l ,.) 1\ (Ph. J. dx}, 1\ ... /\ d.:!/')
= T;,...ikPh...J. dXI, 1\ ... II dxi,. /\ dxi, 1\ .. I\dx}'
(t l < . .. < iR) (j 1 < .. . < i.)
= Tp,...ihP;'...i,) dX11 /\ ., . /\ dtih+'.
(l, <... <i h +.).
Apparently, multiplication of forms coincides with exterior multi-
plication of the corresponding tensor fields. Thus, A (Mil) is provid-
ed with the structure of an algebra with unity element, and multi-
plication is associative (because multiplication in the algebra
A (dx l , ... dx") is associative), but not commutative. The al-
gebra A (Mn) is decomposed into the direct sum of linear subspaces
A(k) (Mn) = {Th...ih dxl ,/\... /\ dxih}. The number k is caJled
the degree of the form W(R). The elements of WIll) are called homoge-
neous elements of the algebra A (Mn). It is convenient to interpret
differential forms W(h) as multilinear skew-symmetric mappings.
If a 1 . . . ., ah E T.M". then, assuming dx i (a) = a i (the ith coordi-
nate), i.e. omitting the factor dt, we obtain
W(R) (ai' . . ., ah)
=(T Il ''' lh (x) dXI, /\ . .. /\ dx;h) (a" ., '. aA)
= T ll ...,,. (x) dx[il (a l ), ... ' dx''') (a A )
=T f ( ) Ii, I,.]
1...i A X 41 ...,' a h
T   T  
=- [I, ..1,.Ja1 ... II,. = it...I,.1I1 .,. a k .
Thus
W IR ): T " X,.. X T" -.. R,
h
w(l» (8 0 (1. 02 . . . ah» = (- f)" CJ)(k) (ai' . . ., all)'
Let x', . . " x" be Cartesian coordinates in R n . Let also ro<") =
dx' II. . . /\ dx" be an exterior form of maximal rank and let
a" . . ., an E R" be an arbitrary collection of vector arguments
in RI!. We write vol n (ai' . . ., an) for the n-dimensional volume
of the parallelepiped n (ai' . . ., an) spanned by ai' . . " an (see
Fig. 5.3).

5 2. Simple Tensor f'lolds
313
( a ... a )
Lemma 4. Let A = ."...n be a matrix constructed of the coor-
an '" on
dinates of ai' . ,., an' Then dIJt A = vol n (al> ..., an)'
Proof. For n = 1 the statement is evident. Suppose the formula is.
proved for k 1/. - 1. Since vol n (aI' . . ., an) remains unchanged
under rotaIons. the vectors al' . , ., a n -l can be arranged so as to.
n(."... In'
Figure 5,3
lie in the plane spanned by ell . . ,. e"_I' where e1' . , '. en is It
basis in R n . Then the matrix A takes the form
A'
H' I .
o . . . . . ,0 I ann
Hence.
det A = det A' = (det B'), (a')
= (vol II (a 1 . . . "a"_t»' (IL') = vol n (a l . . . .. an)
(see Fig, 5,4). Here (a') is the projection of an onto en' The lemma
is proved.
Lemma 5, Let R n be referred to Cartesian coordimdes xl, . . .. x"
and let win) = dzl f\ . . , f\ dx" be a form of maximal rank, TM1/.
win) (a1. . . " an) = vol n al' . . ., an)'
Proof, Since dzl (a) = a I we have
w(") (al' ' ,.. an) = (dx l f\ .. , f\ dx") (a l . .. '. an)
= ck!l (al)' ,., . dx"] (a,,)
= at i " '" . a"] .... det A.
where A = ( a :.: . a ) . i.e. det A = vol n (all" ., an)' The lemma
an '" an
is proved,

31-1 5. Tel1$or A l1aIY$/' and Riemannian Geometry
Thus, the value of the standard form dx l A . . . A dx" on any col-
lection 810 . , ., an is equal to the Euclidean volume of the paralle-
lepiped spanned by ai' . . .. an' In particular, the volume of a pa-
rallelepiped (in such an interpretation) is not a scalar; for instance,
it changes the sign under an odd permutation of 8], ' . '. an (it is
an "oriented volume"). Thus, we have represented (1)= dx] A . , . Adz"
as an n-linear functional of a]. . . .. an' This interpretation of vol-
ume as the value of an £xt\!rior form (of the arguments ai, ..., an)
Figure 5.4
is rather fruitful; it leads to many interesting geometric facts, For
.example. such an interpretation implies the invariance of a multi.
pIe integral under a change of variables.
A similar interpretation is rossible. with forms of a lower degree,
Let, for example. a form (1)( ) = dz'l A .., A dxl" be given in R....
Figure 5,5
It is required to find the value of (dx lJ A... A dzi,,) (8 1 , ' . ., ah).
where a 1 . . . ., ah is an arbitrary set of vectors in R". Exercise:
prove that (I)(h) (al' . , ., ah) = vol n (b 1 . . . ., b,,), where
vol n (b I , ..., b h ) is tbe volume of the k-dimensional parallelepiped
located in the coordinate plane Rh (el\. . . ., 9/,,) and spanned by
the vectors bit ' , '. b" E R" wbich are orthogonal projections of
the vectors al' ' , " a" on the plane Rh (el" . . .. e,.) (see Fig.
5.5,),

5.2. Simple Tensor Fields
315
H the form W(A) is the combination wI. .ikd:(;'t f\. . . A dxlk,
its value on ai' ' , ., 8A will be the combination of the volumes
vol n (b l , . . '. b k ) with weights WS,..,lk'
There is a relation between exterior form pi maximal degree and
the volume of a bounded domain on a Rie8nnian manifold. The
reader is supposed to be familiar with the definition of a multidi-
mensional (multiple) Riemann integral. Let M" be a smooth
Riemannian manifold and let D be an open domain in M" for which
there exists a diffeomorphism onto an open bounded domain U in R n
reCerred to Cartesian coordinates Xl. . , " x", Of course. this diffeo-
morphism exists not for every domain D on Mn. but for simplicity
we shall confine ourselves to "sufficiently small" domains. Let el} (x)
be a metric and let g (x) be the determinant of the matrix (CI) (X».
Definition. The volume oj a domain D c:: M" is the number
V(D)=voID= r... rY C(x) dx l f\... Adx n .
J U(,,} J
where Xl, . . ., x'" are Cartesian coordinates in the domain U (x) c:
R n (see Fig. 5.6).
x.
Je'
FIgure 5,6
This definitton needs substantiation, Le. it is required to demon-
strate that in simple cases this formula gives the same values of the
volume as those obtained from other considerations,
(1) Let Mn = R n and let U = D c: R" be a bounded domain in
R", gl/= fJ lJ , Le, g (x) = 1, so that
vol D= I ... r dxl ,.. dx.,
J U(:<) ,)
which coincides with the customary "Euclidean" definition of the
volume of a bounded domain in R n . It is assumed here thatthesym-
bol <k l . ' , '. dx n implies, in addition to its formal meaning (see

316 5. T ""or A nalllsis and R temanlliall Geometry
the definition of the Riemann integral), also the following: if dx l . . . .
. . ., dx" arc considered as infinitesimal quantities, du = dx l . ..dx"
is the infinitesimal volume of an infinitesimal parallelepiped with
edges, d:r;1 ..., d:r;" measured in Cartesian coordina tes Xl, . . ., x n .
Thus, the volume of a bounded domain U (x) c: R" can be represen-
ted as the sum of infinitely many volumes of infmitesimal parallele-
pipeds calculated by the ordinary formula (the volume of a rectangu-
lar paraJlelepiped is equal to the product of aU its edges emerging
from one vertex).
(2) Let Mn be a smooth submanifold in RN and let giJ (x) be the
induced Riemannian metric on Mit c: RN (the ambient metric is
Euclidean). Also, let Xl, . . ., x n be curvilinear coordinates in the
neighbourhood ola pointP onM" and let adomainD lie in this neigh-
bourhood. Then we may assume that the volume of the domain D,
vol D, is the sum of infinitely many volumes of infinitesimal paralle-
lepipeds (which are no longer rectangular) {TIh} obtained when D
is subdivided by the coordinate planes xi = const (with "infillite-
I";;I"
simal spacing") (see Fig. 5.7). Let us consider an infinitesimal "cur-
Figure 5,7
viii near" parallelepiped ilk' Since Mn is a smooth submanifold in
RN, we may assume that nIt is well approximated by the "linear"
parallelepiped n" contained in T pM", where P is a vertex or n h
Figure 5.8
(se Fig. 5.8). The sides of n h are directed along the coordinate lines
{x'} through P. Let a1' . ", an be the velocity vectors of these lines

5.2. Simple Ten.or Field.
317
(i,e. 81 is the velocity vector of the line along which only the co-
ordinate xi changes, while all the others are fixed). Since III is in-
duced by the Euclidean metric, vol 11 11 = vol fill coincides with tho
Euclidean (n-dimensional) volume of nil imbedded in R". Thus.
vol D   vol nil. Let at P there be given, in addition to curvilinear
(j)
coordinates Xl, . . "' xn, orthogonal coordinates yl, . . .. y"; this
means that we define Cartesian coordinates in T pMn, Let 01' . . ..
. . '. en be unit velocity vectors of these new coordInate lines at the
point P. Then in the coordinates yl, . . '. yn we have g'l (y) =
OJ/ at P (generally. only at one point). Let (x) -+ (y) be a coordi-
nate transformation and J the Jacobi matrix at P. The matrix J.
n;
Figure 5.9
operating in TpMn. sends e 1 , . . .. en into 810 ' . ., an' We now
prove that vol fill = V g (x) da", where da" = d.:r;1. . . " dx"
is the Euclidean volume of a rectangular parallelepiped with sides
dXI, ' . ., d.:r;". Consider a parallelepiped 11" spanned by ai' . , " an'
Then vol 11 (al' ' . ., an) = V g (x), where vol denotes the Euclid-
ean volume, Indeed, since in the coordinates yl, ..., y" the ten-
sor gll is of the form glJ (y) = BIJ, we have (gll (x» = J (gl} (y»JT =
JEJT = liT, i.e, I (x) = det (CI/ (x» == det (liT) == (det 1)2,
and hence det J = V g (x). Since J transforms the orthogonal coordi-
nate frame e l , . . " en into 8 1 , . , " an, we have, according to
Lemma 4, V g (x) = det J = voll1 k (8 1 , . . ., an)' The statement
is proved.
Since 11 1 is spanned by (dXI) a,. . .., (dx") 8" (see Fig. 5.9),
"01 IT" (a l . dx!, " , an ,dx")
= (val 11;' (ai' . . ., an»' dx!. . .. . dx" = 'V g (x) dx!. dx n .

318 5. Ten.sor A /Ia/gs', and Riemannian Geometrv
Because tho volume of the domn!n D consists of the volumes
ITh' we obtain vol D = r ... (' y g (x) dx l . . . dx". which is Wh011
J U(..) J
was required.
(3) Let M2 c: R3 be a smooth submanifold given by the radius
vector r = r (u, v), then by the definition of the area of a domain
D on M2 we have
vol D= }  V g(u, v) dudv, g=det(: )=EG-f'21
U(u, )
i.e.
volD=j JYFG-Fdudv=   Ilr u ' r.J Idudv,
U(u, )
where [ , ) stands for the vector product and j I , J I for its magni-
tude. Thus, the formula for the area of II domain coincides with one
of the formulas of classical analysis.
(4) Let a submanifold Mn-I c: R" be given as the graph of a
smooth function x" =1 (Xl, ' . ., X"-I). Find the volume of a bounded
X O
en
R"" 1(x1,.".;en 1)
Figure 5.10
domain Dc: Mn-I. We may assume that vol D = l; (do), where do
is the infmitesimal volume of an infinitesimal parallelepiped. Pro-
jecting this parallelepiped onto the hyperplane Rn- l (Xl, ' . ..
. . ., X"-I), we obtain the infinitesimal volume dl<> = dx l . . ,dx"
related to do by da =  = --.!.....-, dx l . . . dx n - l (see Fig, 5.1.0),
cos 0: coo 0:

5.2. Simple TeMor Field"
31
Here CL is the angle between the normal m to Mn- 1 and the vector
en Let us find cosa. Evidently, cas a = 1:i'.7, = (en. m), where
en =(0, ",,0, i),
grad F
m= ,gradFl '
and
F (x', . . ., x n ) = T"- 1 (.1" . . '. X"-I) ,
g Tad F (- /"" .. " - I"n-h 1),
.. / n-I
IgradFI= V 1+  (/,,1)2 ,
j1
1
(e", m)= 11 "-I '
1 +  (/ 1)2
iaS x
.. / ,,-I
da= V 1 +  (/,.1)2 dz l , ... . d.z"-I,
1=1
Thus
vol D = : .. 
U(.l, 'Ut ;I;'R-1)
.. / ,,-I
V 1 +  (/,,1)2 dx l .., dzn- I .
1=1
Does this lormula coincide with the general formula suggested
above for calculating the volume of a domain? Let us find the in-
duced metric on M"-1 in the explicit form. We have
ds2= (1 + (I,A! (dZ I )2+ 2f",lf=, dz l dx i ,
( 1+(/",)2. f"d"J )
(g/1) = ' . ,
f"d"J 1+ (/"..)2
n-I
g (x) = 1 +  t1 (verify I) , For example, if n= 3, then g (z) = 1 +
1=1
I + t;. Problem: calculate the area of a circle of radius r on
a Lobachevskian plane and on a two-dimensional SpheT('.
Thus, we have given saveral evidences for the general formula
vol D = r .,. (' y g (x) dx l . . ,d:r;n which p(\rmits the calculation
j U(,,) J
of the volume of a bounded dOfilain D on a Riemannian manifold;
we have assumed, however, that D is contained in a single chart,

320 5. Tnsor A nalysis and RIemannian Geometry
While deriving the general formula, we have proved that the
number vol D can be represented as
vol D= (' ... (' V g (x) . (dx ' A . . . A dx n ) (a f dt l , ..., an dt") ,
J U(,,) ,}
i.e.
(dx 1 A '" A dx n ) (aldt l ,..., andt n )
= d;l:(J(a\ dt l ) . . . dx"! (an dtn) = dtl!, " . dtin.l = dtl, ... . dt",
where al' . , " 8n are tangent velocity vectors to the coordinate
lines {x} at the point P and dt t , . . " dt" are infinitesimal displace-
ments from P along these lines; these displacements form the in-
finitesimal parallelepiped n... (a 1 dt l , . . .. andt"). The formula for
the volume can. therefore, be written as
vol D = (' '" (' y g (x (t» (dz l A . ,. Adz") (81 dt', . . ., a" dt") ,
.) Ul,,(I) J
or in a compact form
(' ...  y g (x) dx l A ... A d;l:".
J U(:o:) .
Apparently, the integral (' '"  V g (;1:) dx l A ... A dx" does
J (l(x) J
not depend on the choice of the local coordinate system. Indeed
for the transformation (;1:) -+ (y) we have
r . d (' Y g (y) dyl A ... A dyn
J U(J1) J
= I' ... (' {detJy).dyt A '" A dy"
J UM J
= I' ... (' (detJ.x).detl ) .dY'(;I:) A d' Ady"{;I:)
J U(u(x» J \ "II
= (' ... (' (detJ,,)det (  ) ,det (  ) .d;l:1 A.., Adx"
J U(v(,,))J "y z
=  ... (' det J" dx l 1\ .,. A dx n
U(J1(:r)) J
= (' ... r V g (z) dz l A ... A dx n .
J U(!I(:r)) J
It is sometimes convenient to consider the integral
(' ... \ V g (;1:) dx l A ... A dx"
J D .

;; ,2 S itnple ten..r Field.
3.1
as an "t<xtl!l'ior dHfarcntidl form' stich that its application to tht'
fmiJ of vectors at (x) dll, . . .' an (x) dt" ghes a number, thp volume
of domain D. Let us considel' Another structure related to skl'w-
symmetric tensors of ranI- 2.
Definition. A skew-symmetric scalar product is said to he defmed on
11[2" if there is given on 111 2 " an exterior 2-form 00(2) = Wi} (x) dx! f\
dx! whose matrix (WI! (x» is non-singular, i.e, det Wi!;/= O.
(Such 2-forms are called non-singular.)
The n<?n-singulariy of w/L impJies the existence of the inverse
field, Wi}, i.e. WjJW}h = 6. The form of maximal rank, Q, can be
associated with the form }(2). namely, Q = Q(2 n ) = W X , . .).: 0)
(n times).
Theorem 1. The skew-symmetric product W(2):W(2) (a, b) = o)/Jaib J
is non-singular at a polnt P E M2" it and only if the form Q(2 n ) does
not vanish at this point.
Proof. We proceed by analogy with the proof of the formula
dcr = Vi dx l f\ . . . 1\ dx n for the Riemannian metric. The proof
of Theorem 1 is thus reduced to verifying a similar formula for the
skew-symmetric scalar product. Since Q is the form of maximal rank,
in the coordinates Xl, . ., x" it becomes Q = { (x) dx l f\ . . .
. ., f\ dx", where { (oX) is a smooth function on M2". That Q does not
vanish at the point P means that f (x) ¥= O. If we demonstrate
that (f (x) = V det WI} , the theorem will be proved. Let us consider
the matrix (WI}) at P. In the neighbourhood of P we can choose new
corodinates yl, y" in which (wi! (y» (at the point P) takes the
form
/I 1
I}
-t f)
\ n 1-: i I
0
f}
n
We recall that any skew-symmetric matrix can be reduced to this
form (irrespective of whether it is singular or non-singular). If
(ool! (y» is non-singular, it does not have zero eigenvalue. LeL
(0)1j) be non-singular and let J be the Jacobi matrix of the transform-
ation (x)  (y), then A (x) = JA(y) JT, where A (x) = (wiJ (x»
alld A (y) = (oo!} (y», i e. £let A (x) = (det J)2, det A(y) = (det J)2

.12 5. Tt",., A naly.l. a,ld Rimannlalt GtOmlrv
'11111, hence, det J = 'V det A (x) = V det WiJ (x). In the coordinates
y', . . ., y" the form Q, at the point P, beeomes Q (y) =
dY'I\.../\ dy". Passing to Xl, " ., x", i.e. making the substitution
(x) --.(y), we obt ain Q (.7:) = d",t Jdx l /\... !\ dx", so that
f (,1:) = V det OO!J (3;), which is what was required. Thus, the state-
ment is proved for the non-singular matrix (00/), If det wI} = 0,
the matrix (wo) does have zero eigenyalues, i.e, at the point P
its canonical representation W = dx' 1\ dX"+1 does not contain
c,ertain variables (corresponding to zero eigenvalues). Hence, con-
idering the nth exterior degree of the form w, we find Q = O. The
theorem is proved.
For n = 2 we have (O() = OOI2dxl 1\ d:z:2 + W34 dz3 1\ dz',
Q(') = (t! /\ W =  (2g 12g34 d:r l 1\ dz 2 1\ dx 3 1\ d)
= (CI2g3) dz l /\ .. , 1\ d,
fZ (x) = det (OOIJ) = (gI2g3.)2. f (:r) = Cl2g" = V det (wlJ)'
5,3. CONNECTION
AND COVARIANT DIFFERENTIATION
5.3,(' THE DEFINITION AND PROPERTIES
01' AFFINE CONNECTION
Thus far, we were dealing only with algebraic operations
over tensors, while differentiation was not touched upon. And this
is no accident; now we shall try to explain the reasons for our cau-
tion.
The necessity of differentiation of a tensor field frequently
arises in applied problems (see, for example, the delinition
of the divergence of a vector Ih:ld). First of all it is ordinary
differentiation of the components of a tensor in curvilinear
coordinates, Let us consider, for simplicity, a covariant
field TII" t h . Fix a cUl'vilinear coordinate syslem Xl, .. ., x" and
construct a set of functions P. II " =  T ' " I f., i.e. P arlial deriv-
-, ...  dot'" ... 
,Itives (in the usual sense) of the components of a tensor ficlci.
Perform this operalion in each cOOl'dinate system. This means tJU\l
in cach system with the set of th(' components T 1 ;... II. we asso-
i!
ciate a new sel P ,. " j ' =- --:- T i , t " Do these sets form
ex. . II k i}za. I' It
n lensor lie}d P? III other words, arc the' transformed Hnder
rooldillato snbstilutioll like ll.'nsor? Let 1I verify Ihis fact b

5.9, Connection and Couarlant DIfferentiation
323
direct calculation. The substitution (:z:) -+ (:z:') leads to the trans-
formation
P ",. I;...I
=
ar. . II I,.
1,...1,. =  ( 3=-, .2=.-T. . )
lJza.' 8 z o.' a.2:1t .. 8z iA '1. ,JIL
a h a h b .
= -a (T I1 ... III ) "77  . . . _ I '
ax uZ az" az II
I
a ( azll  )
+ T 11 ''' 11I 8z'" oz'; ' " ozl,.
0%(1. az'l az'lI
= Pa: itu.iA -;-a:- --:;- . .' 
uZ 8z" iJz'I<
a ( iJzll 8z l11 )
+ Tit... I,. _ a ' --:-:-. . .  .
iJz iJz" iJz'lt
Thus,
u:p a%lt a I,.
P ..' .. = Pa'it ,,.____ -;-.., +8a"t' (T, x, x'),
a. I\... lit . '0' a:ra. iJz1t Dz'If. . I." II.
where
a ( "zl, ilzlJt )
8",':1;..,1;' = T I1 ... I ,. iJza' -:I;"'"""""'i' .
u:C iJz It
For an arbitrary substitution (x) -+ (:z:') the quantity 8a';I;...I is,
in general, non-zero, i.e. Pa;ll..l,. life not. transformed according to
the tensor law. Hence. ordinary (non-covariant) differentiation
&:1 is not a tensor operation: it does not turn a tensor field into
another tnsor field, The expression for Sa';I;...I also shows what
assumptions are needed for the operation a1 to be a tensor one:
iJ' is a Lensor operation only for a linear subsUtution (x) - (x');
in this case 8a'I;...i;.  O. To bring further discrediU (from the
tensor standpoint) on the operation a:" we shall consider a vec
tor field TI referred to a curvilinear coofdinat system (x) and cal-
cilIate the divergence of this feld by the formula divT=

324 5, Tmsor A /lalysis and Riemannian G.om.try
2} :: which is an analogue of the definition of divergence in
i
a Euclidean coordinate system. Is this expression a scalar, i.e. is
it invariant under an arbitrary substitution (x) -+ (X')? Lel us
verify the equality div T(x) == div T(z.), whero (x) I\nd (x') are arbit-
rary coordinate systems. We have
.  aT" iJ ( azi' t )
dlv T ix .) = lJ ,az" == o;r- a;I T
I
oT/ azk iJzl' Ti azl> as,,:"
= '{J;it '"iW'  + '"iW' arl> az l
=  : +R(T, x, x')=divTtE)+R(T, x, j,'),
i
where
, ! azl> a''':i'
R (T, x, x) = T azl" azl> azl .
For an arbitrary coordinate substitution R (T, x, x') does not van-
ish, i.e. the definition of divergence given above is not invariant.
The explicit relation for R (T I x, x') shows that R = O. for example,
under linear substitutions. Thus, under non-linear substitutions iJ: i
is not a tensor operation.
Our aim is to learn how to differentiate invariantly tensor fields
in curvilinear coordinates. This is necessary if only because local
coordinates on a smooth manifold are "almost always" curvilinear.
Thus, it is required to find an operation (denoted by V, nabla) hav-
ing the following properties (we assume for the time being that
Mil = R"):
(1) in Cartesian coordinates in Rn the operation V must coin-
cide with ordinary differentiation { a:/ },
(2) the operation V must be a tensor one, i.e. if T is a tensor field
on R" (referred to curvilinear coordinates), VT is also a tensor field.
We begin with examples. Let us consider in Rn a vector field TI;
let (x) be a Cartesian coordinate system and (x') an arbitrary
system. Let us write out conditions (1) and (2) imposed on V.
. aTI
In the system (x) we have (VT)j = dzj ' and in the s'stem (x')
" az i ' iJ:z) .
we have ( "i7T»)'=- iJ I -."... (VT»), The problem is tlJcrcfore to end
z iJzl

/;3. Conneclion and Covariant Differentiation
325
the explicit form of V and calculate the components of (VT)r in an
arbitrary coordinate system. Straightforward calculation yields
( VT ) i: = az l ' az  (  TA' )
} ilzi IJz" a:r. J 8:r. A '
dz i ' 8zi ozt iJTA' azo.'
=8X1 azi" az A ' azo.' -;;;J
-I- az i ' ii"' Tk' ....!- ( ozi, )
o:r f az' iizi {Jzk
= /ji',6'  + TA' az i ' 02z1
A , a:r.o.' uz l iJ:ci' azk' ,
",?T " ;' /' i)zt' 02 z 1
('VT)}, = ,)..i' -I- TAT7'k" fj'A' = azr' 1J:r.J'iJz k '
Thus, wo have obtained the functions f}:A' which measure the devia-
tion of 'V from ordinary (Euclidean) differentiation, i.e. we have
learned how 'V acts on a vector field,
Lel us now consider a covector field TI in nn. To find the explicit
form of 'V on TI it is required, as before, to solve the system of equa-
lions
( T) aTI .
'V iJ=- iJ '
ZJ
az l iJzi
('V T);. i' = -.,.,.. --:';"' ('V T) iJ'
az' 8 z 1
We have
IIxl iizJ 0 ( a:r./t' )
('VT)!.), =----".  _ a ITA'
oZ' ax' az ' z
-= IJ:r.,', 'Jz. az A ' aT A: 'Jz' -I- az, az T A ,  ( !f... )
ux' az ' liz' axa. 8:r/ ax' iJzI 8z' az l
k' a.' ,)T A' 82 Z k' IJz' azi 8T I' -A'
=- 5 i , 5 j , ---:' -I- Tk' -,-.......,-. -..,.,- = = ---,. -I- Th,ri'j',
iJ..o. az J az' iJz' a:r) ax}
- k' ,J'zk' 8zf ,Jzi aT i' -It.
where f;,i'= ------t.--,.,-.--,,-. Thus, ('VT)I'i'=- a ..-I-TA,ri'J',
az1 iJz o:r;' ozJ :<1
where r:J' arc the functions which measure the deviation of 'V
f rom ore! inary d H!erentiatioll on d covector field.
Lemma 1. The equality :i' = -r.> is valid,

326 5. Tem.r II. naty.l, Ilnd Riemannilln Geomelrl/
axi' i1zY' ",
Proof, The identity --"'-V = 61. is obvious. Differenliaticon
az 8z
with respect to xl" yields
81 z l0 IJz ,. 8z'- asz" az"-
.-+-, ,-.,.,.=0,
8z'" iJz'" az'" a:!<' iJz""az'O IJzP
i.e. r,, + r'k' = O. The lemma is proved.
Thus, the action of 'iT on vector and covector fields (in cllrv!.
Jinear coordinates in R") is of the form
"aT" i'
('iTT)j. = a:rJ' + TkT'i'l
8T i . k'
('iTT)I'i' =---,,-.- Tk.fi'J"
az 1
Let us now consider the action of 'iT on operator fields, i.e. on
fields of type (1, 1). We have
('iTT)} = iJ:k (T]) ,
( 'iTT ) i: .=i- ( : )
J  az' a"r az' azk aza.' azl 1>
IJzi' iJzJ 8z/J azl az 1l ' a: az" ,
= -a;r- azJ' a;;:7" iJza.' az J IJz'I' aT
'8zi' azJ az/4 aSz l {JzQ' a"p'
+ ,,' iJzl iJzJ' fiF' az q ' azo.' iJ,," iJz J
Ct' iJ:r: I ' Bz J Bri' IJzl aiz P '
+ T p' 87 iJz" fiF a",a.' 
a I' p' r l ' T " r '"
=----p- (TJ,)+T i , p.It'- p' J'k',
iJz
Thus, we have found how 'iT acls on T}.
Theorem 1. Let M n = R", let (x) be Cartesian coordinates, and Itt
(.:t') be arbitrary cur/Jilinear coordinates. Then there exists in R n a
tensor operation 'iT ckfined on an arbitrary tensor field T;: :::; by
the formula
It
( T ) 'i.. Ii.. _ a ( T i ;, ,.ii.. ) "r';' ,i=q'.. ,ihrl;
'iT JI' ..J;':'" - "'{j';? i\, ..i;' +.LJ i;.. .J;' q...'
.=1
.l!, T i;.. .ii.. r Q :
- o' " , z'i.cc.',:
'-=:It )1" .1..=Q .' "P

5.3. CQnnec/ion and CQvariant DiOerenlialioll
327
where the functions rj:q' are transformed under the subsWutlon
(x') __ (x") as
" 8z i ' iJzl' iJ"A' t' az l " aOz l '
rj"A" = Dz.' (,,,:1" DzA" rj'' + a"i' azl"iJ,,'"
Prool. 'The explicit expression for 'il on the field TI" A is ob"
'1 . '(1
tsined by making the above calculations for vector, covector, ant!
operator fields as many times as the rank of T J il'''A. As a use-
I' ..II'
ful exercise, we recommend the reader perform all the calculations.
Lel us examine the transformation law for rh, We have
j' 8T" , j'
'ih,T = a-l" + Tp r P'R',
. ar i ' 1''' I"
'ilh"T"=--;;:- +T r p '/<"
a;J:
_   ( azl' r" ) + az1" Tp'rl"
- a",I<" az A ' aT iJ"P' 1'"1<"
_ a:z!<' 8z l ' ar l ' T" az' alz" p' {I;J:P' I"
- D"A" ar i " a:z!<' + iJzl<' 8zA'a,,}" + T {)zP' rp'A;-
i" iJ:J:'c Dz(" i'
'ih,T =--;;:-''il",T
iJz a;J:
= o",A' ');J:'" ( arl' + TP'r': , )
DzA" (1z;' az'" P II; .
Comparison of these two equalities shows that
TP' ,)zll;' az;" r l', , = TP' fJzR' DO;J:" p' fJzP' ,/"
{Jzl<" a;J:" " II; fJzA' azA' a"P' + T az p ' 11"11;".
Since this identity must hE' satisfied [or any field T" we obtain
rl' r" az p ' 8",11;' Dz'" fJ:zP' a:z!<' a o ,,'"
,,"A" = p'A' D"p' "zA' aT - iJzP' iJzA;' "azA'azp' ,
According to Lemma 1,
ao"l" fJ:x;P' iJzA' _ aoz'" az l "
a:z!<' az 1" ' az p " '""ih!'" - a;J:1'" az A ' . az A ' =
aOz I' ozl"
iJzP' fJzH' . 8",1' .
The theorem is proved.
We have proved the theorem on the existence of "tensor differen-
tiation", but only for the case M" = Rn, i.e. when Mn is provided

J28 5. rensor A nalysis and Ricma1m£an Geometry
with Cartesian coordinates, This has enabled us to calculate expli
citly the functions rr, which measure the deviation of 'i1 from or-
dinary Euclidean differentiation iJ o " , namely,
.7'
i' I)z;' iJ'z; o",,' or l iJx;
r l''=- iJx l h - - ,
iJzi'iJz' -  ox;' -;;;F'
We tacitly assumed that in R n there exist "preferred" Cartesian coor
clinates in which the operation 'i1 coincides with ordinary differentia
lion. Let us now turn to an arbitrary smooth manifold and define the
operation V axiomatically, relying upon the properties of V in Rn
demonstrated above,
Definition, Covariant diflerentiatiOl! 'i1 is said to be defined 011 a
smooth manifold Mn if for each smooth atlas there is given in each
chart a set of smooth functions rk transformed under coordinate
:=mbstitution according to the law
r!' ox l ' IJzi iJx ri + ozl' a"x l
J"' = (1x; tlxr I)zk' ;A iJzl ' uzi' oz' .
Then 'i1 is given by the formula
k
( 'i1T I"'k -- ( T'"  ) "Til.,,;.i.=q:...iAri.
»1 . 'P'''- -- 0"'''- )'"")p + L.J )," ')p q'"
.=1
p
 r i ,,, ik '1
- LJ il....},c'J... jpr,,,
.=1
The existence of manifolds Mn on which the operation 'i1 is valid
has already been proved, We can therefore set M n = HI!; then
r;k are given by the formulas derived above. Essentially, the set of
r}h does not form a tensor. These functions are called Christoffel
.ymboLs, They are transformed by the tensor law if the substitution
(x) -+ (x') is linear, In the definition of V the system (x) (not primed)
is no longer a Cartesian one because we now deal with an arbitrary
manifold Mn. The Christoffel symbols (or V) are sometimes said t()
define an affine connection on Mn.
Definition. The torsion tensor of the affine connection r is a tensor
defind in each coordinate system by the relation g}k = r)k - rl j .
Lemma 2, The set 01 functions gjk does form a tensor,
Proof, Under coordinate substitution, r}k are transformed as
T,i' dx i ' ax l or i ox;' i)1zl
. j'' =- "8rf" {j;F' ax' rl + a;r {jzi' iJz! .

5.8. Connection and COllarlant Differentiation
329
Alternating q:,., with respect to the lower indices and taking into
account that the "non-tensor term" is symmetric in j' and k' we ob-
tain
i' _ ozl' ax! , i
2j'k' - 7iil o",i' oz,.' Qjk.
The lemma is proved.
Definition. The connection (or covariant differentiation) V, is
called symmetric if the torsion tensor is zero.
The connection introduced in Rn is symmetric; this follows from
the expressions for V in terms of partial derivatives, On an arbitra-
ry M n , the Christoffel symbols need not necessarily have the form
of second-order partial derivatives of the coordinates; such a form
arises if only Euclidean coordinates exist locally on M", In a fixed
coordinate system. V is represented as a combination of operations
Vk. covariant derivatives with respect to individual coordinates (ana-
logues of partial derivatives 0:" ).
Proposition 1, Covariant differentiation (connection) V satisfies
the lollowing relations:
(1) the operation V = {Vk} is linear,
(2) lor an arbitrary tensor field rm the set of functions V I< TU! =
(VT)j.\J) jormll a tenor field,
(3) ilthe lensor field is scalar (i.e. a smooth function f OT/.
.ll), then Vf = {Vkt} = { :1" } = grad f.
(4) on a vector field T 1 the operation V is of the forms,
T I aTf '" i
V" = a",k +T r"k.
and on a co'ector field Tithe operation V is given by
T aTt T r '"
V k I =a;;;-- a Ik.
(5) the operation V satisfies the Leibniz formula
V k {1'm. p\?} = (VkTgi)P\1 + T!}!. (Vl<p[l).
where TU! and P! are arbitrary tensor fields.
PIOO/' Properties (1)-(4) immediately follow from the definition of
V. 1 t remains to prove (5). Let us consider the simplest case: one-
of the fields, ru! 01' Pf!?' is 1'('3]ar. Then (5) follows from the

330 5. TOMar Ana/ysls and Rt<mannlall Geom<try
Leibniz formula for scalar functions. Lel now T and P be vector
lields r ' , pJ. We have
"h(Ti.pi)= iJ:k (Ti,Pi)_:-To.pJrh+Tipa.rk
= (  Ti ) pi + TI ( Pi ) + T,.pJrk
fJzh iJzk
= ( : + T«r) pi + T f (  + pao.A)
= (VIIT I ) pi + TI (V"pi).
For arbitary fields P and T property (5) is proved by repeating tile
above reasoning (after the sullstitution of multi-indices (i), (I) for
i, j) and using the definition of '\7, It is left for the reader as a 1Iseful
exercise to perform all calculations, The proposition is proved.
Theorem 2, Given on M n the operation V = {Vd satisfying (1)-
(5) (see Proposition 1). Then for an arbitrary tenor field rH! the fol-
lowing identity is valid:
0.
T (U T ll...I,. ,) (T i ..i ) 'CI T i,...;lqr;...lo. f lv
V" w='I7" J 1 ...J1'= iJz" i:oo.J + £.J J,oo.i p . .11
=I
P
_ 'CI T I ,. .11' . . OJ
LJ ),.. .;J,=-... 'lpfJ,II'
..I
i.e. V is covariant diDerentiation in the sen.se of our definition (see
Ilbu/le). The algebraic properties (1 )-(5) uniquely define the operation '17,
i.e. it can be introduced axiomatically with the aid of properties (1)-
(5).
Proof. On scalar functions and on tensor fields of rank 1 the opera-
tion V II is given by (1 )-(4). It remains to define VI< on an arbitrary ten-
sor field. We now prove an auxiliary lemma: any tensor field can he
decomposed into a linear combination (with smooth coefflcients)
of the products of first-rank iields. Indeed, any tensor field is a multi-
linear mapping
T= Q J j, . J !el.  '"  elL  e i . 0 ... 0 eiq: (T.)q x (T*)"_R,
1" q .
where era' eia. are fields of rank 1, and a l ll"' are smooth functions.
,...Jq
This definition is invariant, so that the decomposition refers to an
.arbitrary fixed coordinate system (in another system the decomposi-
tion will change). Let us choose a coordinate system x\ ' . ., x n
!lnd Jet {T} (i. is faxed) be the set of the components of the tensor

5.3. Connctlon and Cava,lant DID'ent/allan
33!
el., and, correspondingly,'let T' U. is fixed) be the set of the compo-
nents of the !ensor e i .. Expressing T in this system, we obtain
T= a!" '-!Tf'.. . TakTS. ." Tlq ( ...!.... )
/ 1 " .lq' II< P, Pq iJz'1..
( a ) ( iJ ) ". ( iJ ) II q ,
... 8%";'  oz ... a? '
i.e.
T=r<;::::;ql< ( iJ:"'l ) 0 ...  ( 8:"k )  ( :z t
o ' .. 0 (:,,, )lIq .
where
T(f,l" .Ctll = al" 'h1 . . , TaIlT;\ . . , T ' 9
p,.. -lIq 1,' ..19 I, III Pt IIq'
Thus, any multilinear mapping is completely defined by the coeffI-
cients of decomposition relative to the basis. Let us now return to
tbe formula for the operator V on Tm. Consider, for simplicity, a
field of rank 2 (for an arbitrary field calculations are analogous),
Since V is linear, it is sufficient to verify formula (5) for the monomi-
Ills TIJ = ex (x) T,Pjo where ex (:c) is a smooth function. Calculate
V h (TI;) under tbe assumption that VII (T.) are already known (see
(4». By the Leibnlz formula (5) we have
Vh(TIJ}=Vh (a.(x)T,P j )
= ::h T/PJ+a. ( :i -Tprfh)p;+a.T/ ( := -PqrYk)
= iJ1I (a.T/Pj) - a.rfl<T pPJ-a.rYIiT/Pq
= 8: 11 (T lJ) - r111 T p} - r1I1 T IQ'
which is what was required. The theorem is proved.
Definition. Let V be an affine connection on M n , Local coordinates
xl, , , '. x" are called Euclidean for 'i1 if in these coordinates rh(X) =0.
Snch coordinates may not exist if, for example, V is non-symmet-
ric. Indeed, in this case Qh is not zero. If there existed the coordinates
in which r}... (x) 5$ 0, the torsion tensor Qjk would vanish in this
coordinate system and therefore it would be identically zero in any
coordinate system (becanse Qjll is a tensor).

332 5. Temor A nalY$ts and Riemannian Geomttry
Essentially, the concept of covariant differentiation (affine connec-
tion) does not rely upon the concept of a Riemannian metric; the
only fact we have used is the existence of a smooth structure on M n .
The Riemannian metric and connection are, therefore, two distinct
structures on M". In particular, Euclidean coordinates for the af-
fJneconnection and Euclidean coordinates for the metric gll (i.e. the
coordinates in which gll = 6'J) are distinct concepts. Thus, the
Riemannian metric and covariant differentiation do not, in general,
define each other.
Let a metric glJ be given on M n . Then, among all afflfie symmetric
connections we can choose one (and only one!) connection which is
"consistent" with this metric and is completely derIDed by this met-
ric. This is an important class of connections calJed Riemannian con-
nections,
5.3,2, RIEMANNIAN CONNECTIONS
Let glJ be a metric and V = {r1k} an affine connection
on M".
Definition. The affine symmetric connection V = {r}I<} is called
compatible with the metric gu (or the Riemannian connection) if
V(gfJ) eE O.
We know that V is a tensor, and this means lhat if the identity
(Vk(gj/) E; O} holds in one coordinate system, it will also hold in
all other systems. Thus, the tensor gll is "constant" with respect to
the Riemannian connection in the sense that the covariant deri\"a-
ti \"0 of this tensor is zero. It follows (for any lensor field) that Tm:
VI< (gIJTII>S3 glIV" (Ti\), see the Leibniz formula (Sec. 5.3.1,
Proposition 1, relation (5». In particular, V commutes with raising
and lowering the indices. We now prove that Riemannian connec-
tion does exist and is unique.
Theorem 3, Let glJ be a metric on M". Then there exists a unique
'ymmetric affine connection compatible with glJ and .uch that
r! __..!.. ;a: (  + {)gJa _ (lgjk )
)h 2 g azl dz" fJ3:(J.'
Proof. Suppose the exislence is proved. Let us demonstrate that
the connection is unique. By definition, V h (g;J) - O. Since
V is defined in terms of rk, it is suflicient to prove
that this system uniquely gives r}h as functions of glj and
iJ glJ 0 agl} r a: r a
a;a:. We have Vhgij= ,  = g(J.j ik+g"'i jA. Circular permu-

 3. Connection and Covariant D ifferenltalion
333
tatioll 01 i rHlices yields
+  =I'M+r/,jh
ag' T"' r
+ aJ) =. i,AJ+ ".11
aC)I, T"' I'
a;r- = ! A,JI + J,ki
(ijk)
(kij)
(;ki)
where 1'1, A) = gi"ri' Adding the first two identities and sub.
tracting the third one from the sum, we obtain
agll + ag", aflJ 21' 2 r'"
axA a,) - a;r = loJA = :,'" iA'
(recall that fi.; = rj). Hence,
r'" -J... Ia. ( OBI) + OCAI _ 08i" ) .
it< - 2 g az k 8z; ox l
We have used the rule for solving the system g!J.1I = Ta. = Qt>.
Since gall is a non-singular tensor, there exists the inverse tensor
gGt/l such that g!J./lg/l v = 6X, whence
g/lvgaaTa. = gIWQII' l\Ta. = gllvQ, Ta. = gfla.Q.
We have obtained the formulas' which express I'M in terms of g'l
and its derivatives. This means that if the initial system has a solu-
tion, it is unique, To prove the existence of the connection, it is sufii-
cient to calculate r}k by the formulas derived above. Reversing the
manipulations, we obtain "h (81/) ;;;;; 0, The theorem is proved.
Remark, Let, on M", there exist a coordinate system x', . . ., x"
in which g'j = l\1J. The existence of such a system is equivalent to
the existence of the system in which (glj) is a constant matrix (inde-
pendent of the point in some neighbourhood). Then under a linear
transformation, this matri:l. can be reduced (simultaneously at all
points of the neighbourhood) to the fOI'm l\1). Thus, a coordinate sys-
tem may be called Euclideall if in this system the matrix of the
metric tensor is constant. Wilh this remark in mind, we turn back
to an analysis of lhe Riemannian connection. A local coordinate sys-
tem is Euclidean, from the connection standpoint, if and only if in
this system {r}" == O} (in some neighbourhood). Statement: A
coordinate system is Euclidean, from the standpoint of the Rieman-
nian conneclion compatiblf.' with gij if and only if this system is Eu-
clidean from the sLanrlpoinl of gl) (i.e. glj is locally constant).
Indeed, if gll is locally ronstanl, then rJk E: 0, by Theorem 3, i.e.

334 5. TtI'Isor A nall/sl. and Rlem4nnlan G.om.try
this system is Euclidean for the Riemannian connection V as well, Con-
versely, if r=o, then, by Theorem 3,  = g",}ffk+ g.. I rjk =0,
i.e. Of} is constant in this :"ystem, which is what was required,
Let us consider a smooth hypersurface yn-I c:.: R" defined by
the graph x" = f (Xl, '" I ;Z;"-I) and Jet Po E yn-I be an arbitrary
point. Let also T p , (V'I-!) be parallel to the plane R,,-I (Xl, .. '.
oX"-I) , i.e. af (P:) = 0, 1  i  n-i, We have already calculated
8z
gl} at Po relative to this special coordinate system, We hsve
81} = 61J + 1,,1 + I"" whence !It I = 0, i.e. r1< (Po) = 0, where V is
v% p.
the Riemannian connectioll compatible with gl). Generally, rlk
are equal to zero only at the point Po; f}k need not necessarily be ze.
ro in a neighbourhood of Po,
We have aJready noted that an attempt at defining the divergence
n 6
of a vector f1eld by the "Euclidean formula"3::;;:r (Ti) generally
i_It/%,
fails because this expression is not a scaJar. We now demonstrate
that the existence of a Riemannian connec-tion permits a correct
definition of the now divergence. !\ote that the concept of diver-
gence (a change in an infinitesimal volume) implies the existence of a
metric on M n , otherwise the conc.ept of voJume is devoid of sense.
Let T be a vector field on Mil. provided with Riemannian connec-
tion. Set div T = VITi. It follows from the properties of V that
dh'T is a scalar function on Mil., Let us find the eApJicJt form of
diy T in terms of go Bnd the /ield components,
Statement 1, The following relation is mUd:
6T' () -
div T = '";i';T + Tt< -c;- (in V g),
QZ 8:.:
where g is the determinant of the matrix (gl})'
Proof. According to Theorem 3, we have
d ' T T I 8TI T t< r '
IV V, =""ii%"i""+ "'i
_.E!.+ Ta., f gi p ( Ogpa. +_ iJ g ",, )
- 8",1 T 0",1 a:%'7. 8zl"
because
ri .!. 11' ( ag  + iJgp _ {)g", )
"'"" = 8 g (J;tR. aza. (}",I"

5.8. Connoction and Covaria"t DtfJernltalion
33r.
Furthermore,
d i v T =  + 1. T'!: ( giP iJha. + glp !!E.!.. _ glp {J1Ia.l )
iJ.x1 2 iJzI iJza. iJzp
= iJTI + .!. T" g ip iJlIl"
cJzl 2 iJz" '
jnce
g lp iJlfpC1. _ g l)' U[("'I _ 0
iJzl (Jzp - .
I t is required
to find glp !!!lP..., We assert that
{J:z:'¥-
glp allp = 2 -.!..- (In Vi).
{Jz'" {Jza.
Hecall that AlP. is the minor (with sign) complementary to g,p in the
matrix (g'/)' This implies that it is sufficient to verify the relation
..!.  A .!!!P.. = -.!....  ....!L
If l;p iJza. vi vi iJza. '
(I,p)
i.e. to prove that
 6 iJI1IP_ iJll
..:::J l/J(j;fi' - iJza. .
(I,p)
The determinant g = det glJ is the slim of homogeneous monomials
of degree n, n = dim M rI . Fix g,p in g and collect in the sum g all
the terms containing g,p (as a factor). Then g = , , . + g'J'R ,I . ;-
+ . . " where Rip is a polynomial of degree n - 1 which does 1I0t
include glp' The pair of indices (i. p) may be arbitrary, but in order
to fmd the coeflicient of another element g... we should "scatter"
the preceding sum and arrange the determinant g as a new sum,
pic1dllg up the terms with ga.: g = . . . + g..R,,-p +, , .,
Let the pail' (i, p) be fixed; then R,P = ill'" Indeed,
onsitler the standard expansion of g by a column (or a row)
 = t gla. Il ,a. (this is true for any fixed t). Since gll' appears
«=1
"
only ill the term gll,A", (in the slim :2j C'a. A ,..), R'I'= A']J" Cal-
a.=1
culating .!.L, we see that 'Jill !!. appears in the final rlation for
Oz dz'"
t! with the factor ll iP ' while the remaining terms of this sum,

336 5 Tensor A nalysts and Riema'lnla/1 Geometry
i.e. (A tp ) =U-gir,t1 lp , do not contain the function giP- Hence.
2L =  iJgtl' t1 iT " which is what was required, Thus,
(J;i"" {J:r!'
(i,p)
aTI iJ ,/-
divT=a;r-+T<% ar' (In V g).
5.4. PARALLEL DISPLACEMENT, GEODESICS
M.t, PRELIMINARIES
Let us consider a smooth manifold (not necessarily a
Riemannian one). In many particular problems there is a need 10 com-
pare vectors applied at distinct points; for example, it is required to
compare two tangent vectors lying in distinct tangent spaces. In
the cae of an arbitrary M" such a comparison is rather difficult be-
cause T",M" and TyM" are distinct and may be identified in many
different ways. In certain specific cases, where, for example, M" =
R", we can use the operation of parallel displacement or transla-
tiOn which permits the comparison o[ vectors applied at distinct
points. Formally, this procedure (for M n = R") can be defined as
follows. Let us consider two points P and Q and let a be a vector at
the point P. Consider also a smooth curve i' (t) through these points,
" (0) = P and l' (1) = Q, and translate the vector a along i' (t)
in such a manner that the vector remains parallel to itself. This op-
eration generates along l' a'vector field a (t) whose components are
constant (withrespectto t) and t1qual to the components of a at the
initial moment. In particular, the derivatives of the comp.onents of
the field a (t) with respect to t are zero. The vector a (1) applied at
the point Q does not depend on the curve l' along which a is translat-
ed; we may therefore say that the parallel displacement in R n
does not depend on the path (Fig. 5.11).
For an arbitrary M", however, this simple scheme does not work,
This is primarily because M n cannot be covered with one chart, i.e.
a unique coordinate system, common [or all poiuts, cannot be de-
fined on Mn, Suppose first that A-f2 is smoothly embedded in R3 and let
P, Q E M2 be a pair of suff1ciently close points nil .12. Let a E TpllJ2
and let i' be a curve from P to Q. We may suggest the following rule
of parallel displacement on JV[2: consider a as a vector in R 3 and
effect along l' an ordinary "three-dimensional" translation. In this
case a turns into b at the point Q. but b need not belong to TQ kJ2.
This drawback can be eliminn(ed by orthogonally projecting
b onto T Q M 2 and defining the projection :!b as the parallel displace-
ment of II from P to Q along y (see Fig. 5.12). The result of this op-
eration does not depend on the tJ'ansJation path', but the operation has
II drawbal'k; it is defined only in a small neighuourhood of P. If

5.4. Parallel Displacement. GeodesIcs
337
we translate a "by a larger distance". the vector nb. which was called
"pat'aUel to a", may be a zero one. This occurs. fOI example, on
S (see Fig. 5.13) if the translation path is a quarter of the meri-
dian PQ. There are many arguments, according to which a parallel
Figure 5, tt
displacement leading to a zero vector should be rejected, The pro-
cedure may, of course, be more accurate: to move along 'l through
infinitesi!llal steps and after each step to perform orthogonal projec-
tion of the vector thus obtained onto T "I(1)M2, It appears that such
Figure 5,t2
an operation can be defined correctly, and it does define a "parallel
displacement" on M 2 c. R3. We omit the details, for this procedure
employs the embedding of M2 in R3. I t would be desirable to elabo-
rate a general concept of parallel displacement which would not rely
upon a particular embedding of M n in RN,
It is worth noting that in order to define parallel displacement,
we have to fix a smooth curve along which tile operation is per-

338 5. Ten:;or A nall/s" and Riemannian Geometry
formed. Such a curve arises quite naturally; otherwise, the plu.ase "to
transJate a vector from a point P to a point Q" loses its meaning,
Roughly speaking, to "translate" a vector, one has to displace it
Figure 5.13
along W, thereby tracing a trajectory. Imagine that we are displac-
ing a vector a in some M n , It is not clear a priori whether the result
of this displasiment depends on the path along which the vector is
displaced, It may depend. the parallel displaceD1ent along the cent-
c
.:.:f.%/:..'.$.."":I.:.:.l.",':,:,,:,.":
':""""ii"""""''" C
",,  z' '':' ',,., ' : :"' ''''' ''2'''
::'::':'::::':-.:'"::!':::":$:':':":'.
c
.;"z!%?Jgri .
Figure 5.1'
ral Hne of a Mobius band being an example (see Fig. 5,14). In this
case the result of the operation depend on the peth because M
is non-orientable. With an orientable menifoJd, too, it is not olJvi-
ous that the parallel displacement is path-independent.

5.4. Parallel D!lplacemcnr, Geotkslcs
339
5.4.2. THE EQUATION OF PARALLEL DISPLACEMENT
If we recall the definition of lhe directional derivative,
we see that this defInition relies upon the possibility of comparing
the values of a tensor field at nearby points in the same coordinate
syslem, so that once covariant differentiation has been defmed, we
can define infinitesimal shifts. Thus, we sha\l define parallel dis-
placement on M", proceeding from the operation V.
Let P and Q be arbitrary points on M n connected by a smooth (or
piecewise-smooth) trajectory y (t) such that y (0) = P and y (1) =
Q, and' let ;. be the velocity field along y (t). The components of
this field in the coordinate system ,1;1, ' , " ,1;" will be denoted by
{£A}, 1 =s;;; k =s;;; n. Let. on M", there be given an affine connection de-
fined in the coordinate system by the set of partial derivatives V =
{VA}' We define the covariant derivative of a tensor field T =
{T!n along the vector field ybytheformulav; (1) = {V.;.T!Wn,
V:lpJ = tAv" (T;l) or, conventionally, V.y = £I<V". This operation
is called covariant differenttation along a curve.
Definition. Let y (t) c: M" be a smooth curve and let a field
T = {T 1 } be given along this curve, This field is said to be parallel
awng y (t) relattve to the connect£on 'iJ if V.y (T) ..... O.
"I
By virtue of the definition of 'iJ,£I<'iJ", we may say that the field
"I
T which is "parallel along y (t)" has covariently (.onstant coordi-
nates along y (t), This is an enalogue of the Euclidean case, since,
for V = {Vk = 81< } and M n = R" our definition of a parallel field
is reduced to the ordinary definition of parallel displacement,
Let us fix coordinates .:r;I, . . ., x" and write out the condition that.
the field T is parallel, We have
'iJ.y (T 1 ) = I<VkTI = 0, "= dz:e(t) t
) ( I ) .. ( )  ( !.!!... + T " r l ) - 0
'I (t = ,1; (t , "",1; t, de 8z" P" - ,
d:z:" .E!!... + TPr' = dT! + TPr' tk" =0
III (}:z:" dt "" dt 1''' dt '
. dTI I tk"
Definition. The equatIon df" + TJlr P""dt = 0 IS called the equa-
tion oj parallel displacement along a curve V (t),
FOI' diflerent curves V we obtain different equations of parallel
displacement. This equation (a system of n first-order equations)
permits one to find the paraJlel field components r l , Since y (t)

340 S. Ten.or A naly.ls and Riemannian Geometry
.. h f . dzA(t} k W
IS given, t e unctIOns dI are nown. e now turn to a particu-
lar problem of parallel displacement of a vector. .
Let y (t) be a smooth curve from P to Q and let a = {a'} E TpM"
be a vector defined at P. Our task is to construct at the point Q
a new vector bE TQM n which we might call a "vector parallel to
a", Let us consider the equation of parallel displacement along y.
In this equation the functions rA and dz) are assumed to he known
and it is required to find the unknown functions {Ti(t)} (the compo-
nents of the paraUel field T (t» such that the condition T i (0) =
a i be satisfied at the initial moment. As is known from the theory
(If ordinary differential equations, the system dJ,i + T"r /;tA = 0
has a solution which is unique and can be continued up to Q,
Definition, The vector b = T (1) E TQM", which arises at Q,
is called parallel to the vector a E TpM n along the curve y (t);
"1'(0) =P, '1'(1) =Q.
Apparently, b depends, in general, on the curve y along which
it is displaced. If M" = R n , the vector b is parallel to a in
the ordinary sense, provided for V we take the Euclidean connec-
tion, i.e. put r1A E; O.
We now consider the properties of parallel displacement on a
Riemannian manifold M n . Let V be a Riemannian connection.
Theorem 1, Let a, b E TpM" be arbitrary vectors and let y (t) be
a sTfl()oth CUl"Je from P to Q. Consider the parallel displacement of a
and b along V (t), This operation preserves the scalar product of vectors,
!.e, if 8 (t) and b (t) are parallel fields along I' (I), a (0) = a, b (0) =
d'
b, then iii (8 (t), b (t) 53 0, where (, ) is the scalar product in
T y (/) M" generated by KI}'
Proof. We include a and b in the parallel fields T = a (t) and
R = b (t), T (0) = a, R (0) = b, Let us consider the function
f (t) = (T, Rho!l) = (gIJTIRi) (t), i.e. the scalar product along
'I' (t). Differentiation yields
dt) = V y f (t) =VAf
= SVA (gfJriRi) = "gfJVA (T I Ri)
= SglJ (VAT!) RJ +A g,jT i (VARi)
= KuRi (IIVIITI)+ goT I (AVIIRj)
= BfJ Rj (V,r i )+ glJTi (v.Rj) 53 0,
y Y
because V, (T) = V. (R) = O. The theorem is proved.
y 'I

5.4. Parallel DI.plfJcement. God.lc'
34t
The converse is also true: let, on a Riemannian manifold M n , there
IJe given a symmetric affine connection in which the parallel dISplace-
ment along any curve preserves the scalar product, then this connec-
tion is Riemannian. Indeed, from the proof of Theorem 1 we obtain:
tit TiR' ('ihglj) E3 0, i.e. VJtgi! _ 0.
o Thus far, we have considered parallel displacement along a smooth
curve, it is a simple matter to define such a displacement along a piece-
II
p
Q
Figure 5,15
wise-smooth curve, Indeed, let y (t) have a cusp such that the curve
is smooth and has 8 non-zero velocity vector on the right and left
of the cusp (Fig, 5,15). Displacing vector a along y (t), we approach
the point A to obtain at this point vector b parallel to a, Take b
as the initial position of a new parallel field along A8 and repeat
the procedure,
5.4.3, GEODESICS
Definition, Let M" be provided with an affine connection
(the metric may not exist), A smooth curve "i (t) is called a geodesic
in, the connection V if V.(y) = O,.where y is the velocity field of the
"I
trajectory "i (i).
In other words, a geodesic is a trajectory along which parallel
displacement of its velocity vector generates a velocity field along
the entire curve: namely, a velocity vector turns into a velocity vec.
tor. always remaining tangent to the trajectory. Let us derive the
equalion of B geodesic.

342 5. T...or A na/ysl. and RttlPlannian Geomtlry
We have
O=V y (\» = { d;: (V ' ) = OJ.
if Tf=, then =; +TaTAr,,=o or
 -I _ ri .!!!!:.. = O.
at" CIA dt at
(0)
Equations (G) are called the equations of geode$ic!;. Their solutions
are sets of functions Zl(t), . , " z"(t) which define the trajectory,\, (t).
a geodesic. System (G) is a system of n second-order ordinary differ-
ential equations end its solution is uniquely determined 111" the ini-
ti I . I 1 (0) _ P I h P (P i P " ) dz (0) _ i
a 'ia ues:c _ ,were = ,. . ., '-at -- a ,
8 E TpM" (2n constants in all, n constants deline the position of the
point P through which the solution passes, and the other n constants
define the velocity vector at this point). The familiar theorems of
the theory of ordinary differential equations imply the following
statement.
Statement t, Let P E M" and a E TpM n , Then there e.ris a geo-
desic '\' (t) suck that y (0) = P and y (0) = 8, and thi, geodesic is uni-
que.
Proof. We introduce coordinates :e l , . . " x" ill the neighbourhood
of P. Tc find the geodesic. we should find a solution of system (G):
the existence and uniqueness of the solution is ensured by the theo-
rems of the theory of ordinary differential equations.
Corollary, Two geodesics. which touch each other at a point,
coincide.
Let us consider a Riemannian manifold M" and a Riemannian con-
nection V, We consider also geodesics generated by V and examin
a parallel displacement along these geodesics, Let y be a geodesic, 'I'
the velocity field, and T a vector field parallel along y. Then at each
point of 'I' (t) there is valid the number cos ex (t) = (T, y)/ I T I A
I y I . where a (t) is the angJe between the vectors T and y.
Lemp1a 1, The parallel diSplacement 01 the vector T a long the geode.ic
'V pr,eserlJes the angle a: a (t) EE canst.
Proof. According to Theorem 1, all pairwise scalar products are pre-
served. i.e. in the identity (T, y) = I T I I y t cos 0: (t) both the
left-hand side and the magnitudes I T I, I y I are preserved.
In a multi-dimensional case this condition is insufficient to unique-
ly define parallel displacement along a geodesic. For A{l, however,
we obtain, by virtue of Lemma 1, the following: let l' be 8 geo-

5,4. Paralltl Dtsplacemtnl. Geodesic.
343
desic and let a E TpMt (Fig. 5.16), then the parallel field T (t),
for which T (0) = a. is formed by the vectors T (t) having the same
length as T (0) = a and subtending with the vector y (t) the same
angle ex as the angle between a and .;, (0). Since parallel displace-
ment along a geodesic has already been defined, we can defme paral-
lel displacement along any piecewise-smooth curve '\', To this end,
p
Figure 5.16
'I should be approximated by a broken geodesic (composed of smooth
geodesic segments). and the parallel displacement should be effected
along each of these smooth segments, keeping the angle ex constant.
Under parallel displacement along a trajectory which is not a geo-
desic, the angle between the displaced vector and the trajectory ve-
locity vector may, in general, vary. Let us consider several examples.
(t) If Mt = R', then parallel displacement along a smooth curve
is carried out in accordance with the general rule: namely, the paral-
lel fIeld has constant components relative to a Cartesian coordinate
system.
(2) Let Mt be a right circular cone in R3 with the verte" angle
a (see Fig. 5.17). To deal with a smooth manifold in Rat we assume
o
Figure 5,17
that the verte" is "punctured", Let 'I (t) be the seclion of the cone by
a plane orthogonal La the axis, OP = r, and the vector a E TpMt
points to the cone vertex. Displace this vector along V (t) IIntil it
returns to the point P and find the angle through which a rotate:<.
We shalJ use the connection compatible with the Induced metric

344 S. r.mor A nalY8u and Rr.mann/an G.ometry
on the cone. Since the metric is Euclidean, the cone can be
developed by cutting it along the generator. It is suf!icient to deter-
mine the rotation of a under its paralJel displacement on R along
1'1 (see Fig. 5.18). We have: PS = r sin , the length of y (t) =
o
Figure 5.18
2n.PS = 2nr sin {. the length of PQ (dashed arc) = 2nr - 2nrx
sin  = 2nr (1- sin) = rcp, i.e. q> = 211 (1 - sin;). Thus, the
rotation angle cp is equal to 2:n: (1 - sin  ). Here we have used
r1 _ O. For MS with non-Euclidean metric the procedure is more
complicated and requires the calculation of the Christoffel symbols.
Let us discuss the geometric meaning of these coefficients from Ihe
standpoint of parallel displacement. Writing Va" (8), where
8" and 8 are coordinate vector fields, in the coordinates Xl, . . " :J!'
on M", we see that the tensor Va" (iJ) is again a vector field.
Lemma 2, The identity Va" (iJ) = r'iJ !I valid,
Proof, We have
Ax (Y) = Z, Vx (yi) = XQ,V a (YI) = ZI, lVa,,(OIl)]" = aVq (T).
where
o,,={a:=6}. op={1i=6}.
aV'J (T)=c4 ( 8q T + Trq) = agTWrq = 66r =r;",
Le,
"Jd" (op) = r"o,
The lemma Is proved.

5.11. Parallel Displacement. Geode'/Cs
35
This statement can be Interpreted liS follows. Let tiS consider II
frame 0 1 , . . ., On at a point P and make an infinitesimal translation
of the vector oil along oa.; in this case all experiences "rotation", and
rk are the coefficients of the decomposition of 01\ in 0 1 , . . .. On.
(3) Let M" = R n be referred to Cartesian coordinates xl, . . ., x",
then r} = ° (the connection is Riemannian). The equations of geo-
d esics are of the form
d'
dti"" = 0, 1    n, Le, :x,C = aat + /P,
where faa, ba.) = const, Thus, the geodesics are straight lines
and on1y such lines.
(4) Let M' = 8 2 in the standard metric. Choose on 8'&
spherical coordinates (e, cp) in which ds 2 = de' + sln 2 e dq>2,
here the north pole is e = 0, To ftnd geodesics, it is
required to calculate r}A' Since (gl}) = ( sjne)' we have
(verify I) ri = - ; sin 2e, ri = cot e, r}k = 0 for all other com-
binations of indices (it f, k); here :&1 = e, :.:2= q>, III = 1, 812 =
g21 = 0, '22 = sin 2 e. It follows that the equations of geodesics are
die t. 2 ( dcp ) 2 0 d'cp dcp de
of the form"'"'iJ'ji"-Tsm e tit = ''dii"+ cote dtdt=O' One
of the solutions is cp = COIlst, e = t, i.e. a meridian emerging from
the north pole, Thus, one of the sections of a sphere by a plane
through the centre is a geodesic if by the parameter a we mean the
arc length t.
Proposition 1. Let 8'1. be endowed with the standard metric. Then a
curve on the sphere 8 2 is a geodes of the Riemannian connection if anti
only if it is a plane section oj the sphere through its centre.
Proof. We first prove that any central plane section of 8 2 is a geo-
desic (with respect to the natural parameter). This fact has already
been established for the meridian Yo (see above). Let us consider an
arhitrary "equator" y, i.e. a plane central section. Since each equator
is uniquely determined by an orthogonal straight line to the plane
defining the equator, there always exists a rotation which transforms
l' into Yo.
Lemma 3, Let f: M" - M n be the isometry of a Riemannian mani-
fold Mn. and let 'I be a geodesic oj the Riemannian connection. Then
the image oj y under the isometry f is also a geodesic.
Proof, Apparently, isometry preserves the Riemannian connection
and, therefore, it also preserves the equations of geodesics, i.e. f
Lransforms a solution of the system into another solution, which is
what was required.

3G 5. TenJor A na/lI.ts alld Riemanllian Geometry
Returning to the proof of Proposition 1, we see that 'V is a geodesic.
Conversely, let y be a geodesic on S, Consider the velocity vector V
at an arbitrary point of y and draw through this point the equator
along the velocity vector (one and only one equator passes through
any point on s in a given direction) (see Fig. 5.19), Since y and the
Figure 5.t9
equator are solutions of the same system and since these solutions
touch each other, they coindde.
(5) Let M' = L, be a Lobachevskian plane referred to the standard
metric ds 2 = dX' + sinh'xdcp2. We now find geoqesics of the Rie-
mannian connection. The metric takes this form in polar coordi-
nates (X, <p) on a two-dimensional plane, Let uS consider the Poincare
model with the metric (1 _ r 2 )-' (dr' + r 2 dcp2).
Proposition 2. All circukzr arcs meeting the absolute at right angles
(In particular, all the diameters) are gecdeslcs of the Lobachevskian
metric in the Poincare model. Any geodesic of the Lobachevskian met-
ric is of the abovt form,
Let us derive the equations of geodesics. Since the metric of L,
is obtained from the metric of S' by substituting hyperbolic functions
for trigonometric ones, we have: r2= -  sinh 2X; r, = coth x'
rl<=o for the remaining combinations of indices (I, i. k), here
d'X t, ( dCP ) 2
.zl = X, .z2 = <po It follows that dii"'" - T smh (2X) , Cli" = 0,
+cothx' ; ,  =0, One of the solutions is cp=const, x=
t, i.e. a straight line through the point 0 on the plane. Since
(X, cp) are also polar coordinates for the right-hand sheet of the hyper-
boloid (a pseudosphere of imaginary radius), the stereographic projec-
tion maps the straight line cp 0:::- (j)o. X == I j nto one of the diameter>

5.4. Paralltl Displactmenl G.odtsicR
34;
of the unit circle. Thus, we have proved that the diameter '\'0 on the
Poincare model is a geodesic
Let liS now prove that IIny circular arc orthogonal Lo the absoilite
H /I geodesic. \'\'0 shall ue Lemma:-l. It is required to prove Lhat nll'
surl! circle i mapped under an isometry into '1'0' To this end, we go
. . d3"'+dyl
u\"er to the model 011 the upper half-plane with the metric --.-.
y
We recall that there exists a homographic transformation (an iso-
meLry) which sends a Imil circle into the upper half-plane. In this
case the circumference is transrormed into the real axis and the
diametor Yo into a straight line orthogonal to the real axis (Fig. 5.20).
0= ((!:!) (D)
((C)
Figure 5.20
The y-axis on the upper half-plane may be regarded as a geodesic, for
it is the image of the diameter '1'0 under an isometry. Thus, any
slraight line orthogonal to the real axis is a geodesic, since the shift
Z -- 1; + ex, ex E R, is an isometry. It follows that any circular arc
meeting the x-axis at right angles is a geodesic, for it can be trans-
Formed into the y-axis under a homographic mapping: first, the shift
z -+ Z -I- IX, a. E R, and then 1; -- -=:., a E R (see Fig, 5.21 ) . Wo
I-a
have thus proved that all straight lines orthogonal to the real axi!!
and aU circnlar arcs meeting the x-ax:is at right angJes are geodesics.
Let 'l he an arbitrary geodesic (on the upper half-plane), it is required
to prove that this geodesic coincides either with a straight line
orthogonal to the x-axis or with a circle orthogonal to the absolute.
Choose a point P on 'l and consider the vector'; Dl'aw through P
8 circular arc which meets lhe a!Jsolule at right angles and has the
same velocitr vector (see Fig. 5.22) As we have already demonstra-
ted, y coincides wilh this arc.

348 5. T.nsor A muysi. and Ri.malln/an G.om."y
As an example. we shall consider the perallel displacement of a
vector on a Lobachevskian plane elong the trajectory V (t) delined
on the upper half-plane by the equation y ""' Yo - const, i e. along
S'
A
0= A'
Figure :1.21
tbe straight line paraJlelto the x-axis. This trajectory is not a geode-
sic and, therefore, parallel displacement along this trajectory does
not preserve the tangent velocity field. Approximate 'V by a broken
y
}{
Figure 5.22
geodesic.(see Fig. 5.23), A qualitative picillre of parallel displacement
of a is shown in Fig. 5,24. The vector a rotates .about its origin,
Remark. Geodesics on a Lobachevskian plane can he found in a
more elementary way: by integrating explicitly the equations of geo.
desics. It isconvenienl to perform integration on the upper half-plane.
The Chrisloffel symbols are of the form (verify!): r: t =  I
rl =- -.!... r: a ""..!.... the remaining symbols v8nbh. The equa-
!I Y

5.4. Parallel Duplaccment. Geo<kliel
349
.. ' .. y1_;'
tionS of geodesics arc x = --!-' y = -'----V' whence
y2_;' ,v '
_.z_.y
II II
d'y ii;- ;'y
-U = ;.
(;)'
= _..!... ( 1 + 1 ) = _1.. (y' + i),
II z' Y
y"= _.l-(y' +1), yyW +y'. = -1, (yy')' = -1,
y
C d y' z' D
yy'= -x+ , y y=(-x+C) dx, T= -T+CX+T'
x-2Cx+y=D, (x-C)+y2=CZ+D.
In this calculation it was assumed that x =p O. If ; = O. we obtain
straight lines orthogonal to the real axis. and if x =1= 0 we have, evi-
dently, circles orthogonal to the absolute.
y
K
FIgure 5.23
Problem. Displace a vector a along a plane (non-central) section of
S (see Fig. 5.25).
(6) Let M2 = T 2 be a two-dimensional torus, It can be trans-
formed into a Riemannian manifold with a locally Euclidean metric.
On the circles 8 1 (q» and 8 1 (\j» we introduce the coordinates q>
and \j>. respectively; then (cp, \j», 0:::;;; cp, \j>:::;;; 2n, are coordinates on
the torus, and the metric (in these coordinates) takes the form
dlp2 + d\j>\I. This metric may be considered as induced by the Euclid-
ean metric from R4 under the embedding of T' in R(  CII given
by I (cp, \j» _ (ei<P. eill» E Ct. Since ds 2 (R4) = duj"z+dwdw, we
have cJsZ (T2) = dq>2 + d\j>2.
Thus, since in the Euclidean metric rjk = 0, only the images of
straight lines on R 2 (cp, \j» under the factori7.ation h: R2 (q>, \j» _
T' (Ip, \j», h (cp, \j» = (cp, \j»' mod 2n, are geodesics on a torus, i.e.

.-150 5. Te.zgor A nalllsis and Riemannian Geometrll
T2 = RZ/Z E9 Z (see Fig. 5.26). The are no other geodesics
on tho torus. The geodesirs are divided into two classes:
closed and unclosed. I t is r.onvenient to depict geodesics on
y
I
Q
o
x
Flgule 5.24
a torus as straight lines on R' (<p, '1» with a fixed lattice Z E9 Z =
(2nm. 2nn) , m, n E Z. Let us consider a sheaf of straight lines
emerging from the point 0 on RO (q>, '\» and lind geodesics which are
Flgule 5.25
the images of these straight lines. Apparently, the geodesic through
the point (0, 0) on a torus is closed if and only if the inverse image of
this geodesic (a straight line) meets some "integer" point (2nm,
2nn). m, n E Z. It follows that the geodesic on T2 through (0, 0)
is unclosed (homeomorphic to a straight line) if and only if the cor-
responding straight line does not contain "integer" points except for
CO, 0). This can also be expressed in terms of the slope of the straight
line l to the x-axis: namely, a geodesic is closed (homeomorphic to a
circle) if and only if tan IX = X/Y is rational (here (X, Y) are the
coordinates of the direction vector of the straight line l (see Fig.
5.27»; and a geod(jsic is unclosed if tan IX is irrational. I n Fig. 5,27
tho straight line passes through the point (2n.3, 2",,2); after factori-

5.1. P"r4llel D!Jpl4cemenl. Goode,'c,
351
zaUon we obtain on the square 0 =:;;;; q> =:;;;; 2n, 0 =:;;;; 'i' =:;;;; 211 a colleclion
of segments-the images of this straight line (see Fig. 5.28). Fig-
ure 5.28 also shows the trajectory which arises on the toru!! aUer
'1
2.. /{
"i'
U.' .:
V 2.- op
/
-1..-...-.
'"
...
.,;;; {if'!'!?b&?;;:;/;r::
Figure 5.26
the fundamental polygon is glued in accordance with the operation
of the group Z e z, Figure 5,29 illustrates the geodesic with ratio-
nal (tan a. = nlm) and irrational slope. In the latter case the straight.
.:,
1/
..,. 17"
V
>I' -1 (3,2
t.:J..:
.. .:
0 a 'I'
Figure 5.27
segments are everywhere dense on the square (under factoriza-
tion the segment of the strsight line I passes as close as possible to
any point). The trajectory tbus obtained on the torus induces an ir-
rational torus winding, the entire torus being the closure of the tra-
jectory' (see Fig. 5.30),
We now apply the concept of geodesies to particular geometric
problems.
Theorem 2, Let M be one of the following manifolds: (a) RI, (b) 8 2 ,
(c) La (Lobachevskian plane) provided with standard metric8 and
let (!J = Iso M' be the group of all isometries of M2, Then eack map-
pIng g E  is defined by three continuous parameters, i.e, dim (!J = 3.

352 5. Tensor A noll/sls ond R I.monnloll Geometry
By @J we mean the complete isometry group, i.e. the group of
diffeomorphisms preserving the metric. The group @ = Iso M n can
be defined for any smooth Riemannian manifold. This group can be
transformed into a topological space: namely, two mappings KlI gi
1
. ....
4
: . : .: "... .
. . - ," . "0 ',. .
. .
. '. .
., . ...
Figure 5.28
are assumed to be close if they are close 11:1 diffeomorphisms of M n ,
i.e. for all points x E M" the points gl (x) and g (x), g.. g. E @J,
are close,
Theorem 2 is a particular case of a more general statement which
is presented here without proof.
Let M" be a compact, smooth, Riemannian, connected, closed mani-
fold and let e = Iso M", then dim @J :::;;; I1(Rt f ) . i.e. each mapping
g E @J is defined by not more than "("if) continuous parameters.
This general theorem will not be used in the sequel. Although we
shall prove Theorel1) 2 only for the three manifolds mentioned above,
the reasoning remains true for an arbitrary manifold, in the course
of the proof we specify the places where a particular form of M2 is
employed.
Proof. Let Xo E Mn and let H (x o ) c @J be the set of isometries
such that Xo remains fIXed. Evidently, H (xo) is a subgroup. It is
called a stability subgroup of the point Io; for distinct Xl and x. the
subgroups H (XI) and H (x) are, in general, distinct. Let hE H (xo).

5.4. Parallel D up/aument. GeOtustc.
353
Since h (xo) = xo, we have dh (xo): T «,M" c::: T x.M". Construct the
mnpping A: H (x o ) -- @$ (n, R), setting A (h) = dh (.:1:0)' Apparently,
dit E @$ L (n, R) because h is II diffeomorphism. Furthermore, '" maps
H (xo) into the subgroup 0 (n) c::: @$L (n, R). Indeed, we may assume
/
Figure 5.29. tan a is rational number on tbe left, and it is irra-
tional on tbe right.
that the coordinates chosen in the neighbourhood of .:1:0 are such that
CiJ (xo) = /jiJ' and then glJ (.:1:0) defines in T x.M" the Euclidean
metric. Since h is an isometry, dh (.:1: 0 ) preserves the Euclidean scalar
product in T :r,M", Also, '" is a homomorphism of H (xo) in 0 (n).
o
Figure 5,30
Indeed, A. (hI 0 h,) = d (hI 0 hi) ("0) = dh 1 (.:1:0) 0 dk a (xo). And all
the more, '" is a monomorphism, In fact, suppose dh = E (identi-
cally), it is required to prove that h (x) = x for any x EM". Since
M" = M2 is one of the manifolds S2, R2' or L 21 and two points 011
each manifold can be connp-cted by a geodesic. The statement is
obvious for a plane. On S:, according to Proposition 1, the geodesics
are equators, which proves the statement. If M2 = L2' we consider
the upper half-plane, the construction of the geodesic is shown ill
Fig. 5,31. This statement also holds true fOl' manifolds of a mOL'S
general type (the proof of this fact is omi tted).
So, let dh (.<0) = E, connect an arbitrary point x E M" with ;to.
Let y (0) be Ihe velocity veclor of 'V (I) at the poinl "0' Since h is
an i50m811)', (he image of l' uuder It is <I geodesic, and since dIL (xo) =

354 5. T.n,or A nalll,i, and Riemannian Geometry
E, the geodesic '1'1 = h ('1') has at Xo the same velocity vector as y.
Wp have already pointed out that two tangent geodesics coincide.
Sinn! 'I' may be considered as the natural parameter, h does not
al tel' this parameter along '1', so tllat x remains (under h) at the same
(listallcc from .:io, i.e. h (x) = .:i' And this is wllal was requited. Thus,
Q
p
Ie
Figure 5.31
H (xo) is .I closed subgroup in 0 (n). Closeness of H (xo) follows from
the fact that the isometry, which is the limit of isometries preserving
xo. also preserves Xo. For M2 == S, R2' or L 2 we have, therefore.
dim H (x o ) :;,;;; dim 0 (2) = 1. For an arbitrary lIr it can be shown
that dim H (xo)  dim 0 (n), i.e. dim H (xo>  n (n - 1)/2. We
now tllrn to the group e. Statement: any isometry gEe i$ defined
by the image of the point xo, i,e, g (xo), and by the differential dg (xo):
T ""M" -+ Till",.) M". Indeed. consider the correspondenceg -- (g (:to),
dg (xo» and let (g1 (xo), dg l (x o » = (g2(XO), dg(xo»' whence
gl (.:ro) = g. (x o ) and dg) (x o ) = dg. (xo)' Consider then g (x) =
(g;') . g (x), x E M n . We have
g (.xo) = (g;') 0 g. (xo) = x o , i.e. g E H (x o )'
dg (x o ) = d «g;') 0 g.) (xo) = «dgl)-l 0 (dg.» (x o )
= (dg) (xo»-) 0 (dg. (xo» = E,
whenc(: g i2 E on Mil, i.e. g (x) = x, g) (x) = g (x). Since g (xo)
. n(n-1)
IS defined by n parameters and dg (xo) by not more than 2
n(n-j)
parameters, g can be defined by not more than n +  =
n(+j) parameters. For JVJ' = S. R', and L. we have dim @J :;,;;; a.
On the ether hand, it was shown in Chapter .} lhat each 01 the
groups Iso R2, Iso S2. and Iso L. contains a subgroup whose trans-
formations are .Ilso defined by three parameters. Since these sub-

5.4. ptJrall.1 DupltJcern.nt. Geodedet
355
groups are open and closed, we have dim Iso M2 = 3 for M2 = 8 2 ,
R2, and L 2 . The theorem is proved.
C1)rollary, Let Is1) (Mn)o be th., connected component 0/ the unit
element in Iso 1>1". Then for M2 = 8 2 , R2, and Lz the group Iso (M2)O
comcides with the three-dimensional groups constructed in Chapter 4,
i.e. Iso (8 2 )0 = SO (3), Iso (Lo = SL (2, R)/Z2' and Iso (R2)O
coincides with the group oj all linear isometries oj a plane which preserve
orientation.
Theorem 3. Any two-dimensional smooth, compact, connected, closed
manijold admits triangulation.
Prooj. Let M2 be provided with a Riemannian metric (viz., embed
!vJ2 in a Euclidean space) and consider geodesics on M2. To prove
the theorem, we need the following lemma.
Lemma 4. For each point Po on a Riemannian manifold M" thue
e;rist a neighbourlwod U and a number e > 0 such that; (8) any two
points in U can be connected by only one geodesic of length smaller
than e, (b) this geodesic depends smoothly on the initial and terminal
points.
Proof, We first recaH the theorem on ordinary differential equa-
tions; given a system  = F ( U,  ), where u = (u l , "., '4")
and F is a set of n smooth functions defined in the neighbour-
hood W of the point (UI' VI) E RZ". Tben tbere exist a neighbour-
hood U of (u" VI) and a number 8 > 0 such that for each point
(u o ' vo) E U the equation , = F ( U,  ) has only one solution
t_ u (t) defined for It I < e and such that u (0) = u"
d:O) = Vjt this solution depends smoothly on the initial condi-
tions. Let Po EM", then, according to this statement, there exists
a neighbourhood W of Po such that for each PEW there is valid
a mapping expp defined as follows. Let a E TpM" be a vector of
length not larger than e; draw along this vector a geodesic 'Va (t)
referred to the natural parameter t and associate with a the point
'Va (1) denoted by expp (a). We obtain a smooth mapping of a ball
of radius e in M n (differentiability follows from the existence and
uniqueness theorem) (Fig. 5.32). Let us construct the mapping F:
V __ M" X M", where V is a neighbourhood of the point (Po. 0)
in the manifold T.M", i.e. V = {(P, a), P E U (Po), I a 1< e},
and F (P, a) = (P, expp (a)). The structure of a smooth manifold
in T.M" consists of all pairs (P, a), P EM", a E TpM TI , if Xl,. . .,;r"
are coordinates in the domain U c: M", then each a E TpM n can
uniquely be represented in the form a = tio;, where 01 = .!.. I{
{Jz p.
The functions (Xl,. . ., x", t 1 ,. . ., t n ) form a local coordinate system
in the opel! set R c: T.M". We now prove that the Jacobian of the

356 5. Tell<or A 1I/11116is and Riemann/en Geomdrll
mapping F is non-singular at (Po, 0), Let {.xi, :rat 1  i :::;; n, stand
for coordinates in U X U c: M" X M", then
F.. ( iJ' ) = "a I +  . F. ( aJ ) = a j .
(1%, (1%2 o;q
i.e. the lacobi matrix of F at the point (Po, 0) is ( ), i.e.
il is non-singular. The implicit function theorem implies that F
maps diffeomorphically the neighbourhood W of (Po, 0) E T .Mn
onto the neighbourhood R of (Po. Po) E M n X M n . The lemma is
proved.
Figure 5,32
This lemma can be proved differently. It is known that a dynamic-
al system called a geodesic flow, exists on T .M", Let us consider a
point (P, a) E T.M", according to the existence and uniqueness
theorem. only one geodesic 'I' (t) emerges from P along the vector a,
i,e. there arises the velocity field y, and we obtain the trajectory
(P (t), y(t» = r(t) in T .M". These velocity vectors form the geodesic
flow. The integral trajectories of this field can be extended infinitely,
in particular by the distance e (which is the same for all points in
T .M"), This proves Lemma 4.
We now proceed with the proof of the theorem. Since M 2 is com-
pact and closed, it can be covered with finitely many small disks.
According to Lemma 4, we may assume that each of the disks is
such that any two of its points can be connected by a single geodesic
of length less than e. where I! is sufficiently small, Covering M2
with a rather dense network of points {Pd, we can connect points
inside some disk by a geodesic, thereby subdividing each disk into
triangles satisfying the triangulation requirements. It is important
however that the smooth subdividing curves arising in a disk should

5.5. Curvature r",..or
357
.
also remain smooth in the coordinates of any other disk in which they
can be located if the points they connect lie at the intersection of
the disks. The subdividing curves are smooth because the solutions
of the equations of geodesics are smooth, so that the triangulation
can be extended until the entire M is covered. The theorem is
proved,
] n the proof we did rely upon the fact that M is two-dimensional.
5.5, CURVATURE TENSOR
PRELIMINARIES
Figure 5,33
Figure 5,34
The connection V defines paralJel displacement, and this formula
can be interpreted as follows: t describes paraUel displacement
(by an infinitesimal distance) along the coordinate line :Lt¥.. Let lIS
fix a point P and consider the parallel displacements Vh"'lI and
VIVA' whele V.. = Vg. (Fig. 5.33), This is the case of displacements
along the coordinate Hnes Zh and xl by small distances a. and .
The terminal points Q and Q' will, in general, be distinct. This
effect can also be noticed with another displacement. Let liS consider
the motion shown in Fig. 5.34. Generally, this small "parallelogram"
is non-closed, i.e. we cannot come back to the point P because M"
is "curved". This "curvature" can be measured, conventionally of
course, as the difference VkVI - ViVh = 8. Jf M" = R" and we
use Cartesian coordinates, then Vk VI =VIVh; if M" is arbitrary,
this commutator need not necessarily be zero. A clear example is 8 2
referred to the coordinates (e, cp) (see Fig. 5.35). Here ::I::a. are dis-
placements along meridians, and ::I:: are displacements along paral-
]els.

358 5. Tensor A nalysis and Riemannian Geometry
Simple examples show t.hat. curvature may be of different types.
We shall ilIust.rate this for 8' and for a Lobachevski:1II plane £2'
Figure 5.35
Figure 5.36
Let us consider, on S? and L2' a geodesic y and draw orthogonal
geodesics from e(lch point of its segment.. We now examine. the
behaviour of this sho.!f of trajectories orthogonal to y. Figure 5.36

5.5. Curvature Tensor
359
shows the qualitative picture. On 8 2 the sheaf con\'erges (along oath
directions) at two points. Ihe north and south poles. On Lz tlte
sheaf "diverges" and the distance between the extreme geodesic::'
N
A
s
Figure 5.37
8
R
r
C D
tends to infinity. It can be seen from Fig, 5.37 that on Lz geodesics
diverge on both sides of the segment A B because the arcs CD and
RT hav!' infinite lengtOs. Different behaviour of geodesics on S'
Figure 5.38
and Lz can also be seen for a sheaf of geodesics emerging from a single
point (see Fig. 5.38). We recall that 8 2 and Lz have distinct Gaussian
curvatures: for 8 2 the curvature is constant positive and for L. it is
constant negutive, We shall demonstrate below that the Gaussian
curvature (for M2) is closely related to the properties of the operator
'V"4'V"i - V,"h which measurp.s the "curvature" of M2.

360 5, Temor A nailisis and. Rttmaltnlan Geomttry
5.5.2. COORDINATE DEFINITION
OF THE CURVATURE TENSOR
Let M n be referred to 10l:aJ coordinates Xl, . . ., x" in
some neighbourhood of a point P; consider VhVI- VIVh and apply
this operator to the field T = {TI}. The ronnel:tion V is symmetric,
and straightforward calculation yields
T 1 iJT' T " r i
VI = ""8;f' + "h
T ' alT' iJTP I iJ I (T P ) r l
VI,vd ')= iJzhoz/ +rl>l+TP"'8zii"'(rl>l)+VI ph
(T i ) r p a3TI 8TI' r l TP 8 ( 1'/ )
-VI' h/="ifZii7j;! + -;g; 1'1+ iiiA pI
aTP P i r p i aTI r p T q r i 1' 1>
+ iJzl ph+ Tq q,r,,"-a;;; 1<1- qp kl,
(VkVI-V;Vk) TI
T p [ 0 r i 8 r l ] r p r p ) arl
= d%i< pi-a;r pk -( 1<1- ih 8;p"
+ Tq [r:,rh- rhI',- I'prf, + rprrlJ,
Since rk = rLJ. we have
(V k V,-V,VA) T 1 = r q [ a:k rl- 0:/ rh.+r:,rh-r:Ar/] = TqR,kh
where
i ar1 arh pip I
Rq,l<l = "8"%i1 - ---a;r + I' ql r ph. - r qh. f pi .
Lemma 1. The collection of Rtkl forms a tensor of rank 4.
The proof is obvious, because V = {Vh} is a tensor operation.
Definition. The tensor R, AI is called the Riemann cwvatwe tel/sor
of a given connection V.
If M n = R n , this tensor is equal to zero. Indeed, it vanishes in
a Cartesian coordinate system and, therefore, it vanishes in any
other coordinate system due to the tensor transformation law, There
exist however Mil where R'hl is non-zero t.:ee below). We recall
that the coordinates in which rh a 0 are Euclidean for a given
connection. This implies that the following lemma holds true.
Lemma 2. Let M be provided with a symmetric affine connection.
If the Riemann curvatwe tensor of this connection does not vanish (ill
a certain coordinate system), no Euclidean coordinates (in the neigh-
bourhood oJ a poil/t) can be defined on M".

5.5. Curvatur< Tensor
361
Jf such coordinatl's l'xistl'd, rh would vanish in this coordinate
system. and therefore the curvature tensor would also vanish, TIllis,
the curvature tensor is 811 "obstruction" to defining Euclidean coor-
dinatl's (for a given connl'ction).
5,5,3, INVARIANT DEFINITION
OF THE CURVATURE TENSOR
In the preceding subsection we have constructed the
Riemann tensor in a particular coordinate system. Now we give
an invariant definition, Let X, Y, and Z be arbitrary smooth vector
fields on M" (with symmetric affine connection). Let us construct a
"curvature operator" R which associates a new vector field with the
triplet X, Y, Z. It is convenient to handle the fields as linear differen-
tial operators, 50 that bl'low we shall write just X instead of X.
Definition. Put R (X, Y) = "V-,,"Vy(Z) - "Vy"Vx(Z) - Vrx,y}(Z).
Thus, R maps T" X T" X T.. into T". where x E Mn.
Theorem t, The mapping R is trilinear and, there/ore, defines a
fourth-rank tensor,
Proal. For linear combinations of the arguments with constant
roefficients trilinearity is obvious. What is required to be proved is
that a smooth function I (x) can be taken out of the symbol of the
operator R. If we prove this fact, R is completely defined by its
--
values on the basis fields. oa. = ( a:a. )' 1  a.  n. Let us con-
sider the mapping (X, Y, Z) - (X, Y, / (x) Z), where I (x) is a
smooth function. It is I'equired to prove that R (XY). (fZ) =
I.R (X, y).Z. We have
VxVy(fZ) - "Vy'Vx (fZ) - V[.T, y} (fZ)
-= V x «Vyf) Z) + V::c (I VyZ)
- Vy«Vxl) Z) - Vy(l V;\:Z) - ("ViX,YJf) Z -/V[x,y]Z
-= (VxVy/) Z + (Vy/)Vx Z + (V..d) VyZ + t (VxVyZ)
- (VyV_,d) Z - (V..d)VyZ - (Vy/) 'VxZ - I (VyVxZ)
- (V.TY-yx/) Z - I (V[,y]Z)+ {(X (Yf)- Y (XI) - (XY - YX)j}
+ f{VxVyZ - Vy.V."Z -Vtx,yjZ} = 0 + j.R (X, Y) Z,
because 'V_"I = X (I). Let us verify that R (IX, Y) Z = I.R (X, Y) Z.
The following relation holds true:
R (IX, Y) Z = V/x'VyZ - VyVfJcZ - V[/x,y]Z.
Evidently, V'x = jV;", since
(V,.\') T = (fX)k VkT = I{Xk VkT}'= J (Vx T ).

362 5. Ten,o. A nalll,i' and Riemannian Geome/ry
Furthermore,
ux, YI = I (X'}') - Y (fX) = f (XY) - (YI) X -I (YX)
= I [X, }'J - (YI) X,
whence
R (IX, Y) Z = f ('Vx'VyZ) - 'Vy(f 'VxZ) - 'V/{X,y] Z + 'V(y/)J'Z
= I ('Vx'VyZ) - ('Vy/)'VxZ -I ('VY'V.A-Z) -1'V[X,y]Z
+ (Y/) 'VxZ = I.R (X, }') Z + 0 = I,R (X. Y) Z.
which is what was required. The formula R (X, In Z = I. R (X, Y) Z
can be verified in Ii similar wa)'. The theorem is proved.
Let us relate the invariant definition of the curvature tensor and
the coordinate definition. Consider the basis fields 01 as differential
op!!rators and decompose X, Y, Z in these fields: X = Xioj, Y =
YJojo Z = ZAo". We obtain R (X, Y) Z = XlyiZ k , {R (01, oJ) Uk};
i.e. R (X, Y) Z is completely determined by R (al) I) Ok' Moreover,
R (oj, oJ) Z = 'VO/'V0I Z - 'Va} 'Yilt Z - 'V[I1"l1j] Z.
Apparently, "Va, = 'VI (by the definition of 'Vo,), i.e. R (ai, a}) Z =
("VI"VI - 'VI"V,) Z - 'V[l1j,lIj)Z. Since fal' oIl = 0, we have
R (a" a J ) Z = ('V'''VJ- 'VI'VI) Z. Thus, we have obtained (in a.fixed
coordinate system Xl, . .,. $") the "coordinate" definition of the
Riemann tensor, and this proves that both definitions coincide.
5.5,4. ALGEBRAIC PROPERTIES
OF THE RIEMANN CURVATURE TENSOR
Theorem 2. For any three snwoth fields X, Y, and Z on
M" the loll owing identities are satisfied:
(1) R (X, Y) Z + R (Y, X) Z = 0, Rj.Rl + R},'A = 0 (skew
symmetry in the argurMnts X and Y),
(2) R (X, Y) Z + R (Z, X) Y + R (Y, Z) X = 0 (the Jacobi
identity) or in the coordinate writing Rt"l + Rtfk + RI, I = O.
The correspondence between the indices (j, k, l) and fields X, Y, Z in
Rj kl is as follows: j - X, k -- Y, 1 -- z.
(3) If the connection V is Riemannian, we have (R (X, Y) Z, W} +
(R (X, Y) W. Z} = 0 lor arbitrary fields X, Y, Z, W, where (,) is
the scalar product induced by the metric gij; in coordinate notation:
Rl},kl + RJ/,Al = 0, where Rl},kl = gla.R'f.k"
(4) If the connection 'V is Riemannian, {R (X, Y) Z, W) =
{R (Z, W) X, Y), i.e. R'J,kl = R"/"I'
In terms of the components R iJ.kl (i.e. after index lowering) the
Riemann tensor is skew-symmetric with respect lo the indices in
each pair (I, j) and (k, l), and is symmetric with respect to the pel"

5.5. Curvature Tensor
363
mutation of the pairs (when the indices in each pair remain in their
own places).
Proof. (1) Evidently, the relation R (X, Y) Z = Vx"ilyZ-
"ilyVxZ -'V{X,y)Z implies that R (X, Y) Z is skew-symmetric with
res pee 1 to the pair X, Y.
(2) We first prove that for symmetric connection the relation
"il.\:Y - VyX = [X, YJ is valid. Indeed, we have in terms of the
coordi nates
V xY - "il yX = XiV'tY - yIV/X = {Xi ( , + YPff,,)
-y! ( +xpr! )} 
IJzl 'I' IJz
= { Xi aYk _yi + xiypr -yiXpr } 2-
    
= { Xi i)Y _ y i .!E:... }  = [X Yj
iJzl IJzl IJz "
because yiXpr = YPXirI' f1 = r}I' If X and Yare commu-
tative, then Vx}' = VyX. Let us now prove the Jacobi identity.
B)' virtue of Theorem 1, it is sufficient to verify this iden'Uty only
for commutative fields X, Y, and Z (e.g. for 0" 0" Ok)' It suff1ces to
demonstrate that
Vx'.:;yZ - 'VyV'xZ - V[X,yIZ + VzV'x Y -V'x'VzY
- V[Z,xIY + 'i/yVzX - VzV'yX - V[Y,Z!X = 0
The fields X, Y, and Z are commutative, so that the identity in
question follows from the relations of the type 'VxY = VyX. Con-
dition (2) is thus proved.
(3) It is required to prove that (R (X, Y) Z, W} + (R (X, Y) W, Z)
= O. It suffices to verify that (R (X, Y) Z, Z} = 0 (taking into
account polarization of quadratic forms). As before, we assume that
rx, Y) = O. Then, (R (X, Y) Z, Z} = «Vx'i/y- VyVx) Z, Z}. We
need to show that ('V xVyZ, Z) = (V'yVxZ, Z). Consider the function
(Z, Z) = f and calcuJateX (I) = X (Z, Z) =V x (Z, Z) = 2 {VxZ, Z}.
Furthermore,
YX(j) = 2 Vy('VxZ, Z} = 2 (Vy'VxZ, Z) + 2 (VxZ, VyZ}.
Simifarly,
XY (f) = 2 (VxVyZ, Z) + 2 ('VyZ, VxZ).
According Lo the symmetry of (, ). WB have ('VyV xZ, Z) =
("ilx"il1'Z, Z), which is what Was required.

354 5, Ten.or A nalysts and Rirnallnlan GeomIr/l
(4) Let us consider the octahedron shown in Fig. 5.39. Four of
its faces are shaded, and each vertex is marked with a scalar product.
We assert that the sum of the products at the vertices of each shaded
"
< R{X,YJZ, W>
c
IJ
Figure 5.39
face is zero. Verify this for, say, face (ao:c). Using the symmetry rel-
ations which have already been proved, we have
(R (X, Y) Z, W) = - (R (X, Y) W, Z),
(R (Y, W) Z, X} = - <R (Y, W) X, Z),
(R (X, W) Y, Z> = - (R (W, X) Y, Z>,
i.e., according to the Jacobi identiy,
o:+o:+c
= (R (X, Y) Z, W) + (R (Y, W) Z, X) + (R (X, W) Y, Z>
= - (R (X, Y) W, Z) - (R (Y, W) X, Z> - (R (W, X) Y, Z>
= - {R (X, Y) W + R (Y, W) X + R (W, X) Y. Z) = 0,
Similarly, we can verify that the sums 0: + b + d, c + d + P.
and a + b +  vanish. Let us consider the identity 0 + 0 = 0 =
o = 0 + 0 and write out each of these zeros as follows: (a + 0: +
c) + (0: + b + d) = 0 = (a + b+)+(c + d + ), whence 20:=
2, i.e. (R (X, Y) Z. W) = (R (Z, W) X, n, which completes the
proof of the theorem.
DefinltioD. The Ricci tensor of the Riemannian connection is the
tensor RJI = Rj,/h i.e. the tensor obtained by contrar.ting the
Riemann tensor with respect to a pair of indices. The Ricr.i tensor
is symmetric (verifyl),
Definition. The scalar curvature R of a Riemannian manifold is
the function R (x) = gIRhl' i.e. complete contraction of the Ricci
tensor with the tensor inverse to the metric one.

$,5. Curvalur Temor
365
Apparently, Rl is a second-rank tensor, and R (x) is a scalar
function. For many particular problems it is useful to express a
Riemann tensor explicitly in terms of g,} and its derivatives.
Theorem 3, On a Riemannian manifold the following identity holds
true:
R 1q . U = gl"R,u
1 ( (Jlfl/ ol/q iJalfl O'flql )
= 2"  +a;fij;i - ozq ozr - ozl iJz"
+gmp(rrp -rrf,,).
Proof. The coordinate representation of the Riemann tensor im.
plies that
i or:, or pip I
Rq.I=-a;l+ rq,r pk-rqrpl
( ar, p r ' ) k
= """'8%iI + r q/ pI< . [ ,11,
where [k, II denotes alternation with respect to indices k and 1 without
division by 2. Furthermore,
, ( iJr/ r p I )
O=K.IRq.I</=R.q,I<I=g.1 + Qlrpl< (k, I]
( iJr' , )
= go! oz: + r p"r [k, l},
where. stands for the pair of indices (ql), The expression in paren-
theses may, formally, be treated as the result of covariant differentia-
tion V" of the set {rH (the symbol. is ignored for the time being).
The set {rl} does Dot form a tensor, but in each given coordinate
system we may also consider a tensor with the components r; (in
other systems this tensor will have components distinct from r;,
although this fact is insignificant for differentiation in a given
system), Since g., induces index lowering in the "tensor" r;, this
operation can be performed under the sign of covariant differentiation
because the tensor g,} is covariantIy constant. It follows that
e = ga/V" (r) [k, lJ = V" (g.lr:) (k, II = V" (r.,.) [k, I],
where
r - r - 1 ,,, ( iJl/"'1 iJl/a! 8f1q/ )
.,. - I,ql- 2" g.,g {j;q -I- oz - iJ z t10
=1. ( agll + 81/.q _!!s!. )
2 ozq 0,,;1 oz"

366 5 Ten.or A nall/.i' and RiemannIan Geometrll
Substitution into the initial formula for R. q . A1 yields
R.,Ar=VA (r..)[k, I) = ( iJ::: -r..,.rf.) [k, II
= ( a;ql _ r ..,qrr.) [k, lJ
1 ( oOIl.1 + (}ol1.q uo gql ) [k IJ
=2"  ozA{}zl - {}zlt{}z. .
- C..prlr. [k. II
i ( U"1111 + o"gqlt lI'g .A iJ.gql )
=2"  iJzl{}z' --
+ g..p (r:"r: l - r:lr.).
The theorem is proved,
Corollary 1. If the Riemann curvature tensor dces not vanish in a
certain coordinate system, no local EucItdean coordinates can be defined
on M n . i,e, the coordinates in which ClJ is a constant matrix (or, which
is the same, r11t :::: 0).
The proof follows from Lemma 2.
We can also reason as follows. Let us consider the tmnsforma-
. i' IIzl' ( (}zi I \ u'z l )
tlOn law r .",=- o I r;lt+ , . . For the coordinates,
,.. z oz} uz j oz"
in which r}:,.'E3 0, to exist it is necessary that the following relations
(i.e. equations for the coordinates zi') be valid:
aSzl oz; ozk i
iJzj' ozlt' iJzi' ozlt' r jA .
The necessary condition for the solvability of this system is given
b th . d ' t ' 0 ( OIZI ) 0 ( o'zl ) d th '
y e I entl les ---;- ., It' =---, ., " an IS
aza. UZ' oz lJzA oz' o:z;<r.
in turn imposes the conditions on the right-hand sides of the system,
It can be proved that the fuUilment of these conditions is equivalent
to the vanishing of the curvature tensor (verify I).
5.5.5. CERTAIN APPLICATIONS
OF THE RIEMANN CURVATURE TENSOR
For a two-dimensional Riemannian manifold the curva-
ture tensor is especially simple (problem: what is the meaning of the
curvature tensor for a one-dimensional manifold?). Let us consider
the scalar curvature R (x), a function on M2, Since .this function is
a measure of "curvature" of MZ, there is every reason to believe thaI.

5.6. Curl1tUure Tensor
367
it is related to the Gaussian curvature which is also known to be-
responsible for the "curvature" of M2.
Theorem Ii, On a two-dimensional smooth Riemannian manifold
the identity R = 2K holds true, where R (P), P E Mt, is the scalar
curvature and K (P) is the Gaussian curvature.
Corollary 2. Since R (P) is completely defined by gJI> K (P) is alsl)
completely defined by gl1; in particular, K (P) croes not change under
Isometries of MS in R8 (i.e. under surface bending).
This corollary is not trivial, for the definition of K (P) relies upon
the second fundamental form which describes the embedding of
M2 in R8. A straightforward verification of the invariance of K (P)
under bending is not simple either and can be performed most easily
only after studying the properties of the Riemann tensor.
Proof of the theorem. By virtue of Theorem 3,
1 ( o'gfl iJ'gqk
R{q,k/=T {Jzqozk + azl8z1
+ gmp (r;J.rfl- rrfk)'
o"g/k
8zq8z1
iJ=gq/
8z1 iJzk
Let us introduce in R8 a special Cartesian coordinate system, choose-
on MI a point P, and define AP in the neighbourhood of P as the
graph z = I (x, y), where (x, y) are Cartesian coordinates in TpMs.
Since TpM' = RS (x, y) is the tangent plane, grad f (P) = 0, i,e.
gIJ(P) = (6'J + l",d",J) (P) = 61} or r/t. (P) = 0 because (8g 1 l8:r!') IF-
= 0 (verifyl). As the tensor R,q,lII is algebraically symmetric, it
has only one principal component, Ru.lI' The others either vanish,
or differ from R12,lI only in sign, or coincide with RlI,l2' Writing the
Riemann tensor in the coordinate system (x, y), we obtain
R _..!.. ( 2, o'gtt _ o"g., _ olg" )
12,12 - 2 iJz oy oz' oil'
1
= 2"1 2 (f ",lp)xl/- (f)zx- (1:)1/1/1
= (f..",f ll + 1..1"'11)/1- (f1//",p)",-(f",I"'/I)/l
= I",,,,I/I+ '",,,,' 11 /1+ ''''/11''/1+1",1'''/1/1
- 1"/I1",u- 'ulzxl/-/zp{"u- 1..1"'1/11
= I I",,,, 1"'11 1 =K,
{"'II 1 1111
whence R I2 ,I2 = K. Calculation of R yields
R = g"'R kl =' gk' R,"I = £I"ga9R qk 'fJ.l = Rf2'12 ( ::t: tiel, gaq).

368 S. Tensor A na/ysis and Riemannian Geometry
where
1: :f: lI'igaQ =g22 g J/ _g2Ig21-r gUg22 _gl?gI2
= 2 (g22gll_ (gl?)?) = 2 det (g /1)-1
=.!. t with g = del gil-
g
2
Thus,R=g-RI2'1?" so that R=2K because giJ(P) = 6 11 , But R
and K are scalars and their values do not depend on the choice of
the coordinate system, hence. in any system R = 2K.
We have demonstrated that the behaviour of K (P) under bending
differs from the corresponding behaviour of the mean curvature, and
therefore the Gaussian curvature is an "intrinsic invariant" of a
surface. Let us consider several examples.
(1) For the Euclidean metric d+ dy2, R = 2K = 0,
(2) for the spherical metric dr 2 + ( sin2.L ) dcp2, R=2K=';",
ro ro
i.e. the scalar curvatUre is constant and positive,
(3) for the metric of the Lobachevskian plane dr 2 + (sinh  r. dcp2
we have R =2K = -  ' i.e. the curvature is constant and
negative,
(4) for the conformal Euclidean metric A (x, y) (dx? + dy2), where
'" (x, y) is a positive function, R = 2K = - i .1 In A, where .1 is
the Laplacian. This formula can he proved by straightforward cal-
culation.
In three dimensions the Riemann tensor has a more complicated
structure, The number of the principal components increases up to
six: R u ,13' RlIl,n' R 31 . S2 , R 12 ,12' R 13 ,13' and R U ,23; the other com-
ponents R1J,kl either vanish or coincide with the principal compo-
nents, or differ from them only in sign.
The "intricacy" of R IJ .R1 depends on the number of principal com-
ponents. There is only one such component for M9 and six compo-
nents for M3. It can be shown that for a manifold of an arbitrary di-
. th b f . . I t . N n'(n2-1)
menSlon n e num er a prmClpa componen s IS = 12 .
for n __ 00 the ratio of N to the total number of components (i.e. to
n') tends to 1/12 (problem: find N).
Of great importance in geometry is "curvature along a two-dimen-
sional direction". Let us consider a Riemannian manifold M" and
let X, Y E TpM", Suppose these vectors are so chosen that the area
of the parallelogram n (X, Y) constructed on these vectors is unity
in the metric gu. Then, the curvature of AI" along tile two-dimensional

5.5. CU""tur< Te...o
369
direction 0' defined by X, Y is the number R (a) = (R (X, Y) X, Y)
where X and Yare arbitrary vector fields in the neighbourhood of P
such that X (P) = X, Y (P) = Y, i.e. they coincide at P with
the vectors X, Y. It is required to prove that R (a) does not depend
on the way X, Yare included in the vector fields X, Y.
Lemma 3. There is valid the formula R «(J) = RaJ,;,1 XJ XkyIYII.
where Xa., ya are the coordinates of the vectors X, Y. Here RIMI is
a tensor, and R (0') does not depend on the way X, Yare included in
X, Y.
Proof. We have
[R (X, Y) z] =R'.pqXPyqz;,
R (0) = ga.aYII [R (X, Y) y]a. = ga.llYIIRf.IX;Xlr.yl = RaJ,,XJXlr.yIY/l,
which is what was required because Xi = X' (P) and yi = yi (P).
Definition. A Riemannian manifold M" is r.alled a manifold of
positive (constant, negative, 1:ero, etc,) curvature if its curvatures along
all two-dimensional directions are positive (constant, negative,
zero, ete,),
To justify this definition, we sbould compare it with the "two-
dimensional" definitions of the positive, etc., curvatures in terms of
the Gaussian curvature,
Lemma 4. Given a Riemannian manifold M, and let K (P) be the
Gaussian curvature, R (P) the scalar curvature, and R (0') the curvature
along a two-dimensional direction (X, Y) = (J at a point P E M.
1
Then R (0') = K (P) = "2 R (P).
Proof. Let us r.onsider the coordinates ($, y) in which coordinate
lines are orthogonal at the point P. We may assume tbescalar prod-
uct induced by the metric on TpM to be Euclidean. Then,
R (0') = RJ.Ir.IXjXkYlIyl
= R12,t (XZX1YIYI- XZXIYZYI + X 1 X 1 }'2yZ- XIX2YIYZ)
= R1Z,I2 (XZYI- YZXI)Z = RIZ,S2.1 = RI2'IZ'
because xaYJ - yaXl = (the area of the parallelogram IT (X, Y)
spanned by X, Y in the Cartesian coordinates (:I:, y) in T p (M2),
Since R12'12 = K (P), we obtain R (0) = K (P), which is what was
required,
Thus, the definition of the curvature along a two-dimensional
direction suggested above is a natural generalization of the concept
of the Gaussian (scalar) two-dimensional curvature of the Rieman-
nian metric. Evidently, the curvatures along any. two-dimensional
direction in R n vanish, i.e. R" is a manifold of zero curvature.

370 {; Tensor A na1ys13 and R l.mann/aJI Geametry
The Curvature along a two-dimensional direction (1 = (X, Y)
admits a dear interpretation, which is presented here without proof.
Let us consider at a point P EM" a two-dimensional plane H spanned
by X, Y and draw from P geodesics Yz (e) along each verlor
Figure 5.40
Z E H, Apparently, these geodesics form (locally) a two-dimensional
surface M1 C M n such that the tangent plane to it at P coint'.ides
with H (see Fig. 5,40). This surfat'.e is caUed a geodesic surface. On a
geodesic surface there exists an induced metric which admits defi.
nition of the Gaussian curvature; it appears that R (a) coincides with
this Gaussian curvature.

Chapter 6
Homology Theory
Thus far. our interest has mainly beer. focused on the Jocal prop-
erties of a smootb manifold, Le, the properties which can be studied
independently in the neighbourhood of each point P of a manifold 111
nnd do not depend on the way 114 is defined as II union of charts. In
many problems, however, it is insuffirient to know only the local
properties of ll'l.
Let us consider, for example, the following problem. Let 114 = 8 1
he a circle with the angular parameter Ip chosen as a local coordinate.
We now try Lo find a smooth function I on 8 1 such that the identity
: =' g (Ip)
(1)
is satisfied, where g is a smooth func,tion on 8 1 , If we solve this
problem in II small neighbourhood of a point P E 8 1 with the r.oordi-
nate 11'0' any antiderivative I (<p) =  g (tp) dip of the function g
may be II so)nlion. Generally, the prblem does not always has a
solution on 8 1 . Indeed, any smooth function on 8 1 can be identified
wit11 a 2n-periodir. fnnetlon of one real variable. Then the function
f is a solution of O\lr problem, provided f (!p) =  g (<p) dip and f is
periodic, Any antiderivative of the function g ('an be expressed in
q>
Lerms of the definite integral f (<p) =  g (Ip) dip + C. The r.on-
<P.
dition that t is periodic can, therefore, be written in the form
<P'T"' g (Cj') dq, = 0 or r g (If) dq>=O,
<D. 0
(2)

372 6, Homology Theory
2n
For instance, if g (q» == 1, then ) g (q» dq> = 2n"* O. Thus, for
o
g (q» 51 1 problem (1) does not have a solution on 8 1 , Condition (2)
is the necessary and sufficient condition for the existence of a solu-
tion of problem (1). Here we see that the existence of a solution
depends essentially on the structure of a manifold as a whole, Sec-
tion 6,1 deals with those properties of a manifold that govern the
global behaviour of a function or a mapping on manifold.
6,1. CALCULUS
OF EXTERIOR DIFFERENTIAL FORMS,
COHOMOLOGY GROUPS
6,(,1. DIFFERENTIATION
OF EXTERIOR DIFFERENTIAL FORMS
Differential calculus of exterior differential forms is
somewhat simpler than differential calculus of arbitrary tensor
fields. If a connection is valid on a manifold, the covariant gradient
of a skew-symmetric tensor field is not, ill general, a skew-symmetric
tensor field. It is natural, therefore, to define the gradient of an
exterior differential form as the composite of gradient covariant
component and alternation with respect to an indices of the tellJjor
field, If the connection is symmetric, the Christoffel symbols do
not appear in the formula for the gradient of an exterior differential
form, i.e. the definition of this gradient does not depend on the
choice of symmetric connection on the manifold, Indeed, let in the
local coordinate system (Xl, . . ., x n ) the differential form m have the
components {mi., ' . ., mi,.}, Then the gradient dm is defined by
(dco)J" ....J+l= (-1) lcrI Vcr(JII+I)CO<l<1,), ...,<I(j,,). (1)
<1
where
OCt>i ll ...,, a. ex.
V,WI.. ., ,i = 8",1 -r'i/'>ClCi" .... ill -.,. - rli"W1h ...,0:' (2)
Substitution of (2) into (1) yields
ilwoU,), .... cr(J A )
(dw)J., ...,j+l=  (_1)l cr l iJzcr{}k+l)
cr
"
-   (_1)1 cr I r(jk..)O(j.)(j)<I(j,) '" cru,_,)cr(j.+l) ... <I(i,,"
o ,.a1

6.1. Cal.ulus of Exlerior Dit!erenlial Forrm
373
The second term vanishes because for fixed sand 0\ there exist exactly
two permutations of indices (ii, " , h+1)' G and 0'. such that
o (it) = 0' (il)" , ., cr (i,-I) = a' (i'.I) and a U."'l) = c' (i'H), . , ..
o (jh) = 0' (h), and also a (iUl) = G' (i.). cr (I,) = cr' UH1); the
parity of these permutations is distinct. Thus, since the Christoffel
symbols r} are symmetric in the lower indices, the two terms corre-
sponding to the permutations cr and c' are cancelled out, so that we
obtain
( d ) . -  ( _1 ) la l
0) J.. .. '}k+l - LJ
a
QID"<i.), '... ,,(J)
0" a(J+I)
(3)
In formula (3) many of the terms appear several times. Indeed, if
cr (iHl) = cr' (iA"'I)' then 0 = a' h, where h is the permutation
which leaves IHI in its own place, In this case
_1)1 a'1 oCt)a'(J. ': . a'(J) (_1)1"1+1 hI 8(o)a"(J,)... "h<J,,)
( ax" (1+1) 8z a (J"+1)
= ( _1)1" 1 8(o)a(j.),., <I(J,,)
ax,, (J "+I)
Bearing ill mind what has just been said (and trying to make gradient
and exterior product compatible operations), we ratain in the com-
ponents of the gradient dw only one term out of identical kl terms,
We al'rive therefore at the following definition,
Definition 1, The gradient of an exterior differential form (0 is an
extel'ior diffel'ential form d(O whose components in a local coordinate
system (Xl, . , " x") are written as
(dO»j" ' '" j"+1
+I aCt)
= 2; (-1.)'" J. . , ,j'-I'+1 ., ,i"+1 , (4)
.==1 8:l-
FOlmula (4) differs from formula (3) by the factor and sign.
Theorem t. The gradient of an exterior differential form satisfies
the following conditions: (a) d (WI A C11 2 ) = dW1 A CO 2 +
(_1)de IP ' l w r f\ dco z , (h) d (doo) = O.
Proof. Let (Xl. . , " x") he a local coordinate system and let (01' 00 2
he two extel'ibr differential forms of degrees p and g, respectively,
Then the components of WI and W z do not vanish only for those
sets of indices (il' . , .. ip) and (ii' ' , "fq) for which i. =1= i" f. =1= 11
fol's::/= J. If i 1 < i z <, . , < ip. I = {il" , "ip} andiJ < 1 2 <, , .
..' < Jq, J = {II' .,', jq}, we pllt WIoI = COI,I.,.", ip' (0.,.. =
W2, j" ", jv' Similal'iy, if 1 1 < . , . < lp+q, K = {II' ' '" Ip"'9}'

374 6. Homoloffll Theory
we put (Wl!\W 2 )X={(,)I!\W.)'.....,' p . Q . By formula (1) of
Sec. 5.2, we obtain
«(,)1 /\ «'a)K =  ( - 1)10(1)' W I ,IW 2 , J. (5)
where summation extends over all Ihe subsets I c:: K consisting of
p elements, I = K"J, and CJ is the permutation which places the
indices of the set I into the first p positions retaining their order.
Formula (4) for the gradient can also be rewritten as
(d",)r= L; (_1)'0(1)/ OIDI''!;) (6)
OX' ,
lEI
where a (i) is the permutation of the /jet I which places index j
into the first position, retaining the order of the remaining indic.fts.
Applying (6) to (5), we obtain
(d ("'1/\ "'J)K
= L; (_1)' 0 1 1 )1 a « (J) I!\ ID.)X>((}
lex 0,,1
=  (_1)'0(1)' l; (_I)ICJ(I)1 i1 (IDI, 1"6)2; K'I'(;)
lEX I ox
=  (_1)'0(1,/,1)1 o (IDJ,I;ID 2 ,J) ,
IC-IU./'U{I) 0:<
In the Jast sum summation extends over all possible partitions of
the set K into three disjoint subsets fi}, J, J, and a (I, J, i) is the
permutation which puts index i into the first place and interchanges
the sets I and J retaining the order of indices within them. Thus,
(d (6)1/\ Cl)2)h:
 ( 1) 101(, I, J)I i1 (IDt, 1-"'2, J)
= LI - ----
iJz/
R/UJU(i)
 (_1)1 0 (1", J) I 0(,)1; I 00 2 , J
KtUJU{I) o:z
+  ( _1 ) ICJ(;,I,J)I(JJ aID.,./'
LJ 1,/ iJzl
XlU/U(I)
= (d(,)1 A Cl)2)K+ (_I)de g w, «(JJI A d(1)K'
=
It is sufficient to prove prQperty (b) (in Theorem 1) for the basic
forms into which any form can be decomposed. We may assume,
without Joss of generality, that in the local coordinate system

6.1. Calculus of E:<lerior DiD"ent!a! Forms
375
(.:t'. . . .. x n ) the form (j) can b wri Hen as (j) = I (x', . . ., XU) dx! /\ . . .
. . . /\ dx R . If condition (b) is valid for the forms (j), and (j)2. i I is
also valid for their exterior product W1 /\ W.. Indeed,
dd (WI /\ (1)2) = d (dw i /\ W 2 + (_1)de g 6), WI /\ doo 2 )
= ddwi /\ (1)2 + (_1)de s (dW,1 d(1 /\ dW 2
+(_1)degOl'dwl /\ dw 2 + (_1)2de g Ol, WI /\ ddw 2 .
The first and last terms are zero by assumption, and the other two
are equal but have opposite signs. It suffices, therefore, to verify
condition (b) for an arbitrary function I and for the form d:l'-. In the
laHer case dd (dx") = O. since x R is also a function on the manifold.
at
We now demonstrate that d (d/) = O. Indeed. (dl) I =  and also
(d(df))1 = iJ(df)J _ iJ(dl)I =_2L..6aO.
J 8:z;1 8 z J IJz/ iJz1 iJzi iJz!
Theorem 1 is proved.
It should be noted that condition (b) relies upon the remarkable
property of a function of many variables: the independence of mixed
partial derivatives of differentiation order, Condition (b) is, in
certain senses, an ultimate extension of this property to systems of
functions that form the components of a skew-symmPtric tensor.
Examples
1. I n a local coordinate system any exterior differential
form can, by definition, be represented as the sum It) =
 WI, "'11' dxll /\ . . . /\ dXil'. Thus, according to Theorem 1.
1.<.. .<Ip
the gradient of the form It) is calculated by the formula
dw C"  d (WI. .. ,Ip) /\ dxl, /\ ' .. /\ dxlp. (7)
1.<. ..<11'
Furthermore, in a local coordinate system the gradient of a !imooth
function. dl. has the components { ...2.L } , Hence, dl=  dx'
IJz/ IJ:z1
and thorefore
dw=
n (6).
'" " . . . I" d i /\ d I /\ /\ d I
.LJ I X Z · , , . x P.
{J:z;
il<...<ipi_t

This formula is, apparently, equivalent to formula (4).

376 6. Homology Th.ory
2. Let us find a differential form (Jj, which satisfies the identitr
dw = Q, (8)
where Q is a fixed differential form. I n each local coordinate system
equation (8) is reduced to the system of partial differential equations
h+1 oro
" ( -1 ) '+1 I, ... f'_1"1 .. .l hH = Q. .,
 J: 1c...1:Iu.J
.I az;'
According to condition (b) of Thearem t, equation (8) has solutions
if and only if dQ = O. Indeed, applying the operator d to the left-
hand and right-hand sides of (8), we obtain dQ = d (dw) = O. In
particnlar, for dl'g Q ='1 the problem is reduced to finding  a
manifold M a smooth function I such that its gradient is equal to a
given form Q. In a local coordinate system (xl, . . ., .1.") the form 2
is represented as Q ""  QI dXi, and equation (8) is equivalent
1=1
to the system
ot
-=QI' i=1, ..,.n.
ozl
(9)
If system (9) has a solution,
respect to the variables xi. we
partial derivatives of a smooth
entiation order. we have
then, by differentiating (9) with
obtain = ao! . Since mixed
lJzl iJz1 lJzJ
fUnction f do not depend on differ-
aQ, 8QJ
-=-, 1i, jn.
oz;J 8 z 1
(10)
Conditions (10), when written without any reference to coordinates,
are equivalent to dQ = 0, We now prove that conditions (10) are
sufficient for system (9) to have a solution in a rather small neigh-
bourhood of any point P of the manifold M. To this end, we
nrst solve one of the equations of (9), say, ;;1 = Ql' We have
x'
i (Xl, '" I x") =  QI (Xl, x 2 , " ., x") dx l + CPt (x 2 , .... x"). (11)
xJ
where 11'2 (.r, . . ., x") is an arbitrary smooth function of the variables
(x 2 , . . 'f x"). Substitution of the right-hand side of (11) into the

6.1. Calculub 01 Ezt<rlor DiDrentl41 ForlTU
377
second equation of (9) yields
""
ot r oQ, ( I 2 " ) d I +  ( 2
7i7 = J oz' .:z;,.:z;",.,.z x oz, x.
:rA
=Q2(.:z;t, ,.., .:z;").
Taking inlo account (10). we obtain
Q. (Xl, .." x")
",'
=  :t (Xl, ..., x") dx t + ::: (X2, ..., x")
",1
=Q.(x l , "., xn)_Q. (. z2, .... Xn)+(Z2, .. " z"),
. .., z")
i.e.
OCP' ( 2 " ) _A ( ' 2 " )
 Z , . . .,;Z: - .'2 ZO' Z , .,..:To .
The general solution of equation (12) is of the form
",'
fl>2 (z2, ".. .x") =  O. (x, z2, ..., z") dz2+ fl>s (:t', ..,. :1:"). (13)
;:
Substitution of (11) and (13) inlo the third equation of (9) yields
(12)
""
Os(x 1 , ...,.:z;")=   (Xl. .,'. x")dx l
"'&
,,'
+  : (x. z2. .. " x") dz2+ ::: (x', .." x").
:4
Again, taking into account condition (10), we obtain the equation
for the function fl>a
Qs (x t . ,.., x n )
",
=  :: (Xl. .... x") dx l
"'
,,'
+ r (jOa ( , 2 " ) d 2 + 8cp ( _. " )
J 0'" XO' x , "" Z x 8%3 "'-, ..., z
"':
= Qa (Xl, . , ., x n ) -Q3 (:I:, z2, .... x") + Qa (x, Z2, ..., x")
n ( " 3 ,, )+ OCP' ( 3 " )
-.'3%0'X O '%"...% ""8';3:&",.,:1:,
8C1'3 ( 3 .:z;" ) - Q ( . '.xS " )
7J%3 X' I ... t - 3 X OJ Xo' ,..., x .

378 6, Homology Th.orll
Repeating Lhe procedure, we arrive at the sequence of functions
<PA (xl', . . ., x n ) defined by the following recurrence relations'
"A
tj)A (.1!', . . ., x") =  QA (x, ..., x -I, 0 , ..., ;r.n) d0

+Cj)It+I(.1!'I, ..., x"),
Ij)nq 5iconst,
.. "A
f(x l , ...,x")=   Qk(X, ...,:4-"0, ...,x")dx"+CJ'n+l'
AI ,,
The last relation is an arbilrar' solution of system (9), which depends
-on one numerical parameter Cj)n+1 = const. Incidentally,
nonuniqueness of the solution of system (9) follows from
general algebraiC properties of an exterior differential
form. Indeed, if f l and fl are two solutions of system (9),
then d/l = nand dl t = Q, i.e. d U I -f 2 ) = Q - Q = O.
The lunction h = 11 -f 2 has the property Ihat its gradient is
identically zero on 8 manifold M. Hence, h is a locally constant
function. If M is connected, then h is a constant function. Thus,
any two solutions of system (9) differ by a constant, and the set of all
.solutions of system (9) is either empty or a one-dimensionallinear mani-
fold in the space of all smooth junctions on the manifold M.
In the general case, we can say that solutions of system (8) salisfy
the following conditions:
(a) the ne€essary condition for the exi$tence oj a solution 01 system
{8) is dQ = 0,
(h) if (J) is a solutton of system (8), any form (J) + dw' is also a solu-
tion of system (8).
6.1,2. COHOMOLOGY GROUPS OF A SMOOTH MANIFOLD
(THE DE RHAM COHOMOLOGY GROUPS!
Systems of the type (8) are very important for studying
the structure of a manifold. It is convenient to formulate the prop-
erties of such systems within the framework of the de Bham coho-
mology theory,
Definition 2, An exterior difIerential form (J) on a smooth manifold
M is called closed if dw = O. The form w is called exact if it can be
represented as w = dClJ'. The quotient space of the space of closed
forms of degree k with respect to the subspace of exact forms is said
to be the (de Rham) cohomology group of dimension k of the manifold
M and is denoLed by HI< (M).

6.1. Calculus 01 Ezlerior Differential Form.
379
Any exact form 00 is closed because db) = d (d(O') = dd (00) = O.
The space of all exact forms is, therefore, a subspace in the space of
dosed forms The cohomology group H h (M) is a vecLor space (gener-
ally, of an infinite dimension). In terms of cohomology gl'Ollps, the
problem of solving equation (8) is formulated as follows, Let (0 be
a closed exterior dHferenlial form of degl'ee k and let I wJ denote
the element of the cohomology group Hk (Ai) equal to the coest
of the form (0 with respect to the subspace of exact forms. Then the
followi ng statemen t holds true.
Theorem 2, Conl1ider the equation
dw = Q, deg Q = k + 1. (14)
(a) Equation (14) has (1 solution il and only if the form Q it clo,ed
and the colwmology class IQI E Hh.., (!vi) is zero.
(b) Any two solutions (J) and w' of (14) differ by a closed form, i.f'.
d (01 - (0') = 0. The set 01 all solutions of this equation is the coset
of the lorm II) with respect to the sub'pace 01 all closed forms of degree I."
(c) The space of all clo,ed forms of degree I, Is Iomorphic to the direct
sum of the space of exact forms of degree k and the cohomology groul'
Hh (M).
Proof. If w is a solution of equation (14), Q is an exact form and,
by definition. [Q! = O. Conversely, if [Q! = 0, then Q is an exact
rorm, i.e, Q = dw for a certain form (0 which is just a solution of
(14). Let 00 and (0' be two solutions of (14), i.e. dw =- Q and dw' =--
L Then d (00 - 00') = dw - dw' = Q - Q = 0, i.e. CJ) - w' is
a closed form, Hence, any solutioll w' of equation (14) can be ob-
lained by adding a closed form to w. Let Qh (M) denote the linear
space of all exterior differential forms of degree k on a manifold M.
Then the gradient d Is the linear mapping
d: Qk (M) ...... Qh+i (M) (15)
from the space of forms of degree k into the space of forms of degree
k + 1. The space of closed forms of degree k coi ncides wi th the kernel
ker d c Qh (M) of mapping (15), Similarly, the gradient d maps the
space Q -1 (M) into Qk (M):
d: h-1 (Ai) -- Qh (M). (16)
Then the spare of exact forms of degree k roincides with the image of
mapping (16): 1m d = d (Qh-1 (M)l c: ker d c: Qh (M), Hence, the
k-rlimensional cohomology group H (M) coincides with the quotient
space ker d/lm d. In the notation of the latter space the same symbol
d stands fOI' two disti net mappings (15) and (16). This docs not, how-
ever, lead to confusion, because the subspaces we are dealing with
arc considered in one space of forms of degree k, QA (M). Let us con-
sider in the linear space ker d the cofactor H' to lhe subspace 1m d c:

380 6. Homology TI..ory
ker d. Then the space ker d is decomposed into the direct slim of its
sllbspaces: ker d = 1m d ED H'.
We now demonstrate that the space H' is isomorphic to the coho-
mology group H (M). Indeed, let </': H' -+ H(M) be a mapping
which associates with a closed form w E H' its coset r wi E H (M).
If q> III)) = 0, then I wI "" 0, i.e. II) is an exact form, This means that
II) E 1m d, and since the subspaces 1m d and H' intersect with respect
to zero element, we have w = 0, so that the mapping'll is a mono-
morphism, Let x E H(M) be an arbitrary element, then x is the
coset of some form wE ker d with respect to the subspace 1m d. x =
[II)J. Since the space ker d can be decomposed into the direct slim
of the suhspaces 1m d and H', the form w can also be decomposed into
t.he stirn II) = dQ + (1)', dQ Elm d, (1)' E H', so that x = r wi =
r (I)') = Ip (11)'). ThIlS, 'II is an epimorphism. We have proved t!JIII Ip
is an isomol'phism, which completes the pl'oof of Theorem 2.
Examples
I, Let a manifold JIll be represented by an open real in-
terval, M .., (a, b). Calculate the cohomology groups of this interval.
Since the manifold lit is one-dimensional, only the spaces of the
forms of degrees 0 and tare difierent from zel'o. Consider first the
space 0 0 (M) of the forms of degree 0, Any such form is a smooth
function on the interval (a, b), and its gradient df is wdUen as df =
 (x) dx, where x is the Cartesian coordinnte on (a, b). Therefore,
the space of closed forms of degree 0, i.e. ker d, consists of all fUllc-
tions such that  (x)  0, j,e. ker d consists of constant lunctions.
Hence, the space ker d is isomorphic to the one-dimensional space
RI. The space Qo (M) does not contain exact forms and, therefole,
HO (M) = ker d = RI. Let LIS now consider the space 0 1 (M) of the
forms of degl'ee 1. Since any form of degree 2 on a one-dimensional
manifold vanishes, ker d coincides wilh S1 1 (M). Calculate 1m d.
Let (I) E 0 1 (M) be an arbitrary differential form of degree 1. In
a local coordinate system it is written as (I) = g (x) dx. 'If df = w,
then : (x) dx "'" g (x) dx, i.e. ; = g. Hence, lhe function f can he
defined by
'"
J(X)=g(x)dx, xE(a,b), a<c<b. (17)
e
Thus, any form II) E 0 1 (M) can be represented as w = df (for an
appropriate function I). This means that 1m d = 0 1 (M) = ker d.
Then the one-dimensional cohomoJogy group HI (M) defined as the
quotient space ker dllm d is zero, HI (M) = 0, For other dimensions

6.1. Calcu/u. of E:denor Dlffer.nltal Form,
381
k ;;a. 2 the cohomology groups HA(M) vanish because even for
k ;;a. 2 the spaces of the forms of degree k are zero on a one-dimen-
sional manifold M. Thus, for k;;a.1 we have HO(M) = Rl, H"(M) = O.
2, Let M = 8 1 be a one-dimensional circle, As in Example 1,
we find that the zero-dimensional cohomology group HD (8 1 ) = Rl
and the groups 9"(8 1 ) = 0 for k ;;;;. 2. It remains to calculate the
one-dimensional cohomology group HI(SI). Since 8 1 is one-dimen-
sional, the kernel ker d coincides with !.".II (SI). Let <p be the local
angular parametel' of the circle SI, Solve the equation dt = (0) for
a certain form w = g (<p) d<p. The function f (q» must be 2n-periodic

illlhe parameter <po Using formula (17), we obtain! (<p) = J g (q» drp
°
2n
and ,g (cp) dq> = O. Thus, not every form w E Ql (SI) is contained
'"0
in the gradient image 1m d, but only the form satisfying the con-
2"
dition  g (<p) dq> = O. We now demonstrate that the space of the
o
forms QI (8 1 ) can be decomposed into the direct sum of its sub-
spaces: Ql (SI) = 1m d ffi R 1 , the second term consisting of forms
with constant coefficient, g (Ip) EI canst. Indeed, let w = g (<j» dip
2"
be an al'bitrary form, c = ;n  g (cp) dcp. Then the form w' =
o
2tt
g' (<p) dIp = (g (cp) - c) drp Jies in 1m d because ) g' (q» dIp = 0,
o
On the other hand, any form w = g (q» dip with constant coefficient
2"
g (cp) <E5 C = canst does not lie in 1m d for c *' 0, since  c dq> =
o
2nc *' O. Then the factor group H I (SI) = ker d/lm d = QI (8 1 )/
1m d = R I . Thus, we finally obtain HO (8 1 ) = HI (S1) = RI,
H'«SI) = 0 for k;;a. 2.
6.1.3. HOMOTOPIC PROPERTIES
OF COUOMOLOGY GROUPS
Cohomology groups of smooth manifolds possess some
properties which may prove lIseful in desc\'ibing these groups, Wo
note rllst of 1\11 that each smooth mapping of a manifold induces the
inverse mapping of an exterior differential form, namely, t: Al I -+
M 2 . 1*: QA (M.)...... Q (M 1 ),

382 6. Homology Theory
Definition 3, Let /' M 1 -+ M 2 be a smooth mapping and let CIJ E
QA (M 2) be an exterior differential form of degree k on the manifold
M t, The inverse image I'" (00) of the lorm (0) is an exterior differential
form on M 1 defined by I'" (00) (61' . . " Sh) = <II (df (61)' ' 0 ., df (SII».
where 61, . 0 ., Sh E T p (M 1 ) are tangent vecl.ors at a point P of
M 1 and dl (61). . . " dl (k) E T f(I» (M 2) are their images under the
mapping df. '
The mapping f.: Qh (M 2 ) ..... Ok. (M 1 ) is, apparently, a linear map'
pi ng of a linear space.
Theorem 3, The inverse image of a differential form satisfie8 the
lollowing conditions:
(a) (gf)'" =I"'g"'. f: M 1 -M z , g: M2.....Ma.
(b) f*d = df"',
(18)
" (19)
(c) the mapping j* sends ker d into ker d and also cohomology classes
into cohomology classes: 1*: nil. (M 2 )..... nil. (M l ),
(d) if in local coordinates the mapping f is written as a system of
functions yi = r (xl, 0 . " x"), and the form
00= 2:001, .. .111. (yl, 0' 0' ym) dyl. II ..0 II dylk.,
then
I. (<11)= 11 <III, 0'0 ill. (/1 (zl, '0" xn), ..., r(x l , ..., x"»
,dl' (zl, ..., x") II ... IIdl A (Xl, ,.., z"). (20)
Relation (20) is a generalization of the so-called invariance prop-
erty of the differential of a function, which states that the differen-
tial of a function y = f (x) does not depend on whether x is an in-
dependent variable or, in turn, is a function of another variable:
df
dy = (j; (z) dx.
We begin the proof of Theorem 3 by verifying relation (20). Let
SI = (W, ..., 611 = (.) be tangent vectors at a point P of the
manifold M 1 referred to a local coordinate system (Xl, . . ., x").
By definition, we have
I*(m) n. 1 . . . '. 611) = 00 (df (Sl)' . . ., df (611»' (21)
The components of the vt!ctors 111 = df (6/) in a local coordinate
syslem (y1, . . 0' yrn) of the manifold 1v1 2 are of the form
. '" ufj It. .
1J = L.J i1zlt. i = dj1 (St).
It.

6.1. Calculus 01 Ezterior D [Derential Forma
383
Thus,
f'" (w) (SI' .." ill) =  Wi" ...,;" T\{'T\' .. . 111I
= 2j Wi, ... ill dl i . (tl) , . . dill (SA)
= (2j Wi, ... ill. dl;' /\ .., 1\ alII) (ii' ' . ., A)'
Formula (20) is proved, Relation (18) directly follows from (20) in
any local coordinate system. Let us prove relation (19). According
to (20), if
W=WI,...IA(yt, ...,ymJdyl,/\ ...I\dylll,
then
j* (w) =  WI,. ..Ill. (il (Xl, ..., x"), ...) dl' /\ ... 1\ dlll,
We have therefore
dw =  dWI. " . III (yl, ..., ym) 1\ dyl. 1\ ' .. /\ dylll
"" 86)1.... ill 1 ; I III
= .t.J . (y , .." y"') dy 1\ dy , /\ ." 1\ dy ,
ayJ
aWl .
f* cro)=  ..;. 'II. (/1 (Xl, ., ., x"), .. ,) dji /\ afl. 1\ ... /\ at'll
iJy
=  d (WI, ...1 11 )(1 1 (Xl, ..., x"), . ..)} /\ al' 1\ ... 1\ at'II, (22)
On the other hand.
al. (00) =  d (m!,... III (II (Xl, . . .. x") ..,)} 1\ all, 1\ ... 1\ at'lI, (23)
i,e, the right-hand sides of (22) and (23) coincide. Finally, condition
(c) follows from (b). Indeed, if 00 E ker d c: 011 (M 2)' then aCii = 0
and elf* (w) = f*a (w) = 0, i,e. I'" (m) E ker d c: 011 (M 1 ). The co-
homology class [w] E HII(M 2) is the coset of the form 00 with respect
to the subspace 1m a c: ker a c: 011 (M 2 ). The image of this coset
under the mapping 1* consists of the forms 1* (00 + dw'), 00' E
!;h-l (M 2)' According to (19), all such forms are expressed as
/"'(00 + dw') = /*(00) + dl*(w'), i.e. they lie in the coset of the
form f*(m) with respect to the slIbspace 1m a c: ker d c: Qh (M!).
Thus, the homomorphism 1*; Hk.(M 2 ) -+ HA(M!) of the cohomology
groups of M 1 and M 2 is dermed correctly, By virtue of condition (a),
the relation (gf)* = j"'g* also holds true for mappings of cohomology
groups. The theorem is proved.
Theorem 3 is useful in many respects. For example, if two mani-
folds M 1 and M 2 are diffeomorphic, their cohomology groups arc-

384 6. Homo14g11 Thr"
isomorphic. Indeed, if /: M 1 --.- M 1 is an identity diffeomorphjsm,
the homomorphism I"': Hk(M l ) __ Hk(M l ) is an identity isomor-
phism. Thus, by choosing an arbitrary diffeomorphism q>: M 1 -ilf 2
and calculating the inverse diffeomorphism 1jJ: Ma -- M H we obtain
two possible mappings 'Ij11p: M l _ M l and cp",: Ma - Ma, both
being. identity dilfeomorphisms: 'Ij1q> = 1M. and q>1j> = 1 M ,. Using
then Theorem 3, we find tbat CP*'P* = 1 Hk (M,), 'Ij1*Ip. = 1 H II{M,)'
i.e. the homomorphisms cp* and'll. ace mutually inverse and, there-
fore, cp* (as well as '11*) is an isomorphism of cohomology groups.
It turns out, however, that the cohomology homomorphisms t"':
Hk(Ma) _ H"(M 1 ) induced by a smooth mapping I: JIIl 1 - Ma
possess Bven more stringent properties.
Definition 4, Let X, Y be arbitrary topological spaces and let I
be a unit segment of real numbers. The continuous mappigg I:
X X r,y is called a homotopy between the mappings 10 =
j I (X X {O» and 1 1 = / I (X X {1}). If X and Yare smooth mani-
folds Bnd I is a smooth mapping, the homotopy I is called smooth,
.and the mappings /0 and 11 are called homotopic mappings.
Remark, If X is a smooth manifold, the Cartesian product X X [
is not a manifold (rigorous definitions Bre given in the next section).
Yet, the concept of a smooth mapping holds true for a Cartesian
product. Indeed, if Uti. c: X is a chart and (x, . . 't x:) is a local
coordina te system, the mappi ng / on the set Uti. X I is represented
as a function of (n + 1) variables, t = t (x, . . ., x::. t), tEl =
10, 11. The function / (x:', . . ., x;':, t) is said to be smooth if in lhe
neighbourhood of each point (x:], . . ., x, t), 0 < t < 1, it is smooth
.of class C"', and all its partial derivatives can be extended con-
tinuously to the boundary points (.x, . . .t x::, 0) or (x. ' . '. x::. 1).
This definition does Dot depend on the choice of the local coordinate
system, and the restrictions / I (X X {O}), I I X X {i}) are smooth
mappings of X into Y.
Theorem 4. Homotopic smooth mappings /0' 11: M 1 - Ma induce
the same homomorphIsm 0/ cohomology groups I: = Ii: H(Ma) -
H k (M 1 ),
Proof. Let us construct the linear mapping of the spaces of exterior
differential forms D: Qk (M a ) - Qk-l (M l ) such that the identity
(/: -I:) (6.» = (dD :f: Dd) (w)
(24)
is satisfied for any form w E Q" (M a)' If w is a closed form represent-
ing the cohomology class (w) E H"(M 2)' the forms /: (w) and
Ii (00) are also closed and represent the cohomology classes I ([6.>})
and I (I wI). By virtue of (24). the difference I: (6.» -/ (6.» is
equal to (dD :t: Dd) w = d(Dw):f: D (dw) = d(Dw) becausedw = O.
The forms l: (w) and Ii (w) belong, therefore. to the same coset

6.1. CalculuR of E:derior D iff_rent/at For""
385
with respect to the subspace of exact forms, i.e. they represent the
same cohomology class If: (w)J = {f (w)J. Thus, I ([wJ) = r" (f (i)J).
It remains to construct the mapping D satisfying identity (24).
Since the mappings /0 and II are homotopic, there exists a smooth
mapping F; M 1 X J _ M 2 such that F (P, 0) = fo (P) and
F (P, 1) = 11 (P). Let (i) E Qh (M 2) be an arbitrary form of degree k
and Q = F* (€I). The form Q is defined at least on the manifold
M I X (0, i), and in any local coordinate system all the coefficients
of the form Q caD be extended continuously to M 1 X 10, 1J. A form
Q of degree k on the manifold M 1 X J is said to be independent 0/ dt
if this form vanishes on any system of vectors (61, . . ., 6h-l' f,).
In the local coordinate system (xt, . . ., xn, t) the condition that the
form Q does not depend on dt means that it can be decomposed into
the sum Q = ZQ.... .ih (xl, ..., xn, t) dx!' /\ . . . /\ d:/}h, i.e. the
decomposition does not include terms with dt. In this case the form
o can be interpreted as a family of exterior differential forms OM. (t)
of degree k (on the manifold M 1 ) continuously dependent on the
parameter t. If CPt: M I - M 1 X I is an embedding CPt (P) = (P, t),
then OM. (t) = cp1 (0).
Lemma t. Any exterior differential/arm 0 0/ degree k on the mani-
fold M 1 X I admits the unique representation
Q = 0 1 + 0, /\ dt, (25)
where the forms 0 1 and 0, do not depend on dt.
It is sufficient to prove the lemma in each chart U a. of the manifold
M). Indeed, the independence of a form of dt is invariant relative
to the choice of the chart and is valid at each point. Therefore, de-
compositions of type (25) realized in each chart coincide in the inter-
sections of the charts. Since the condition of independence of dt is
invariant under linear operations over forms, it suffices to verify
the lemma for special forms Q = / (xl, . . ., xn, t) dxil /\ . . . /\ dx'k
or 0 = f (Xl, . . " :en, t) dXi, /\ . , , /\ dXi_1 /\ dt for which the
lemma is obvious.
Let us now define the mapping D. We put
I
D (w) =  Q., M, (t) dt,
o
(26)
where Q = F* (w) and 0 = 0 1 +0 2 /\ dt is the decomposition by
formula (25). If the form Q on M) X I does not depend on dt, its
rallient dO can also be decomposed by (25) into the sum dQ =
d"QM, +  OM, /\ dt. Here d", denotes the gradient of QM. on the
manifold MI' We now prove formula (24). The left-hand side of (24)

386 6. HomolollY Theory
can be obtained as follows. First, we calculate the form Q = F* (w)
and then calculate its restrictions to the submanifolds M 1 X {OJ
and M. X (1) under the embeddings !Po: M 1 -+ M 1 X I snd !PI:
MJ-+M J X I. In 8 local coordinate system Lhe embeddings!po
and <PI are defined by the functions
! I=XI
<Po:' ,
x"=x n
I e=O
XI=XI
<PI:
z»=x'n
t=1
In other words, t = 0 or 1 and dt = 0 should be substituted into the
form Q. This means that
f (co) = QI, M. (0). IT (co) = QI. M. (t),
(27)
To lind the right-hand side of (24), we calculate successively Dd (w)
and dD «(I). We have
F. (dw) = dF. (W) = d (Qs + QI A dt)
q
= d",QI, M. (e) + 8TQI, M. (t) 1\ de + d,.Q2. M. 1\ dt.
whence
1
Dd (w) = ) (  Qj, M, (t) + d,.Q2. M. (t») dt
o
1
=QI. M, (1)=QI, M, (O)+d lO ! Q2, M, (t) de. (28)
o
On the other hand
I
dD (00) =dl/:  Q2. M. (t) de,
o
(29)
Comparison of (27) and (28), (29) yields t; (w) -I: (w) =
(Dd - dD) (w), which is what waS required.
Corollary, If M = R n ill an nod/mensional Euclidean Ilptu:e, then
HO (M) = HI and H" (R n ) = 0, k  1.
Proof. For n = 0 the manifold M consists of a single point and
the statement is obvious, since a single-point manifold does not
have non-trivial forms of degree higher than zero. And the forms of

6,2, Inlgrallon of Ezlerlor ForrM
387
degree 0 are functions on M, i.e, just real numbers. Thus, Qo (M) =
Rl, the kernel of the gradient, coincides with Qo (M), and the image
of the gradient vanishes. Hence, HO (M)= ker dllm d = Qo (M)/O
= Qo (M) = Rl, Let now n > 0 and let Mo = RO be a single-point
manifold. Consider two mappings <p: Mo -- M, <p (Mo) = 0 E M,
and '1': M -- Mo. $ (M) = 0 = Mo. Thus, the mapping q> sends
the only point of the manifold M 0 into zero vBCtor, and the mapping
't' sends the entire Euclidean space M = R" into one point Mo.
Let us consider two possible mappings 'I:<p: Mo -- Mo and <p'ljl: M__
M. The mapping 1jJ<p is, apparently, an identity transformation of
Ihe single-point manifold Mo, while W maps the entire Euclidean
space M into zero vector, We now demonstrate that the mapping
q>'IJ is homotopic to an identity mapping of Euclidean space. Con-
struct the homotopy in the explicit form, F: M X 1 __ M, F (x, t) =
Ix, x EM = Rn, t E [0, 11 = 1. For t = t we obtain the identity
mapping I J (x) = X = F (x, 1) and for t = 0 we have W (x) = 0 =
F (x, 0). According to Theorem 4, the mapping qn!> and the identity
mapping both induce the same homomorphism of cohomology groups
Rnd, therefore, the homomorphisms ($cp)"': HI< (M o ) _ HI< (M o ) and
(cp'ljI)"': HI< (M) __ HI< (M) are identical isomorphisms of groups.
Since (Ip'ljl)'" = $"'CP'" and (,pip)'" = CP*$"', the homomorphisms cP'"
and 'ljI'" are mutually inverse isomorphisms of groups. Hence, a
Euclidean space has the same cohomol0fi,y groups Hit (R") as a
single-point space, i.e. H O (R n ) = RJ, H (R") = 0, k  L The
corollary is proved.
This corollary has another formulation known as Poincar's lemma.
Theorem 5 (Poincare's lemma). Any closed form Q on a manifold M
is exact in a sufficiently small neighbourhood of each point P EM,
i.e. Q = dll),
Proof. Consider a neighbourhood U of a point P E M homeomor-
phic lo Euclidean space R". Then we have H (U) = 0 for k  1.
Hence, if Q is a closed form on M, the cohomology class !QJ vanishes,
which means that the form Q is exact in the neighbourhood U.
Theorem 5 is proved.
6.2. INTEGRATION OF EXTERIOR FORMS
Integration of an exterior differential form is an analogue
of the operation inverse to differentiation of a function, Let US con-
sider an example. Let an exterior differential form II) of degree n
be given in a bounded domain V of a Euclidean space R n . Then in
the Iocsl coordinate system (.xl, . , ., x n ) the form (i) is represented
by
(J) = f (Xl, . . ., x n ) dxl/\ dx /\ . . . /\ dx n , (1)

388 6. Homology Theory
In some other coordinate system (yl, .." yn) the form will be
written, according to the transformation law of tensor components, as
tJ) = f(x l (yl, .,., yn), ,.., xn(yl, ..., yn» det (  ) dY' !\ . .. !\ dUn,
{Jyl
(2)
We see that this formula resembles the formula for the transfor-
mation of an integrand function under coordinate substitution (in the
integral calculus of functions of many variables). Namely, the
values of the integral of a function lover a volume V in different
coordinate systems are related by
 f (Xl, ,.., :r n ) d:r l . . . d:r n
v
=  f(:r l (yl, ..., yn), .. ,) I det ( :;; ) I dy' ... dy", (3)
v
Formula (3) shows that it is not in fact the function I that is inte-
grated over the volume V, but a certain geometric object whose com-
ponents are transfonned under coordinate substitution (Xl, . , ., x")_
(yl, . . ., yn) Xi a multiplication by the modulus of the Jacobian of
this substitutIon. This argument caD be used to define the integral
of an exterior differential form.
6.2.t, THE INTEGRAL
OF A DIFFERENTIAL FORM OVER A MANIFOLD
Let us now return to formula (3) and extend it to an
arbitrary smooth manifold M with boundary A set Xc M is said
to be of zero measure, j.1 (X) = 0, if for any chart Uo. and the coordi-
nate homeomorphism (fI..: Uo. c V.. c R the set <Po. (X n U..) c
R is of zero measure in Euclidean space R n . The definition of a set
of zero measure is correct and does not depend on the choice of the
atlas. This can be proved by showing that if j: VI _ V z is a smooth
homeomorphism of the domain VI c: R n onto the domain V z c: R",
then the image f (X) of any set X c: VI of zero measure is also a
O-measure sel. Indeed, if X c: VI is a set of zero measure, fl (X) = 0,
it can be represe!1ted as the union of not more than a countable num-
ber of compac subsets: X = U XI, '" (XI) = O. Theil, '" (/ (X» 
ic:a1
 j.l (f (X ,), and it is therefore sufficient to prove that", (j (X i» = O.
71

6.2. Infegrill/on 01 E:eterlor Forrn&
389
In a loca) coordinate system (xt, . . ., x") the mapping I is rep-
resented as the set of smooth functions y; = II (Xl, ' , ., x"). Thus,
 (/ (X,»  I det ( ::; ) Idx ' ... dx"C  dx l ,,' dx"= Cft (X,).
x, 
where C is a constant satisfying
dati ( ::; ) IC
on the compact X ,. A set of zero measure on a manifold obeys the
ordinary properties of a set in a Euclidean space;
(a) ft CQI XI)II f1(X,) =0,
(b) the image I (Y) of a smooth (n - i)-dimensional manifold Y
obtained by the mapping f; Y _ M is of zero measure.
Let us now define the integral of an exterior differential form (I)
of degree n on an n-dimensionsl oriented manifold M when the form
(I) has a compact support, Let U c M be an arbitrary open domain.
If U is entirely contained in some chart U a. with coordinates
(x:', . , " :1::;), the form (I) can be expressed in the chart U a. as (j) =
fa. (z,:, . . " z:1) dx f\ . . . f\ dx. We put then
\ (0)=  ..,  fa. (x:', ..., x) dx:' . .. dx:;. (4)
ii U
The right-hand side of (4) does not depend on the choice of the local
coordinate system (x':, ..., zl;). Indeed, if U c Ua. n U, it
follows from formula (2)
w= fa. (x (xp, ..., xII), ..., x:; (xp, ..., x8»
( iJZ 1 )
X det  dxb A ... A d.xj!
iJ"'tJ
= /P (xA, ..., xU) dx p A .. . dx/!,
Using (3) and taking into account that for an oriented manifold
det ( II% ) > 0, we obtain
IIz&
) . u'  Ia. (x:', ..., x:) dxf,. .. . dx
=  . i/  ill (xp, ..., xN) dxA . .. dx/!.

390 6, H omolo8lf Theory
Definition 1, Let M be an n-dimensions] smooth oriented meniCold
with boundary and Jet «) be an exterior differential form of degree n
with compact support. Furthermore, let{U... ..., Uo. N } be a
. 1
fimte set of charts covering the support of the form «). We put then
N
) «)=   III/_I' (5)
M II (u/1;/ \ i1 u../
Definition 1 does not depend on the choice of the charts U0. 1 , . . .,
U /1;N' To demonstrate this, we give another definition of the integral
of a form III.
Definition l' , Let M be an n-dimensional smooth oriented manifold
with boundary, let (J) be an exterior differential Corm of degree n
with compact support, and let {!po.} be a partition or unity subordi-
nate to the atlas {U..}. Then
S 111= l1 S '1>0. 111 . (6)
M .. u..
Definition l' is mOre convenient, lor it reHes upon integration only
over an open domain in II EucI1dean space, whereas in formula (5)
integration is perlormed over a more r.omplicated domain, On the
other hand, in Definition l' it is required to prove that the right-
hand side of (6) does not depend on the choice of the partition of
unity.
Without loss of generality, we may asume that the atlas U,., is
finite, 1  IX  N. Let {} be another partition of unity, Putting
N
1/1/1; = CPo. - '1>;', we obtain l: 1/>,. 53 O. I t is necessary to demon-
e<c=1
strate that
N
  'i'..IIJ=O.
..I urlU..
(7)
1'1'-1
We have 1/>N =.' - l: 1/'..; since SUPP 1/>N c: UN' there exists another
.....1
function X. which is identicaIly 1 on SUPPWN' SO that suPPXc:U N .
N-I
Then, X (P) "'N (P) 51 WN (P) and therefore 1J!N = -  x1/>.. ,
..=1
supp X,1/>.. c: U H n U u' It follows that
N N-I 1'1'-1 lV-I
  'P..CJ)= ) "'N IIJ +  ) 1jI/1;1II= -   x1J'o.«)+  } 1ji,,I.
,,I Uo. UN ..",I Ug. ..I UN 0.=\ U"

6,2. Integral/on 01 E.,ter/or Forms
391
Since supp x1\1..(,) c: UN n U,, ' we have  x1Jl..w =  x1\1..(o), whence
ul'o' u..
N 1'1-1
h  1\1..w = h  (1/>..- x1\1..) w, (8)
(Z.tZRl U a. a-I U a:;
The right-hand side of (8) includes (N - 1) terms and coincides with
(7) in which the functions (1/!,. - x1\1..) arE' substituted for 1j1a. We
have
N-t 1'1-1 1'1-1 1'1-1
 (1I'..-x1l'..)=  1I'..-X  1/>..=  1j1..+;("'N
B-.:=-t (X.s:::::ct «ea! IXs::::Ii
1'1-1 N
= 2i "'..+1/!N= 2: "P",=O.
(1:::11 a.-I
Thus, formula (7) is proved by induction with respect to the number
of terms N, Let now there be given two atlases {U..} and {uiI} with
the corresponding subordinate partitions of unity {<p..} and {!pil},
Each partition can be completed with zero functions to the partition
of unity of a new atlas equal to the union {U..} U {UP}. The zero
functions do not alter the right-hand side of (6) and (ormula (7)
implies that the right-hand sides of (6) coincide for the partitions
of unity {q>a} and {q>iI},
Proposition 1, (a) Definitions of the integral by formulas (5) and
(6) coincide.
(b)  P"IWI+A2W.)=AI  0)1+A2  (0).. AI' A2ERI.
M M M
(c) if the orientation of M' is opposite to the oril!ntat/on of M,
then  W = -  ((I.
M' M
Proof. Property (b) is obvious both for formula (5) and formula
(6). It is sufficient, thel'efore, to prove that (5) and (6) coincide for
forms with a sufficiently small support. If supp 0) c: U al .. it
follows from (4) that  ((I =  (oJ and in formula (5) we have
u unu.. l .
N 1'1
  (0)=   W,
i1 \-1 'I i-I
(U.. 1 , J!;!1 U.. J ) U"I.()(UO:I'j1 U,,}
( I-I )
Since aU the sets V.. I . n U"l""'" J' U"'/ belong to the same chart
nd do not intersect pairwise, the sum of the integrals is equal to the

392 6. HomoloEII Theory
j n tegral over the union
U (Val n(va,'<l/V aJ ) } = Va; n ( U (Ua,<UIUa/))
io1. ,=1 G b:::d Jt=IIl
=V al .n M =V(1.lo'
 ro. On the
u al .
other hand, since BUppro c: U al ., the partition of unity can be so
chosen that supp '1'.. n Supp ro = $25 for a * al.. Then 'I'a" =-1 on
supp ro, and formula (6) becomes S (,) = S q>al.w =  (,). State-
M u a " U(1.I,
Hence, the right-hand side of (5) is of the form
ment (a) is proved.
We now prove statement (c). The opposite orientation on the
manifold M can be defined without changing the atlas, but changing
only the coordinate system in each atlas V..: namely, (y:',. . ., y:;) =
(-.1:1., x, ' . '. xm. For the new orientation we have by formula (4)
ro
U'
=5
, .. 5 f (y, " ., y) dy ' . . dy
U'
= S . u' S f (y1. (x,!., .,., x), ...) 1 det ( ::r ) I dx:' . , , .
Since
I.. (x:'. .,., x)=f;'(Y:', (x;", ..., ), ...) det( ),
we have
J, 0)=  f.. (x:', .... x) sgn det ( ::t ) dxl. ,'. 
== -  fa (x:'. ..., x) dx:' . . . dx;l = -  ro.
Hence, the sign of all the terms in (5) will also change if the orienta-
tion of M is altered. Proposition 1 is proved.

6.2. Inlegration 01 E:rterior Forms
393
6.2,2, STOKES' THEOREM.
We now prove the fundamental formula which generalizes
numerous integration formulas of mathematical analysis.
Theorem 1. Let M be an n-dimensional oriented manifold with
boundary 8M and let (j) be an exterior differential form of degree
(n - 1) with compact support. Then
(_1)n  dw=  (I).
M "lIf
(9)
Formula (9) is called Stokes' formula. Before turning to the proof
of Theorem 1, we discuss particular cases of this formula,
Examples
t. Consider in the plane Ri a smooth closed curve I'
without self-intersecti<?ns which bounds an open domain V in R,
Let on r there be valfd a parameter t defining the circumvention
Figure 6,1
direction and, therefore, the orientation of r as a one-dimensional
manifold. Then the closure 17 is an oriented two-dimensional mani-
fold with the boundary oV = r. If the orientation of the domain V
is delined by a linear coordinate system (xl, x), the orientation on
the boundary r will be compatible with the orientation of V, pro-
vided the domain V lies on the left of r when r is traversed in the
direction of increasing parameter t (see Fig. 6.t). Let (I) be an arbit-
rary form of degree 1 on R2. In the coordinates (Xl, x) the form (j)
is represented as IJJ = P (Xl, x 2 ) dx l + Q (Xl, x 2 ) dx 2 . Then the

394 6. Homol0811 Theory
integral of 6> along the curve r coincides with the integral of the
econd kind
) 6>=  Pdxl+Qdx z
r r
r,
=  (p (.1: 1 (t), XZ (t»  + Q (Xl (t), x. (t» . ) dt,
By formula (9) we have
S 6>=  d6>= S (dP /I. dxl+dQ /I. dx2)
r v v
=  (#-) dx l /I. dx.= S S ( : - ::. ) dx1dx z .
v v
Thus, we obtain Green's formula in two dimensions
 (Pdxt+Qd)= S S (-:£ - ::. ) dx l dx 2 ,
r v
2. Similarly, let r be a smooth closed curve without self-inter-
sections in Rat and let this curve be the boundary of a two-dimen-
sional surface V. By formula (9) we obtain Stokes' formula in three
dimensions. Let us consider a form 6> of degree 1 in a three-dimen-
ional space R3
(J) = P (Xl, .:1: 2 , x') rJx1 + Q (Xl, x 2 , .r) dx 2 + R (x., r, x 3 ) dr3.
Then the integral of the second kind over the curve r may be inter-
preted as the integral of the form e.>
r,
f r ( do:l dz l dz s )
J 6>= J P (Xl, xZ, x3) dt +Q (Xl, x2, x3) di"" + R (Xl, ,zZ, x3) dt dt,
r I.
From formula (9) we have S 6> = S d6>,
r v
de.> = dP /I. dx l + dQ /I. dx2 +dR /I. dz3
= ( .!!L -  ) dx' /I. dx2 + ( !!i - .2£. ) dx. /I. d3,03
110: 1 iJz' oz' az
( oP oR )
+ - -....-.- dx 3 /I. dxl
oz' ,,0:' '

6.2, In'egrat!on of Ez/erlar Fornu
395
Thus, the integral along the curve r is expressed in terms of the
integral of the second kind over the surface V
J (P dx!+ Qd:r?-+ Rdx 3 )
r
= U { ( .!!L - !!!.... ) dx. dx 2 -I- ( ..2!! - .!2... ) dx2 dxl
J J liz' IIz' az' al;'
v
+ ( !!... - .E!. ) d dx! }
.z:3 8Z1 .
3. The lasl relation, the Gauss-Ostrogrcuhky formula, is also a
particular case of (9). Let r be a closed surface in R3 which bounds
a domain V, and let ro be an exterior differential form of degree 2.
In coordinates (xl, Z2, x 3 ) ro is represented as
ro = Pdx' 1\ dx. -I- Q dx' 1\ dx3+ Rdx S 1\ dx'.
Then lhe integral of the second kind over the sUl'face r may be inler-
preted as the integral of the form ro. Applying formula (9), we
oblain
 ro = -  dro,
t' v
.=UAAdx2+Adx2A+mAA
( Up aQ aR ) !
= /)z' +8Z1+ dx A dx2 A d,
whence
3) (P dx! d:r;2+ Qdx. d+ R d:TJck!)
r
= - ) ) ( : + : + : ) dx' d:r;2 dz3.
'fhns, we have arrived at lhe Gauss-Ostrogradsky formula wilh the
opposite sign. The minus sign appears because in the classical
Gauss-Ostrogradsky formula the orientation of the surface r, defined
by a pair of vectors (e" e.) in Lhe tangent space to r, is so chosen that
the third vector e. is directed outward to the domain V.
b
4. Finally. the Newton.Leibniz formula  2 (x)ck=/(b)-f(a)
can also be considered as a particular cas: of formula (9). The
left-hand side of the Newlon-Leibniz formula may be interpreted
as the integral of the form dl =  dx over the real segment [a. b]
which is a one-dimensional oriented manifold with boundary, The

396 6. H omoiogv Theorv
boundary of the segment [a, bl is represented by two points {a, b}
which we shall consider as a a-dimensional manifold. Thus far,
however, we have not defined the concept of orientation Ior a 0-
dimensional manifold because such a manifold does not have a tan-
gent vector. But we can proceed in the rolJowing way. A point of
the boundary of a one-dimensional manifold M is assumed to have
positive orientation if the vector (at this point) directed inward to M
gives the initial orientation of M; on the contrary, this point has
negative orientation if the inward vector gives the orientation
opposite to that of M. A zero-dimensional form on a manifold is
simply a function. Thus, the integral of a function over a zero-
dimensional manifold is the sum of the values of the function at
points, the sign of the values being taken in accordance with orien-
tation of these points on the manifold. I n our case the right-hand side
of the Newton-Leibniz formula may be interpreted as the integral
of the (unction f over the boundary consisting only of two points
{a, b}, the point a having positive orientation and the point b
negative orientatioll, Hence, ) I = J (a) -I (b), and formula
8[a, hJ
(9) takes the form  dt = (-1)'  j.
la, h) 810, b)
Prool 01 Theorem 1. Since the two sides of formula (9) are linear
with respect to the form 10, we can decompose w into the slim w =
WI + . . , + w N and prove the formula for the case where the SUPPOl't
of w is compact and belongs to one chart, All the more, it is sufficient
to prove formula (9) for w of the form
w= I (Xl, ..., x") dx. A ." 1\ d:d'-. A d:r!'+1 A ... A dxn,
where f is a function with compact support defined in R. In this
case cU.I (_1}h-1 oa;,. dx l A ... A dx", Suppose first that k < n,
Then the restriction of w to the boundary RO-I c: R vanishes-
because dx n = 0, so that 3 w = O. On the other hand,  dw =
d-I R+
 .  (_1)h-1 :!h dx l .. . dx n . Going over to the multiple integral,
first with respect to the variable :r!' and then with respect to 111\
other variables, we obtain
..
 cU.I =) ...  dx l . . . d:r!'-I d;r!<+s . . ,dx n ) 0;" d:r!'.
R R-J

6.2. Integral/on 0/ £ztorlor For"",
397
The interior integral vanishes due to the following obvious relations:
f .!Ld:l'=f(x ' , .." x") 1":=+"". Thus, r 00=0, Let now k=n.
J ax" ::c t;Q-OO j
_00 a'
Then (J)=f(x l , .00' x n ) dx t 1\ .. 01\ dx"-I,
00 = (_1)"-1 80;.. (Xl, 0 0 . . x") dx l 1\ .., 1\ dX"-1 1\ dx",
We have
 6)=  '';-'1  t (Xl. ..., :t"-I, 0) dx l . , , d:r!'-I.
HJ/-I RO
 00=  . ',,' ) (_1)"-1 8" dx l .00 dX"-1 d:r!'
B R+
...
r r d 1 d "-I f ( 1) "-1 8/ d ..
= J -,;.:. J X... x J - {j%n:t
Ho 0
= r ... f dx l .0' dx"- I « _1)"-1 f (Xl. ,. x") I:::t.)
J R'" J
= (-1)"  ...  t (Xl.. . X"-I, 0) dx l ... dx"-I.
. a1t- 1
Hence,  (JJ = (-1)"  dw, To complele the proof, it is sufficient
a:- s R
to decompose the form 6) into the sum w = ro 1 + . . . + roN in
!3uch a way that the support of each term be contained in one chart
of the manifold M. Let us consider an atlas {U..} and a partition of
unity {q>et} subordinate to the atlas {U et }. Then 1  :E'Pa' (JJ =
q>..ro, supp q>aro C U(J.' Theorem 1 is proved.
Example 5. Theorem 1 can be used to construct a cohomology
c1aS!3 defIned by a closed form CI) on a manifold. We note first of all
that if CI) is an exact form, i e. it defines a null cohomology class
! wJ = 0 E Hit (M), the integral of the form (JJ oller any closed oriented
submanifold We M, dim W = k, vanishes,  w = O. Indeed, since
ro = dQ and aw = jZ!, we have by Skes' formula  w =
W
(-1 )"-1  Q = 00 The converse is also true: if the integral of a
DW
closed form f'" CI) of degree k over any closed oriented manifold W, dim W =
k, mapped into M, f: W -+ M, vanishes, then the form w is exact,

398 6. Homo.logg Tht!ory
We shall verify this statement for a parwcular case k = 1. In
this case, the condition that  w = 0 for any closed curve y means
v
that integral  (I) along an unclosed curve epends only on the
"I
initial and terminal points of the curve '1'. Then the function I. sucb
that dt = w. is sought in the form I (P) =  (I), where y is a "cur\'e
y
connecting a fixed point Po with a "variable" point P. To prove the
equality dt = 61, it is sufficient to choose a local wordinate system
in the neighbourhood of p. say (Xl, ' . ., :/;n). Then, as y we can alwa)'s
choose the curve comprising the following trajectories: th,e fixed
curve '1'0 which connects Po with the origin (0, . , .. 0) and the
sequence of segments connecting the points (0, . " .,0), (x", . . .,0),
(%1, .r 2 . . . .,0). . . " (x', . . ., ,z:n_., 0), (x', . . "x"-'. .:t n ), Putting
61 =  C, dx', we obtain
f(z'. .. '. x n )
"',
=  w+ I C, (xl, . . " 0) dx.
"I. 0
x' :en
+  C2 (xt, :il. . . .,0) d:il+ ' .. +  Cn (Xl, ., " x n ) d:r!'.
o 0
Hence,
iJ/ ( 1 J
a;ii = C" x. ' , ., ;1;", ,., 0)
,,1t+1
+} 8:" C".I(X I , ,.., :J!'.I. . , .,0) th".!
o
.,n
+ ) 8 Cn (xl. ',., X n ) dx" = g" (xl, ,... :J!', . . .,0)
o
,.....,
+ \ 8gk ( ! J.! ) a ,Jotl
 8;v\t1 X....,;I;' ;1;'
.,"
+..,+ 8:" C,,(x l , ...,xn)dr'=g,,(:z;I. ,...:J!'. ".,0)
o
+Ic" (xl, ..., :i'.', ''', O)-g,,(x t , ...,:J!', ,..,0)]
+... + Ic" (xl, .." X")_gdxl, .,., X"-',O)I=g" (Xl, ., .,X")

6.3. The Degru 01 Mapping and lis Application. 399
6,3. THE DEGREE OF MAPPING
AND ITS APPLICATIONS
This section deals with an important geometric version
of mappings, the degree of a mapping.
6,3,1. EXAMPLE
Let us consider a circle s realized as the set of complex
numbers with modulus equal to IIniy. and a mapping f: Sl -+ st.
f (z) = z". This mapping is smooth. Any point of 8 1 is regular fot'
'0
F1gure 6.2
the mapping f. Indeed, in the local parameter cp the mapping f i$
of the form f (cp) = nip, df () = n!;, n '* O. The differential df is
thus an isomorphism. The inverse image of any point Zo E 8 1 con-
sists of exactly n points. .the roots of order n of the complex number Zo0
Geometrically, the mapping f may be looked upon as an n-fold
"winding" of 8 1 onto 8 t (Fig, 6.2). Let us consider an exterior differen-
tial form w equal, in the local parameter, to w = d!p. Then, j.(w) =
d U (cp» = d (ncp) = n d!p = nw and therefore
 f* (w) = n  w, (1}
81 8 1
On the other hand. subdividing the circle into n segments 1 1 , . . " ["
(viz,. by 'Vi). we find that on each segment I" the mapping f is a
diffeomorphism onto 8 1 (without one point). Thus, since the integral
of an exterior differential form is invariant under coordinate trans-
formtions (with positive Jacobian), we obtain! f* (w) =  (0).
o  
Hence, relation (1) can be derived as follows: we calculate the number-
of inverse images of a regular point and multiply the integral of the
form w by this number. which gives the left-hand side of (1).
By virtue of Example 5 (Sec. 6.2), we find that on the circle 8 1
the behaviour of, the homomorphism f*: Hi (8 1 ) -+ HI (8 1 ) in co-
homology groups is determined by the same number n.

400 6. Homolog/! Th<or/!
6.3,2, THE DEGREE OF MAPPING
Definition t. Let f: M 1 - M 2 be a smooth mapping of
ompact, connected, oriented, closed manifold, dim M 1 = dim M 2 .
and let P E M 2 be a regular point. For Q E 1-1 (P) we put e (Q) =
+1 if the determinant of the 1acobi matrix of f at the point Q is
positive, and e (Q) = -1 if the determinant is negative. The degree
of the mapping f (relative to a regular point P) is the number
debf= 2: s(Q). (2)
Qer'(p)
Theorem t. Definition (1) does not depend
(a) on the choice of the regular point P E M 2'
(b) on the chcice of the mapping I in the class of smoothwise h-omotopic
mappings.
Prool. Item (a) is reduced to item (b). Indeed, if P and P' are two
regular points, there exists a continuous family of diffeomorphisms
1
Mt
Figure 6.3
<PI: M 2 ..... M g such that <po (P) = P, <PI (P') = P. The msppings f
and <PI · f are then homotopic and P is a regular point for both of
them. On the other hand, degp (<Pl 0 f) = degp' I. SO let F: M l X
I..... M 2 be a smooth mapping and let P EM 2 be a regular point
both for 10 = F I(M,XIOJ) and for It = F 1(M,x/I)). Then for the
mapping F the point P is regular at all points of the boundary
iJ (M 1 X I) = (M l X {O» U (M l X {i}). .
According to Sard's theorem, there exists (!.if the mapping F a
regular point P' arbitrarily close to P. Then the point P' is also
regular for the mappings 10 and 11' Since p' can be chosen arbitrarily
dose to P. for each inverse image Q E I;' (P) there exists a unique
close inverse image Q' E I;' (P') such that 8 (Q) = 8 (Q'). Hence,
degp 10 = degp' /0' By similar reasoning we find that deg p Ix =
degp'/l' Again, denoting the point p' by p, we obtain that Ihe in-
Verse iml1ge F.l (P) is a smooth one-dimensional manifold with the
boundary {)F-l (P) which belongs to fj (M, X I). A one-dimensional
compact manifold is always the union of its connected components,
nrcles and segments (see Fjg. 6.3). Hence. the manifold F- 1 (P)

6.3. The Decree 01 Mapping and Its Applications 401
can also be decomposed into the topological sum of segments 1 k
alld circles SJ: F-l (P) = U Ik U US:. Each segment 1,. has two
h !
boundary points, Ok and b k . The set of all points {ak, b k } forms the
union of the invelse images I;' (P) and 1;1 (P). We no... demonstrate
that if the pail' (ak, b k ) belongs to the same component of the boundary
a (M 1 >< I), then I; (a,.) = -£ (b k ), and if this pair belongs to
different components, then e (Ok) = e (bk)' Indeed, as an atlas on
the manifold M 1 >< I we can choose the charts U a. >< / with the
coordinates (xt., .., x;;, t). On the segment I k we deline a parameter
cp, 0 .;;;; cp .;;;; 1. First, let the points ak, b k lie on the connected com-
ponent M 1 >< {O}, all E Uu. >< {O}, b k E Uf> >< {O}. Then the following
inequalities are satisfied:
of I 01 I 0
a.p ok > 0,  bll < , (3)
On the other hand, for a sufficiently smaH neighbourhood V;} P
all inverse images from F-l (V) represent. in a neighbourhoorl of / k'
segments parametrized by the same parameter <p, so that F-l (V) =
V >< F-1 (P) = V >< J". If the neighbourhood V is furnished with
cooldinates (yl, . . " yn), in the domain F-l (V) we can choose the
coordinates (y1, . . ., yn, cp). Thus, to calculate the numbers e (a,,)
and e (b,,), it suflices to calculate the sign of the determinant of the
ox! IJz!
matrices ( ay'; " ( IJY )' Since the manifold M 1 >< I is oriented,
the determinants of the Jacobi matrices of the coordinate transfor-
mation (x:", . . .. x, t) _ (yl, . . ., y", cp) have the same sign, irre-
spective of the index Ct. For example,
Q(Zt., ""z,t) 1 _ Q{z&. ...'Z) ..!.!... I
q (yl, ..., yn, '1') all - Q (y', ..., yn) IJIp all'
(4)
Hence, the first factors on the right-hand side of (4) have different
signs for ak and b k . If all E M 1 >< to} and b,. E M 1 x {1}, we obtain,
instead of (3),
.!.!... I >0,
alp ah
.!.!... I >0,
QIp b,.
(5)
Formula (4) implies, therefore, that B (a/<) = I; (b h ). Thus, nil the
points of the inverse images /;' (P) and It (P) spiit inlO pairs satis-
fying the condition: if a pair belongs to the same compolLcnl of the
boundary iJ (M. >< I), it gives zero contribution to the sum (2), and
if a pail' belongs to distinct components of {} (M] >< I), it gives lhe
same contrihution to (2) both for deg p 10 and fof' dcgp II' Tht'ol'crn 1
is proved,

402 6. Homology Tht!ory
Remark. For non-orientable manifolds an allalogue of the degree
of a mapping can also be defined by formula (2). In this case, in-
loact of Thool'cm 1 we should assort ll1at tlte degree of a mapping
does not depend on the choice of the point and homotopy with respect
to mod 2,
6.3.3. THE FUNDAMENTAL THEOREM OF ALGEBRA
The fundamental theorem of algebra states that any poly-
nomial P (z) of degree 1 over the field of complex numbers has at least
one complex root.
There are many various proofs of t.his theorem, One of them rests
on 11sing the concept of the degree of a mapping and Theorem 1. Leot
us consider a smooth mapping P; C. -+ Cl of the MmplQ'x plane
deli ned by
w = P (z) = ZT1 + an_1z n - t + ' . . + a 1 z + ao. (6)
This mapping can be extended to the mapping of a two-dimensional
sphere S'J into itself, assuming S:' to be a cOll1pl(1x projective straight
line CP (1). To this end, we assume thaL the complex para-
meter z is equal to the ratio of homogeneous coordinates on
CP(1) z=..:L for zo:#=O. Similarly, w= for wo*O. Therefore,
Zo Wo
the mapping
WI = z. + an_1ZI-lzo+ ... + alzz-. + aoz:, (7)
_ n
W o - 1.0
cOl'l"cctJy dcfint's the mapping of CP (1) into itself. Mapping (7) is,
apparently, a smoot.h one, Indeed, in lhe chart Zo =F 0 this follows
from (6), in the cha..t Zl :1= 0 as a complex coordinate we can lake
Lhe function z' = . By setting w' = w o , we obtain
11 WJ
ul = (z,)n (1 + an-tz' + ... +a l (z,)n-1+ a() (z')n>-,. (8)
Taking a 5uflicienl)y smaIl 2 > 0, we choose a chart containing the
point z' =- 0 and rloIine this chart by the illequality ! z' 1< e su('h
that tile denominator in (8) he nOll-zero. Thus. the mapping I:
CP (1) -+ CP (1) given by formula (7) is smooth. w now calculate-
the degree of f. Accordi ng to Theorem 1, the mapping f can be re-
placed by a homoLopic onf.'. Let 115 consider the homotopy with rt>-
spcct to the parnmetcr t, 0  t  1, dt>lined by
Wi =-= z + t (an_iz"-tz o + . . . + aoz;).
rI
W o = 20 .
(9)

6.J. I'h Dgrw ", Mtrpplllll an4 lr. App.u"""oJJ,l 403
JUSL like in toe case cf (7). mappings (9) are Bmoolh. At r  0 we
.,tJtain a simple mapping
w.=z:. wj)-=(. (10)
In the lotal coordinates tv = w;/w(}. Z =oil &alzo tOIS b'Jo.Ppil'lg takes
the for-m w - #,"". and, soy. the point w = 1 .is regular. hldeed, cal-
C1.Jlatjog toe Jacobi matrix cf the mopping u =- Re U1 = Re r.',
v = 1m IJ) = 1m #'"". z = % + iy, we obtain
dot ::  d.t ( :  -: : )
es as
( Re...... -Im...-' )
c::=n1.det. ....n21 z-112>O
1m zf. He ,."a-!
ror z =F O. Since the equation in  l has 8)(8Ctry lJ solutiDns. the
degree of mapping (10) and. therelcre, or mepping (7) is equal to n.
Le. deg 1 = n. If the polynomial P did not have roou, the point
w = 0 WQu1d !Jot hel(logto the imegeof I and. hence. the mopping I:
CP (1)_ CP (I) would ha". a regula, p<>it (w = 0) with ."'P'Y
in'\'erse Image. i.e. tbe degree of nUlpplJ]g f would be zero. CODtradit"-
li(Jn pro\le5 the theorem.
e..4. II'TEGRATION OF FORMS
Theorem 2. ut t: M._ M. be(1.moolhmapprng%r!tnl-
ed, c:«nplJd, conntICl«f IIt4nljolds l .and kt m b an t:;dtrwr dlDernaaJ
form. deg (!) = dim fl .0:;:0 dim Mi' n
S f'(w) = degf- S OJ- (1\)
II, M.
Proof. Since th. lelt.had end r;ght-hed .Idea ct (II) are both
Jinear in Cd, it is su!6cient lb verily lhis formula fOJ the form m
whose support lies in a srnalll1eighbourhood U 01 the point Q E M z .
L.t Q. E M, he e reguler point 01 the meppig f end let U. ;)Q.
he 8 sufficiently Brnml ntl'ighbourhoCid. Then lhre exisb III c.on-
tinuous family cf diffeomorphjsms 'PI: f2 --lrt 2 such that f'JID(P) 
Pi. en id.tity m.pplg .d '1'1 (U,) = U;) Q. 'nd..d, conect Q
,Cind Q. by 8 continuous pella y. Wid.cnt Ires of genetR1ity, we may
aBsume that Q 811d Qo belong to llie .8arniPI chart V diffeomorphic to
an, and the pall. l' is a straight Z:iegment in the Joes1 coordinate
ystern. ul u'" (,DnsLruct on \I a vector lield  which is eqHI lo the
tangent 'Vec.tor to '\' and 1189 a compact 9UpPOIt jn the .-:h8J'L V. lQ
this case llae dynamical l'3)'stem corresponding to the yector field Gt

401 6. H_ ."..,.
i e. lhe one.pa.rarnd@{ family of jjdteomorphisms fl'h shifts th(! poinl
Q UJ (he point Qa. and the diffeomorphi,;m fPo is an IdenliLy one for
I = D. The forcus : (w) = Q) Bod  (fc)) = WI are. Llle..efcre. co.
homologous (according La Theorem' of Sec. 6.1). Bnd by StokE\"
theorem we have S (i,):::  (u1" SbniJariy. ! ,- (11) :ri S J (W L ).
HI M. I'dl N.
Since supp 10 c: U. we MVBSUPP 11:1. c:: U.. Hence, S Cd. =:I 5 COJ>
N. Ug
On tbe oilIer halld. the inver.se lmage 1-1 rUJ is Lll6 u:nion of finlle)y
"
many open lIetsJ-I rUa) = U V,. the mapping' being II. diffeomol"A
....
phism on each of these set.9. Tbus. since eupp ,- tQlJ c.,/-I (U.,).
we have
"
[t'("J- I t'("J= 1: I t.(..J
it. r-.(v.) '..IV.
" "
=  (Sind! IYI)  '".= [1: (sgnd! I y, )] I"',
'-J v. I-I v.
=[ 1: o (PJH "',-d.gt- S "'..
Pf:1Q.} U.. v..
6..8.5. GAUSSIAN tlAl"plNG OF A 8VPI':RSU.RFACJ!:
Let. 11.'1 consider a hypersurr.ace }rf in EuclidtJ:n Bpaee R".
dim M = n - 1, deOned by the equatIon F (x) = 0, grad F .,. O.
Then" there are vpUd 011 }..f the Riemannian mel.rlc {gu}1 tbe Form
or volume yj'gf dy'/I, . . . /I, dy'-'. whe,. I g I - det N'J. .nd
the Gaussian c\JnaLul'C K = K (yl, . . .. Ii"). Let us deline 8 new
/orm K do = K (y" . . ., V") YT"i1 dy'/I, . . . /I, dy'-' c.lled the
form 1)1 curvtdw-e of ILyper.luface }"f. Furthermore. tbcl'e i3 valid on
M a ,smooth Inllppiog I: }..f -to- 8,.-1 C R" associating with each point
P tbe lIorma[ to the 81Jrface aL lhts point. This mapping is caUed
phrlC1 mapping. Let g be a (orm of vo1ume on lhe spbere Sn-J.
Then tbe roUowing ste1emcnL bo£ds true.
'Theorem 3. For the sphtrical mapping f: M __ 8,.-1 01 Q hvperBurlac
M the inverse image of the lorm oj voI Q Olf t.he sphere 5"- 1 JJoo
cf}lJ.al to the form IJ/ cun-aturt On }"f. t*n = K do.
Prool. \I\'iltJout loss of generality, we may assume 1.11aL in the
neighbOlu'IJood of 01 poil'll Ph E }..f lJle marJifDld }..f is the gl'nph cf Ih('
tuncLiOIl rI  I (r, . _ ". z"'-J) find at PI} - (0. " "' 0) Ihl' Ilol"mal
to}..f 13 parelhd lQ (}Je axis Ox". Then a( die POUlt PI} lilt? IiItmanniHIl

6 S. rh D n/ MpJ"II II"" Ilr Ap.,t"''' 40S
metric coincide:! with 8" identity malf'i:x. go = ri'I' and the G811&$-
tit" c.urv.aLutc at P D is K = det ( ,'I ; ) On the .spller Sit- 1 c:::
2' a
R", in the J1ciabhourhood nf the po-Inll (Pc) = (0., '. 0, IJ. we
..-hoo,e. the {;oordinf:ltes (.t l . ' , " 1"-1). 'fheu tho metric. 011 S".... 8t
/ (P.,) is also or diagonsl form, g,/ = 6 , /. .'1d therrDtt! the Jorm I.
on the s]Jhcre SlO-1 .1./ (P D } IS eQllal Lo: g = d,rl 1\ . . ./\ ctx"'-I.
We now calculate the spllericBI mapping. Tile LAngent Sp8CV to. the
mamrolll M is generated h]' Lhe tafJgeflt vectors
o
I
..{
o
o
r-
.
r"-1=
o
I
f,/'-.
f.,
TheJI the JJormel vect(lr n lJes t'be coordinate'!!
f..
f..
t
no:::: V t+ J: '+"'+-L
f,!'-1
-I
The inverse imege vi the form D 'Illder !lpbericsl mapping Is cal.
culated by sulJ.\tituLing ..lie dilfllrelllip] of 1J1e MordiDale or the
rJormlill Vec.tor a for dz t .
01..
11+ t:: ,+.. + J ft-J.
({lz.dl...+ ...+/ft-J.dJz1l.-:1)
Y(t+t.+ ..."1 1 .",.,.)1
.
,sJrI(.t! at t}IC pmnL F., tho first partial derivatives vi the [ul1cUon /
\rarll.1!h. If (Po) :;:; O. we hevQ
I "10-1
d  =:II  ..--!2- ax'
V'++ "'+ ':, ".J. ;..1 azi8 z i
f' (11)=" '.,
Yt+f  +...+ t:: "-1
"Ihus. we have
'.1
<I V ' =
1+r.I+...+I'lt.fl-l
'.l'->
/I.../ld"
t' 1+ J: I+...-tt:n-1 .

41006 6. Iimlllllttll TIuDI'W
at P t - Hence.
l'I_t "-I
f'(Q)= (  , 4z' ) /\ .../\ (  47' )
a.J;liJz . 08r-1.iJ:zJ
l' 1=1
=dut ( -.!'L , ) <h' /\ ... /\ dz n -, =K dd.
i1zlb
Tbeorem 3 is proved.
Ai; 8 corollaty or this I.hoorem, we anj'\'e a.. the GaPSS-8onnf'l
thorem.
Theorem ,_ Lt ft.! bt a closed, ompacl sur/ate;n R - Then S K da =
M ,
4:f[). whtrl!' ). 011 an ;n1eg&.
The tbeor-em can be proJ;rd by usifiR Theorems 2 81111 3:
S Kdd=  f' (0) = (dug f) S D=b.dugf.
.N M B
Remark. The GsuM-Bonrlet thll!orem is eveD more general than
Theorem 4. It turns out toot the integer A does llOt. depend en tile
Rum8DniBn metric and u a)wsys equBI to 1 - g where C is Lllc
number of handles on tbe orjented .wrfBCe M.

Chuplcr 7
Simple Variational Problems
in Riemannian Geometry
7.1. f"UNCTIONAL" EXTREMAL FUNCTIONS.
EULER'S EQUATIONS
Vaf'iulionlll prcble t:oll&tilule aile of Ule rn05t impo['-
181lt d.es,"ic., 1)( ID81.hem8ticeJ problems clnsel)' J'(:laled 1'1 !iuc.h ft.md8
mCltlal pllysiul aDd mechsl'JiepJ pbenolJlcna 85 moticm Dnd £labiht:r
For f!Dmplt" we :!!h,\!1 see below that gaod-e.sic lrajeclorles ere so1u-
Lion:rc of (hI! .eorre5pondlng varl81i()r:u1 problem.
Hefore tuming to tIle ghIJenl concept (If the variation of a flIllC-
lional, we !ililail :'ley a lew word! IIbout the functional ilscll. We are
already feJOiliar wit}, the concept Dr 8 funclion 1/ == 1 (zJ. wIlen 11
is 1:1: I"'(!lil nllmher .Hilt! (be argument x can he wriUen as tlje r::l:L of
tlllm.IJ('1"3 (r. _ . "t.r") in II cunihneur coordinate s}'stem on 8 sm(J()U.
manifold. But nOt ever)' phyicql correspondence can be ()),pre-ssed
in lJch a Irnp.e form, For il1stance. we have alread)' considered
tl,e f()lIowi':tI.g cQr!SpDndnce: with eacb fnute segment of 8 limol;llh
.
lIrve "t' (I) there is assDciated ts Jeoglh ': ('v) = i r.y I dl. TILe
c:ol'l'e!JPQndnce l' ft) -  (y U)J is not 8 "'1unction' i t'he GrdinBry
soen"lC , for' in this ca'l!le an 81'bilral'Y smootl, curve acts liS thll "argu-
ment'. The corl'espondence V _  <1') is an important exampl of 8
non-linear funcUonal -deJineu on the space Gf smoot!. tunes T (r),
We may extend lhis ex:mlplt1' to include SQme otber irn(JoJ'lont <:01'-
rospondcJI£os
tet us comdder in RJ'I 8 boul1d domain D w)tl, smooth bo\mdary
iJD and let z\ . . '1 z." be Cartesian cO(lrdin8L. Con!ider on D all
possible smooth vect!lr funetions I (.rl, _ . '. r") = I (.za, = (f (CI),
. . _./'" (» = {I' (,rt)}, \Vhen tJIIB r}lImbers k and n are andependent.
The dom8in D i.s col(ed Lhc rllogo of the puamet.ers Zl  _. Z".
Giyen 'wo Fun.ticlI, It r....)} and II; (z")}. I OS;;; i OS;;; k, then for
arbitruy reaJ number.! Q Ilnd b there exists 8 IlCW vedor functiDn
ar + bg = {al i (z'Z) ;. bgt (zan, i.e. ell smDDth vector functions
on D form £I line.ar spece F. This Splice is infinite-djrncnsjom1. Vari-
OUS functi(m.a.ls will be studied on the spaco J', end "points' or F
(Le vect-or Functions) 81'e jusL the "argumonts" Df funr:liollBls J (fl,.

'D8 t. 1'arlcJf(o"cl Pr""r.frPU 111 R (i'c(l1tIdI\!
t € F. B11l in the simple eXBmpJps he)lIw. lh ('x(stel1ce (). a Iiflf'.81
skucbJro ill F is 0' no significance. ror WI! sl1all not use Lhe oprDhon
or adrlJt.loo or vector [unctions.
While s1udying functiOJ1ul:\1. It js USijfu) to rocall an anaJogy witb
ordinary fttnctii:lflS.
Definition. A functiona.l J delinet1 on 0 spece F (or on a subset of F)
i.u III c:onlinuou!I mapping or F (Dr of its Slimet) into rea) numbers,
J; F - Rl. and t}le mepping is l10t 8ssumet1 to he linear. ]I
J laf + h81 = oJ In + hJ [IIJ (i.a. J i. lin.a,), lna I.nelional J
is cslled Ifnear.
&illrqp1e. Suppose D is e segment on t}le real streight IIn zl = t
and I (t) = If' It). J' It). r (I)) Is a vacto' function on D. i.e. fIt) =
1 ! l) deJines a smooth cune in ft3; in this case F is a linearlspace 01
al Buh urvel!l in 1\3 (the rodiu3 vectors' end g of the curves r (I)
al1d g (t) c.en he udded Blld mulliplied by 8 Jlumher). The functi()fII1
I
J i.. tal<.n e, tbe Integ.el J [II = J I;; (t) I <It = J II (t) I <II. i.e.
. "
the 18nrh of Lhe c\Jrve I (t). 0  e E;. t. This functional is 1'101\-
linear, atnco J [.f + hgl +- aJ III + bJ Igl I!live en oxomplel).
Given 8 8rnaoth funetio11 L (.z.': pij ) which depll!nds (In I.luee
8"eu,," af v.rlobl..: 1', 1,.;; ,.;; n, pi, I,.;;;,.;; k, and qt I,.;;
a.  n. t  I  h. This 'unction is called., Lagrangian. Thus, I\ny
IJmaotb fune-Lion of three group! 01 variables can be ft Legrlllnlrifm.
ut f = cr (xll-)) he a smooth vl?'ctor function on D c 1\910. Let os
construct the func:tl«lnal J rll
Jlfl=  L(zII,/'(zII),/:"'11')da",
whera S dencles a multiple integllli  . D' S (n iDll!I!J over lI-di-
D
mfJn@icllll domain D. doR.....d.r' A '.. 1\ dz n Is an n-dimensicnal
eJementar}' volume in D (i.e. Q CarmiuD exterio[' fOrIn o. EutJidean
volume). Jh ()= J' .zD) are part.ial derivatives. The fUllctionsl
z- az'a
JIII"o be written in compoct fcnn a. JIIJ  L(zII,J'.Jda",
"
whero t.oo argumen19 of I' and J all! omitted. The function L
(Lagt'8Ijghm) is llLuefvre delin{'d [or eacl. II,nctional. The da5:!i tll
hmctionll'!!i J\l5l de(IJI(d illduoe.s \llrlually aB Important c,uIDple:s of
[llndimulll; cncountered ill me{;h&nin, phJlslrs. and in their appli-
cation!.

7.1. FloIndi4P112'. EJl:lrIMGI '""CIIOM. Eulr,.", E<GII.fI'" '09'
Let \U!I cansider tbe '\lnctionilJ ,of the arc length
I ',{
1[11   Ii (t) 14t  V ,,,(... (I)) d.,('J dt(l) dt,
. .
i.e. D = I = 10, 11.0";; r,,;; 1. n  I, I (t) ... (I) = (y' (t), . . ..
y. (t)) V (I) b 8. !5m-oolh curve in a k-dimemlionaJ .spaco with Ll18
RlemanIJian met.t'1c 811 (yL. . , " IlJ. end Lhe L8grll.JgiBn Is (JIL'he"
form
L(...I.I...JL( I,y" ....r/'. -1,-. ...,-¥.-)
J . 'f"
y g,,(y', ...,y Jy&".
If t11e cune 1 {t) (In Llle 18ne R2 is defiJ1ed exp1ic:.itJy. 11 = t (r)...
tl1en
L (x. t, t.)L(f.)= Y I+ / .
T11is [unctionBI i& else defined on tbe curve, lying In 8. SD'loMk manl-
loJd Mil. We I!h.U \lsually c.oruider a .smaU l1eishhotJrhood of a
particmlar curve and assume Llult. loCh] ootdin8te.! yl, . . .. y. ore
vaJid in this neigbbourhood. Thu.s. we ehall deal wiLli 8 k-dirnenJon'
al Euc1idean sJlace Jlrovided "With e Rii:!mIl.TlnI8t'1 [gener.Uy. :Don.
Euclidean) metric. and in this cllse the additi.oD of \l'ec.t6f functions
defin1ng different curves CBn only be performed in SOme nelghhour.
bood O[ 8 fiJl:ed CUf'i'e.
Ano\h.ro>:8mple: tbo!unctionalo!.,.. JIII=  Y EG-I" 4%4g.
o
H.", D (x, y) Is lb. tOng. 01 lb. parameters (z, gJ, and 1=
(ul (%. y). u 2 (%IY)' u'l(%t Y) 16 8 twc-dimBDsionel 8UnaCe In R8I
witb tbe induced metrIc dr-=E o::::2Fd:r.dJl+Gdr;
L=L(I.. !.)Y EG- I" =V (l .. I J( f ., I,. )-II,.,I.)'.
The space F is the Jinest' space cf 811 vectar functlDns (u) (z. V).
US (z, 1/). U' (%. y» de[inet1 00 D We could cons ider 8l1C1t-her fClrm o.
lhe (u""tien.! o! are.: I [II = J S V 1 + j + t: 4% 4y. where D i.
D
II domain (In Jr (z. y). End f (r, y) ..: (z. h'. .z (z. y)), i.e. the vector
function f is given explicitly by the gropll z -= 1 (z, y ) over the
domailJ D io R"" c: R'; L <;;0 L (:r:If' .z) =- V I + z + ,,;. AdditiClJl.
of IillCh vector f\JnctjoJlS means addiLion cf their gnphs in R 3 0vGr D.
But nol every fll.lr1"pc.e J.rt R can he descTlbcd by t1.c grap)! of Esinglo--
valued I11ncLlun.

06.fO '1. ".or/al.Onal l'.-al1krru In Ricolltannhfll Cnrdr
What prohl('m al'(: t)f primary Lllh!re....l In tho .stud}' of JpUI? Let
U8 dil':!cuss enologies witJI ordumry r..ncHon!'!;. Conside-r, for examp'e.
Cunctions oC one (It two v.,riehles, a (t) and a (u, v). The be.he\'iour
()f such f,mcUon. iff lorgc-Iy -determined b)' the rllJrnber and liDcetiDIL
()f the POiflLS '0 {ot (u,. vo») at which Q' (to) = 0 (Dr au (u.. VII) =
all (t4 o . v.) =- (J). j.e. red C( = O. The points "L \Vh.cla grad a = 0
a
t
Flgu.,. 7.L
re caJled t:rllit:td or slahtJnarg poinl6 of the hmdion a. The term
.e:rlmt11 poinu is also "sed :soffilf!limes. For instance, for the func.tion
Q. II:: a (') shown in Fig. 7. t t'be :!Ihtionary points 'ltre; two madm8.
minimllJD. and inflection. If a. ::::::J IX. u. v), the PQints III which grad a.
.... 0 are maximal minima. IInd saddle poinb (or simply 6hddte9) or
CI-alu,vt
 :.. \J
- 
."';
mT"
.
"
FIgU" 7.2
<order two; srtddle of higher on1el1i are aJ050 po.s.'n.bh (Fig. 12). JL i:s
of primary import.nee .n lI1echanics Lo find t110 poIllls at which.
smooth runctiDn Ihe polenlipl. ['fI.;Icbes.a maximum (cDrresj)Qndihg)y.
potential Energy reaches a miTJimtJm) .because at these poillu the
system j in 8hb.ie equilibrium.
Similarly, in the study or J lEi it is 'Wery import'ant to find tbDse
ZlttLtion.ary vectar fUoetloliS r for whicl. Ule fundional J nachc5 a

7.1. f'.ntlion4Z. £zir"mal FUnalon... £...(,,.'5 EVUrdloltl ..it
minim1lm, 8 maJtimum. of he 8 8nddll!. Jt is desU'8ble, however, to
formulate thIs geometric analogy in the "dLf(erenlial Jenguage", that
is, we ilave to give a corrcct der.mtion of the direction a! derivative
of a l\l(lclionel J at a point f E F. As was already noted, ..II cntice.
point:5 vf tlle (unclions a (I) snd a Cu, v) are represented by the
.!jollltion.'J of the eqU8tioil grad tot = O. We need. therefore. to' derive
an e.nr.logut" of this Equot.iou flJf a Cn(jLioDaL Let u! considEr of-Bin
the equatifJn grad 0: = O. If 8 direcl.ion (8 vector) a = (a l . a) i5
def1rled at 8 point (I', v) E G, the derivative of the (unction 0: fu. v)
....itb rpect to the direc.tion .a. is l(i'Von by
4 6u /Ia
da a(u, v),=-a t h+a.'ZB;"=(8, grada)
(see Fig. 7.3). It follow> that grad a (u., v.J = 0 if .nd only il
:. a (u., VII)  0 I()r any direction a at the point (u. 1 voI)}. J(
I'tpre 7.S
a = all}. tbis means Lllat Q; (r.) = (). The derivftti'1/8 oj C (11. v)
witb respect tD II C8" be calculated 8S
4 d Q . = lim!..[a: (r.: + lEa'. v+eQ,-a(u,I1)lp
_D e
where .= (GI 0 2 ) and II is a plJr1I.meter. i.v
4Q t
- d =11m-[a(x+..)-a(x)],
. II:-()"
wilen! :Ii = (u, v) is the radius vector or a point ill Lllc domAin G.
This expression is used to extend tiLe concept of directional derivAtive,
10 a functiDnal J Ifl.

4tZ 7. }'o,.laliond PNLltm.l ;;n Rimul"..,.loO'j: (Jt:t>mttry
Lel 1J:s con.sidec B "POInl" fE" and a surncientJ)' smaH f\Jl1tliOIl
11 e: F such thal 'I IIII'D..... O. TJJe (unctions 11 ar called (Jrlu,.barions
o[ the f\.ll1ClilfO , ul us cvnsider (hesblfL from l)lt! poinl f to e. point
I + 0" (50. Fill. 7.4). Tho lunction .. (we recall llml .. I.D = 0)
FJlUnI 7_4
deJiJ1es the "djrEcti(Jn of translation" from tl'8.'poiDt" f eJ"Bcdy in the
.same way Jill the ector 8 defines tbe dIrection cf trans1slion 'rorn the
point (u li . va) € G t1.9 dil'letenc;e fl"l;Jm an ordinary function being
that ill. thlU ca.se or luncliolloLJ there ,.ce inJiDitely many sur;h "direc-
tions".
Further, quite Imilarly to the case of ordinary fUnCII()f)s. we may
eonatrucl ,be .xpression !. (J II + ...1 - J [III. Pe,lorming a
.
limiting proes!l w.i:t.b respect to e. we ()btlll the num ber
";-J(II =Hm .!..(J[f+O'lI-J/fll
11 It_fI II-
which can D8hITUy be callal thu derivatil." tJf a Junctional J at a
poin! f with re'pd to the directlDn 11. Drawing flll"tber analogy be-
tween fUhdionp)s Bud ordinary funclions, we- KIva BIlDther elinjtioD.
DefiDiUoD. A function fo E l' is calle B stationary (I'renwZ. crlti-
eal) function for the functional J (fr. pfovide-d  J rlol z::. 0 for any
-,
perturbetion 1] s\Jch lbet 'I) ID = o.
U by F wo me8n the Bpace of vector functions r witl. CtJnst.anl
coordinates (I.e. f (1 11 ) = rCDnst)[) tiles!) delinitions turn inLO
ordinary delinitiCins of direcU()l)el cnvative find stalinnery point.
IL Is conveDil!nl &0 reprosent 8 furlctIonal J as a "g.rap"''' over r
(Fig. 7.5). Appar.ntly, if IE}', lb. .el 01 Ivoclinn. I + <.., who"
11 lAD = O. lormr; a Jinear sp8te T (provided I is taken as zero of T);

V.l, FUndlDn4l, 8.:zlr{'mGI FIIllD"". Eul.. £qWGU01I. 413
'he function.l J is restricted to 1M:! Space, From Ihis descriptive
point (If view lhe "poi.11tS" let. wberu d J If D ] = 0 Uor eny "1) ore
minima. mBxima, ()r .sadi'ile points of tile graph oJ J ur re-,tricled to
the 3ubsp.ace T c= 1-". The meaning v( 05uch a restriction of J to T is
quite dear; W went 1.0 study tlte Lchevloul' of J for those P9tll.ltbe.-
.............
.. ........
: ::. . ..........
F
FSgurll 7.'
tion 11 which do 00\ ,p'ler rat tbe botJndary aD. tbll,t. (,. we study local.
difirentje.l properties of the functlo:n 10 uch tl]1l d J 1101 = 0 (for
any 1). In Fig. 7.0 the ends of tl]1 tUnllilre fixed 81. points A and B"
F(pre 1.45
i.e. 'Q (A) = 'I (D) = O. We now derive en explicit form1lla for the
derivative  J (II. (Tbeexpri:ssion  Jill 1.8 somelime! ca11ed the
first I)(jTlotion Qf J IIJ.) We hvc
f-J I'I-lim .!..(Jlf+'I-JI'n_
'1 .-to III:

414 7. VarlallD,,,.:I! P,."bhml in RI,M/J",s)lJn lltrmulr,
Put 6J=Jlf+...I-Jlf\, 'be.
U -  ILCzt'.P+...;, f..+....:,.)-L(zt'.I', I;"'Hc!<>".
D
RepreSEJJUmg the il1tegrand 8B II. TayJor series, we obtail1
. . "
I!J H D 0"'+   D o+o(.)]c!<>'
t) i....t 8J ;_Ia.",,1 J
. "
=0 H [ : ..'+  " 'to]+ .) }da".
D._I ._1 
To i.ntegr&18 bJ' perls. we c.on,lder
D ( "L' ) " ( 01. ) ,81.,
:l" Bjl11 = iJa "7 'J + III ,
.. oF-..
whenGe
6J-e    ( "/. "' ) "0'
b a_I u B.r
. n
+0 r  ( _  2... ( 8£ , ) ) "'''0'+ r o (o,c!<>".
J 8JI 8" DJ )
Dt_& Q_I f!!J' D
Sinc.e ,all the 'uncli(lD3 ere IlSs\lrned 19mooth. .n toe first. integra) we
can :!eporBle Intt-grati()n wjtb re:spec;t 1.0  rrom integration wItIi.
respect to tbe o1.her variabJE's ,z.t (1 =E;;; i =E;;; tI. i '* ct.) (according to
..he theorem -on tbe c}IHnge (II ol'rler of integration). This yields
Q .
 "., ( -!i-..' ) c!<>h- S...! U  ( 'I' ) "zO]"o.
D a;J- lJt _;. ara iJJ
",L......'IZ...7t!'
Q .
(see FIg. 7.7). Since in too inlegT81  the varillbles 0(1. . . .. rr-. . . ..
p
z-a f.J:'IZ IS" omitted) may be aS9'lmt'd '0 bo pl'remll\tlI"i5. integrelion can
be Jlerlormed expliciLly. j.e.
f  ( oL ' ) "0"
J iJ:tI" iJJI 1)
D ...
f ... r [ . (Q}'J'(Q)--!i-(P)'I'(P)ldo-'''''O,
.) "a .J  al,p IJt,p. 'J
aJ ...:11 ....

? I. F!'I!('l",,4t EktrL'lI'Ial PUflClltH'II. Ekll'r". £qIJQlwr!' "t
because 1ji(P)1j'(Q)-O. /" QEDO. Thus,
. .
6J -. r  [ .!!!:...-l: ..!..... . ( 8L )] 1j' da'+ r o(.} da',
J 81 1 z iJf1 j
DI_I a.=L  D
whence
. .
d J(f]= f  [ !!:...-l:  ( aL ) ] 'do'
 J aI' a"'af' 'I,
D i_I a;aL z'&
since lim.!. S Q (eo) do'" -- O. Let fa be a stationory funclioD br J.
_tI aD
o
f4!M, p -------- a
't-'\...
*'"
. ...--i:H...M.:
..",;:«rr- ..
X"
Figure 7.7
Then for any function 1) [11 IDC::OO) thelollowing ident.i'y is eatiefted:
. .
! l: 1j'[ : -  (a )] da....O.
D i=1 _=L.,p.
As [s )(;no'Wn lrom matlU!m8tir.al aoIYSI3, this identity m8ns tb.at
.!!!:..._   ( L ) =0
/I LJ an 8f1 .
_1 ..
(I
Sy!tBm 01 djillerenti::ll eQIJ43tion5 (1) is coUed tbe B1Jdem 01 EuJer
equ.ations lor a furtdiontll Jill. Tbufi. we hne proved the following
important st.alemenl.
Threm t. A JunClioft J o E F it. called alr-t11lal (.srationar.vJ Junc..
lion for tht functional J [IJ i/ tl.nd only il it Iilllu;Jie. Eul,r'& IqU4lU,1Il(
(I).

.f.S6 f. VotI:riallion4l P,..oblm,ll ill RirmAllltiall Cronu!,."
If 8 foncti(lpol 15 nit onHnAty fundlon on a dtJmlun G.. thD con.
dition that R point. Ie E. C; is extremal means that. £!i '"'" 0, '.e.
grad L  O. which is wltat waS to be: eXpl!'c.led. Here J = cL, where
c = const.
7.2. EXTREMALlTY OF GEODESICS
For 8 Jhemannlan manifold a geodesic was defmed as D
trajectory along wbich tran.slalion preserves the nloclty field of
the tnjectory. Hut gdesk.s exhihit another very import.aI1L charac.
terisli£ wInch Cfln be used to rlenno e geodesic. This characlerislit
is reilited to t.be I!J:tremal properties of B special functionel which
is very much ellka the (unctional of len.gthi here geodesicJ act B5
.extremal !olutions to this funclionaL
Ll!t M lII be 8 RiernsnniaD rn.llifold with the metric gfJ and let
.:ct. .. .. r' be locd c.oordjnal&8. Thn tbe trajectory 1 (t) can be
d.r,..d as ,,(t) = (z' (r). ..., z" (I)); tbe ..gm.n! I = 10, 1) i3
chosen as till.! domain D. For (oo\f,;mience. W9 .shaH consider traJec.lc-
ries l' (I) with fixed initial Bnd terminal points; )' (0) - P.. y {lJ =
Q. p. QEM".
Den.IUn.. The runctlo.al
, t
{.Iv)- S IYldl.= {BI,(e);";"dt
o 0
i. ceU.d the funclional of Ihe lenglh oflh. Iro.ftcUJty ,,<'1. The func.
tional
I .
E I,,) =  1 Y ,'dip  B.,(%);"%;dl
is c;aUcd the Junctional I}/ actIon 01 the lra)tc:torJj V (I). The JuncUonals
Land E are distinct. but they BtE: l'elatE!d by the inequality L"::e; E;
there is .fso a rel8lien betwcen their Xlrem.als (see below).
LemmE I a TI int'quaJdy L':  1:.' il valid.
Prool Applying ScI\vjert' meQlJaht.
I , .
q fBdl)',,;;q t'lIr).q g'dl)
to the ru""t,on. I (t) = I ...d g (I) = I;' (t) I. we obui. (L (V))' ,,;;
1:: (,,). Ihe .qudi.)' laking pia.. only for. con.tom funclion /I (r),
j e. if and olll' U ltl pIJI',ftmC'tcr , l.s proportional to lire lI.fC: Jength.
Lt!t liS c(Jllsid('J' (,)'h'enIPls or lhl' runcljonflb E .Qnd L.

r.t. Ezt1YlMIllJi' 01 c.od"U8
417
Theorem t. The extremal! oj the-functionaJ E <V rlr reprertnred bJl
geodesics 'V (t) JJlUam.etrued bJl thtJ paromfter t proporrional to the wc
length s. In po.rUcular. il thtJ imtial cDndltlon I .; (0) I .... t Is .allsfied
elthe ./arling poi..1 P, the ptJI'tJnIder t i. defined "../4...1" end I.t th.
natural parQ.1Mter, '.e. coincides with the- GTe len.gth.
ProaJ. RecaU thaL we always cDnsider paraI!1f1trized ttejectories.
This means that two trajectories descrihiug I.ha same ,geometric locus
and having dLBuent parameters are taken itS distintt trajectories.
According to Theorem t of Sec. 16.1, lhe extremals (If "he functional
E (Y} BaLi.aJy EDler's equ8t1Dn9 1 which in thb C8S t8ke the form
_ d { .!!:.. ) -o 1";'"';;'''
ilzIt di"8-1
and the Lagrangian L iZi eq1Jtll to L (x':' ;.) = 81/ (z) (;'. Calcula-
tion yieldfi
ilL 8811 "'I"'.
':rll = iJzla .z%'
i3L ..
=2g.,
Oz'
 {  ) _2 Bt" ;;+2g.j;l.
de IJzIo h a
-III . ,;,r 2 8,..;;  g . 
"zla Z - a;t -. JrJ = r
.., f ( 8'''1 .I' 8'1' ':--' )
gI&Jz+- 2 2-.z-:I;r c:l.O,
a1 BzIII
'':'' + t _'0 ( 4'IIJ I aAJ tll ) .:--., 0
;A.- "Y"- a T Bzl -""'iij:" s-z  I
te. ;'+ro"'j::::lO. Thus. EnI9Z'.s equations eoincide with the
eq uati()119 of .geode$ic.s j 11 the Riemannian cODiIleclion. The theorem is
pr()ved.
The e:.:tremlll ()Jlhe JunclLoll.) E ('vi is derIDed by Euler's equa-
tion.') B.S a parametried trajectory. Underan arbitrery sm()olh dUlni'e
of lhe parameter 011 a geodesic trajectory. the trajectory fails. in
generel, to remain a geodesir.
Theorem 2. The xtrem«ls 0/ tJ functional L (y) Clrcf rtpreienled by
smoolh trajectories l' ('1 obtained from thl geodate Ira}ectl1rlu bit
QrfJlt,-ary 6nQQlh parameter subslUullons. In parlicurClr, any eztre11l4l
of lhe- functional E (1') (a parametrlud gerJdsic) II an eorlrenaal of
L h). but the converse dots not /wid.

0418 7. y Pn"kmt 11'1 R"'r4nnlCJ" C..,1rUlr;
R01Jgbly spell1dng. the funClionaJ L Cy) hM .'more o , extremal! than
tbe function.l E (J.
Proof. Let. us c.onidl!r EDler 's equ8t Jons for L (v). The Lagrengian
L ;,. of lb. form L (r, ;) = Jf II,P':.!. Wo oh'.in
.!!:-..!. { .!!: ) -0 ttn,
a.,,'f III iJ: 1 - I
]I t.. .s.. (r.,Z'.i:'J-  ( ]I In  (f';'J ) =O.
1IPlzJ Bli"l a
Let y (t) be 8 soluliOD 4Jf thia system. SinclI 'Y (t) is a &mocth curve
JnM ft , tbisc1Jrveadmil!l the D8tufiil.1 parllmewt =.. Then I y (.I) )....t
aJor-g 'V and. therefore. Eu1er"s equations bet-orne
. '," . ( . .," )
... (6,,,, r'J-Tt hO(Bu" r'J -0,
because {B'F4J = f. But these equ8\ions coincide. 8ccording to
Theore:m t. wit.h the equations of geodesics. Thu':!h if e. n&1ural pa-
I'umelet' is in(C'oduccd on an arbitrary solulioD l' (t), "n extrema) of.
L (Y)I this trajectory becomes 8 geodesic. COJl'vol'a:uly, let,. (.I) be an
arhilrary geodesic on M"; acclJ1'diJlg to the precedipg consideration.
it is tin eIt",me) of L (y). Let S I:::: . (t) .be en or1Jitn.I'Y smooth pa-
rarnotel' suh.Utut!oo on V (.J. Then L Iv (s)) = L (V (s (1))). since tbe
ledgth of arc d()es not change under smooth parameter euhstitution;
hence. tho YlIlue of L d4Je.s not change either. Thus, v (If (In il!l egain
an e"lrernal. The tbeorem ls proved.
It is (',onvenient to fnterpret the relationship hetween the extrem&ls
ar £ end L e.s follows. Let tiS consier the space o,M JI or all smooth
C\lrves on M fl Bnd the action. on t11is space l Dr the inftni!e-dimen.sion-
81 group  whose elements are represented by Brbitnry smooth p.a-
Tameter substitutions on a curve. Then each p'Jint 0' this space. i.e.
B curve y generates the Drbi1  (y) of the action of  en D. Since L
is irwariaDt under the schon 01 . tbi05 ftlndiona] ls consta:nt dong
e8ch orbit  (y). l' E D (Fig. 7.8). AI.I5Q. 5ince L (y) is constant elaD&"
each ClTbit, aU poinh on the grbit  (y) elre degenerate in the sense
that lhtllV exist CUI"V.!:I (i.e. parllmetriz.ed traJectones) arbitrarily
c.loe to lhe (;vrvo V, on whic.h L has tbe sal1lO ",slue as on y. The
siL1J8tion iB difforent. howev6r. f()r E. Thi.s function.) v8ries under
paroflmeter 8uIJstitution ond is thereforEJ not. con,Lan1 OD the crbit3
of the action Df (I). ThlJs, tv find QIJ tbe exLremttb of L. om!: should
conider t11e orbits of eJ1lhe extremal:s of the funt'tiol1al E..
Geodesics ere local m.inlma Dr both rundiooa1s L aDd £. i.e. if
We (;(;In:sidcr 8 ame1l perturba.tion 1] of 8 geodesic \'. where the :support
'1 is 81:30 small. the length of the new trajectory 'f + I) is .nol leS3

7.2. Ez'rnnalU.1I 6If GeodtAlI
41.
then that. Clf y. We now lamtull!l18 the exact proMrm). C()nsider a
ompllct Riemannian mBl1lfo)d M'". Aceording to the results of
CII.pter 5. there exbiLs .en E > f1 such tbet any pair of points P, Q
tl>J
t
F.lzure 7.8
in tile lIaU n: (If ndiWil e. cen be t'IInnected l:Jy a I.dlqne geQdle
Entirely [)lima in the ball.
Theorem 3. ut n: and e b lh(! b4U and the IJumbt'r juri menlfoned,
and let y: 10. tJ ... Mn be a Codic vi length leu than 1£ wltte4 cCJnner:tlo
lwo pointl In n:. SJ£pp(Js to: 10. 11 - M" ;« anolher 'mho!h path.
ollrvctiJ1g the .sa.m.t poin (lht palh can also be p6I1tIJDth).
Thtn L (ro) ;> L (y). the t;uczliry taking plan Dnly whm the poinJ
stu V 10, II end (IJ [0. U cvirn:fde (i.e. 10M" tluu two path. cDindde
61 8mooh cu,.lJt. ;n M"). In 'hI, sens, Ihr: grodesic \' i, flu t:hork3J palh
bdwtel: the point P and Q.
Proof. L(ltP be lhecentre of the ban D:. FrlJm ChBpter awe know
that. the smooth mapping exp..,.' B: -+ M" defined by expPo. =I
1'1 (t) is.B difieomorphi:!tm. Here a E T.M". n is the bell cf radius
 in T pprt. Bl1d Y. (t) is 8. geodesic in M'fI &l!ch that 'We CO) = Pr.'O
Y. (0) = 8 (see Fig. 7.Q.. Thi mopping is called exponenUlIl. Sillce
T ,..M- is pro'Vided with metric.induced Kalar product. e:&:p,.. pre-
servee the lengths of veclor.s emerging f['m the point V. We hQw provo
that in n= geodesic.!:! emergiflg from Pr. are QrthQgooaJ 10 the ))YP8r-
sudaces SQ-I = {exp pt a j I_ I = r..ol1st}. indeed, lei I _ a (t) be
an arbitrefY cur\rG in T p).f" such that. I a (t) I = I. It is t"eQuired
to vc.rify tbst. the correspol1ding c.urvu in MOl, t _ eXp,..(ToB. C').
0< To  I.. ere crthogQnal ta tile geodesics T-eXp,.. (r.a (I)) (see
Fjg. 7.10) Let. \13 consider II two.dimensional surface
{ O,;;r<£,
I.-A":;t";:to+A'
/ (r, I) = .'PI'. (ro (I)),

"20 1. V"I'1.lI.G' I'TO"II\I 11'1 R"mall.l'Il"." Cromd'1l
pammet.nzed with two Jlarameurs J r alld ,. Thi'8 l1rf.ace ma.y be
looked upon .s . con. gen.rated by the ..rv. 1(r., I) (Fig. 7.11).
It is I'Iquif9d to prove thet < : " ::) .11:10 at each point of this
FIJUI'f! 1.8
Flj\I" 7./0
81 "
:euda.ce. where "ir end "ji" 81'8 tangent vectors to tbe coordin.a16
I1stwark on ..he $urfete. We have
. ( " ar ) ( 0' BI )
8r lh.7[ -==V, '"iJj .
where V, is th. to".rianl derivatlw. with Jesped to r. A!so,
( "' 01 ) < CI " ) ( at .1 )
V, .7ii = V'W""ji" + .Vr71 .
Si.nce the CD"'.! r_1 Cr, t) ere isodesits and : is the geodesic
.1 ( .1 " ) h
'VeClO'f lield" we ha\'l V r W-O. whence V r -;:; ., '"it :=0. Furt et-

,.,. Ecm.d" oj G,DIk.tn
.2,
more, ( : , V>=( : . V, :) .8ince VI(  )=V,{ : ),
I.e. V.::. c:. ) = V -!. ( : ) because the fields :,. and :
.. ..
.r
"II
.r
Tr""
rip", 7.1t
commute (988 Chapter S, Src.. 5.5'. Hence,
( " 81 ) I ( " " ) t. ( " Of )
I VI i1r c::I"2V, 7;' {Jr =Y8r Tr' 7j;' =0,
. ' "' I I ( .r " ) ., . .
8JnC8 &' = . 7r = canst ..nd &' I. the velocity vector aJ
thegeod..ic r_l(r,/) (... (Fig. 7.11).
Thus. the [unc.tion ( : .) OONi not deptlnd OD t. But fcr
rO we have I(O,O-expp.O=P a . i.e. "f; 1) =0. Wbll11C8
( ; t ::) ....0 for.U,. ",hlen Ia what. was required. Let DOW
(1): (a-,-"I-D;, Po be an 8Jbitrary amElotb curve. Eacn point of
..(/) caP uniquely hs npr...nted io the form .xp.(r(t)a(t),
we'" O<r(I)<. and la(IIf-I, a(tJET.M'. We now pr<w.
.
that  ,.;, (t) I dI;;Io I r(6)- r (0) I, the equ.my takioR pla.e i' only
I.ha f:nr:tion T (J:) is monolt,onl.C end the functwn a (t) is CODstant.
Ouee this Jact. is proved. We immediately obtain 1-hat 8 redial

422 't. Var1'alloM;! P,."IIIm.I In .Rlm.n,.I.,. Orcll7ldr'J"
geodesic is the !lhortest curve connectinG two cODcentrie Bpheres
with cent", at p. (soo Fir, 7.12). E\oc",1 th.t f (r. t)-'a,pr(r.a (,)
and, th...!or., ..(t)-flr(t), t). Wa have   :; .'(t)+ :, . Since
I"(Q 811d :: 8re or1hogonal (:11M above) and 1:1_1.. we obtain
I : 1 2 -I" (t) 1 2 +1 :: 1';;'1r'"(IIII,
1heeq\1Blity taking place if Dnly I :: Ic::zo. i.e. for :') =0. Thus.
. .
S I  jd';;' S I" (IJ I de;;.r r (b)-r(a) I,
. .
the equallt)' taking place if only r (t) l.s monotonic and . (t) is con.
stant. The inequality is proved.
We now turn tG the proof or the theorem. Let (I) (t) be 811 srbitrary
8moot.h path Irom p. to P = expp. (r.) E Da. where 0 < r < e,
FI.-.re 1.12
Flgu.. 7.13
I_ I c= 1. Then for tiny fj > 0 the path tI) (t) mu:st include a BmooU,
l!IesmSI11 frOtn tbe .sphere S;-1. or radius ti to the sphere S-1. Df radius r
(see Fig. 7 13). Ac.cDrdiug to whet hu Deen proved above, the length
of this flegQ1ent is not Jess thaft ,. - 6. By makiug li lend tel zero. we
find that. tho leJlgtn or w is Dot Jess than r. On tho other J1and, the
l!Iegment of the leodes:ie from Pft to P is of length r. The theorem is
proved.
Coron.ry 1. L' w 10. eJ...... Mr. b! a smooth traJectorJl paramtlrtud
by rhenQl.ur"iparnt"'erand let:he length 01 the path frompu = c» lO)
r. '" (.) nQ/ uc..d th. unglh .J t11i .ther pa1h Jr.m P, t. '" (E)
Thm  1. G «",Msie.

1.1. g"rremalltv ..I C.od",cl
'"
Proof. Consider an arbitrary path inside the ball D; delined above,
where Po IS lhe beginning of the path Thell the statement fellows
from Theorem 3.
Delinitlon. A lI'eodeSIC i: la, bI- M" is called nunimal if it is
sherter than any other smooth path connecting its axtreme points
,,(a) and ,,(b).
Theorero 3 assert!! that any sufficielJtly small segmel1t (II a geodesic
is minimal. At the .\Same time, a ..ather long lI'eodeic may not be
minimal For example. we have proved above that allY equator Oil
the sphere S1 i!l II: lI'eodesic. Consider the segment SNP 01 this equator
shown in Fill'. 7.14. Apparently, the gBodesic SNP is 1I0t minimal
beclluse the segment Pa.S of the equator I!lShorter than SNP.
FiJu",7.1"
A minimal (Ionll') geodeic connecting two points P and Q need
not Ilecessarily be Iwique. For In61lmcII:, the north end !louth po)es on
a two-dimensional spbere call be cOllllected by infinitely uumy mini-
mal geodesics. merldiallS.
M another ceroHary of Theorem 3, we now demonstrate how to
fmd all geodesics on an n-dimensional sphere 8" (n 15 erbitrafY). We
migbt. of C':OUfse, froceed by aJlalogy with the two-dimensiOllal case
::O;;i::.seer:S:: I:eD:rIgmt:e t;.(1(
Ja O;::fc i :u;::;yO £;: ;: ::.sa:11 ':jl1
by II tlW-dimt'1Uiorwl plllnfJ lhrough lhe sphtre c:tntrt).
Proof. Let 115 consider an arbitrary two-dimellsional plane R
!ntefsecting sa alonll' the equator y. Let , he 11 reflection in R"
which pfeserves HI! fixed. Then tbere exists the isomotry,: 8" _ 8"
such that tbe set of its fixed points exactly coincides with tha equator
i. Let P and Q he !!uflicieDtly close points en V which are connected

424 Y. Y/U'Ulllon,,1 Pro&>l.nu In 1lIII"n","'n Gu>mtrJl
by II. unique minimal geodesk (I) (see Theorem 3). Sinee B is so bo-
wetry. the curve g (w) throulj:h the points P and Q is abo II. Ij:eodesic
;:hth:':..t n:n:t:=a=i: ;deiilaJ:s t bd:::
dimensionaJ ellse: (lnly one equator P8BBeI through any point {If a
sphere sn in any direction. The oorollary is proved.
UsinS exactly the same reasoning, we can prove that any merid(II.D
of I. .surface of revolution is a Beodesic.
We consider II.D II.rbitrary Riemannian compact mll.nifold. Let
P and Q be arbitrary poin II.nd let C (P, Q) be the space of aU SJI.!{loth
curves 'V connecting P II.nd Q. i.e. 'V (0) = P, V (t) = Q. Two fUl1c-
ilJl1l1.ls, E and L (see above), are valid IJn this epaee. We define the
di8tanee p (P, Q) between the points P and Q as the Infimum of the
lengths of smooth curves through these points.
PrcJpDdtJon 1. Let P fmd Q be sulficientlll dOle pointr of a manifold
spaced at a distance d. Then the tUlicUonat Df action E; U (P, Q) _ R
reoches the abSDlutt minimum d" on the mlnjmal f/todt,lc that connlct,
P and Q.
iL e:::;eeb; :cfa:n:::it ::IMI:':Thl;
meal1S that the vaJue of the functional E on any smooth curve suf-
fLCientIy d0ge (both pointwise and in the seDlle of velocity field) to
the minimal geodesic is striet]y larger than on the minimal geodesic
i'!leU.
Proof. Letybe a mInimal geodesic. 1'(0) -= P, Y (1) = Q. Accord-
ing \0 Lemme 1, E (y) = LR (V) <: L' ([D)  E (w), where w is a
smooth curve from P to Q. The equality L' (w) => LZ (y) holds \rue
if and only if w is also a minhnal aeodesic (maybe, differently para.
metrized to within a scale transformation). At the 8ame time, the
equality P ([D) = E «(0) is velid if only the parameter is proportion.
: :ttt: :i::O:s'-8:!"rinisa::::ases where (U
7.3. MINIMAL SURFACES
In Chapter , we considered minlmel two-dimensional
.surfaces M, I.e. surfaces whose mean cUr'VlI.ture H is zero. Let us
nvw analyse minimel surfaces fruffi the point oJ' viewoJ' extremll.l
hmclivns 101' a functi(lnal oJ' area.
Let 119 cOll9ider a functional of (n - 1)-dimenslonal volume defined
on compact hypersurfac:e1! wlllch are the grapbe of smooth fUDct101!S
r" => f (1 1 ,. . ...1,,-J) with the domainD emlJeddedio Rn-J (r,. ..,
X,,-I). Suppose the domll.in D hasasmooth boundary oD IInd is bound-
£'rl Wo:. 'l.tS). If} Chspter 5 we hsve ."allled thet the (n _1)_
dimensional \lolume of a hypersurface IhH = (x" = t (r, ....

1.8. MInim." S..r/"'.s
'"
Z"-I)} can be wriUen lI1I

vol V"-I"", j Y t+  (/".)"dz'...d$"-'.
where Is''''''' £. _ Since tbe Lagrangian L hi af the form
LV." .. ..Io!'-') 11 ,+.f; (/)'.
Euler'lI equation becomes
'-,-- ( .!L. ( 1+ ' ( .!L ) ' ) - ) O.
l""I".:r J ".:r J 10:1 iI.:r 1
This is a differentia) equation satisfied .by the extremal function
:J!' =1 (Xl, . . ., X".I). Let WI cOn.!l'ider an arbitrary (n _ t)-dimen-
lional hypersurfece ",II-I c R" which hi an extremal fUJI&lioJl for
Figul'l! 7.15
llie functlanal vJ (n - i)-dimensional volume. Since volume is a
scalllr ,.lIId does rwi depend on tbe cbllice vJ coordinlltes on the 6ur-
face. we may work with pllrticu]ar suitable cvordlnate.s. Let ",,,-I
be an extremal surface and let P E p.-I be an arbitrary point. We
set R,,-J (r. . .., %"-1) = TpY"-\ i.e. we deal with the tangent
plane with Cartesian cllordinates; we also express (locally) V".,
as the graph af the funclilln x" = / (r. .... %"-1).
Theorem t. A hyper-sur/oct 1",,-1 c: R" i zlrema' for lhr: JunctionaL
0/ (n - t)-dimtnslOnal volume if'l1./Id only .Jlb mean curualurr: Ii
is IthntlcaUy ZeTO.

426 7. V arlaUonal Probl.tn I" R leman'ltan Geo71klrg
Prooj. We consider the case n = 3, since for arbitrary n calcu-
lations are quite similar. For M2 c: R a we have
.!... ( Ix )  ( f ) _ 0
f)z Y1+r..+n -to ag Yt+/+J - .
Differentiation yields
Ixx+ lul:' + In/ -//xx-j,rI"fVJ+f"lI+ f""I:'
+ f"lIf - fxfll/xlI- fvlllV = 0,
i.e,
Iv: (i +f;)-2fxMx+III" (i+I:')= O.
By virtue of Chapter 4, this equation coincides with H E!5 0:
Thus, a minimal surface can be viewed as a surface defined by an
extremal radius vector.
Let us consider a surface M2 in R3 referred to (II., v). i.e. r (II., v) =
(x (u, v), y (u, v), z (u., v», Then the functi onal of area is written as
(see Chapter 5): S (r) =  V EG - 1"" du dv. Let (II.. v) be COII-
D(u, .)
formal parameters, i.e. the parameters in which the metric on M'
is of the form
E=-G=(r", r u }= (r o ' r.), F=O.
Then
8(r]=  «rl<' ru)(r., r,,})1/2dudv,
D
Euler's equations are written as
a ., a iJ
Tr;" (2%..) + Q;" (2x.) = 0, 8U (2y,,) + 8V (2yo) = 0,
a iJ
Tr;" (2:..) + DU (2:.) = 0,
i.e. ( aa:. + .,: ) r (II., v) = O. Thus, r is a harmonic
(relative to the coordinates (II., un. Hint:
L (r.., r o )= V(:Z:+ y+ z) (x:+ g:+z:)
and aU partial derivalives are considered as independent in the
variation procedure,
As in the one-dimensional case, we consider the inftnite-dimension-
al space F of smooth mappings DB (u., v) in R 3 . On F, there is defined
a non-linear functional of area S rd. its extremal "'points" (i.e.
radius vectors r (II., v» being described by the following theorem.
radius vector

7.3. MInimal Sur/de..
427
Theorem 2. The vectors r (u, v) are extremal vectoT't for SIr} if and
only if their mean curvature H is zero.
The statement follows from Theorem 1. Let us consider one more
functional, the Dirichlet functional D Ir! = } J (E + G) du dv,
D(u, u)
and compare extremal "points" for the functionals D and S.
Theorem 3. The vectors r (u, v) are extremal vector8/or the Dirichlet
junctional, it and only i/ they are harmonic with respect to (u, v),
. ( al al )
t.e. au l + a.;. r (u, v) = 0.
, 8 ( 8L ) a ( aL )
Proof. Euler's equations are of the form au '8r; +"iV "8r; = 0,
where L=E+G={r.., r..)+<r., r.), te, L=+:t':+y+y:+
a' {}'
+Z:,i.e.t1r=O, wherel1=;rur+ava' The theorem is proved,
The coordinates (u, v) may be harmonic for a harmonic radius
vector, but they may not he conformal for the induced metric on
the surhce swept out by this vector. We now consider a special
harmonic surface on which harmonic coordinates are, at the same
time, conformal, i.e. E = G, F = O. Such a surface is, therefore, min-
imal. Thus, certain extremal "points" of the Dirichlet functional
generate extremal "points' of the functional of area. In other words,
if the initial parameters of a harmonic vector are subjected to an
arbitrary regular transformation, each "harmonic point" referred to
conformal coordinates gives rise to a family of minimal radius
vectors. This situation resembles the interaction between extremal
"points" of functionals of length and action in the one-dimensional
variation problem, i.e. on the space of smooth trajectories. Not
every harmonic vector generates a minimal surface. The functionals
D and S satisfy the relation D [rJ  S [rJ for any radius vector, the
equality taking place if and only if E = G, F = 0, i.e. if the coordi-
nate network is orthogonal and the coordinates (u, v) are c onformal.
The proof follows from the obvious inequality Et G  V EG - Fa
which becomes equality if and only If F = 0, E = G.
Thus far, we have proved that a minimal surface is an extremal
of the functional of area, but we have not justified the choice of
the term "minimal surfaces". We did so for the one-dimensional case:
we proved that a geodesic-Is a locally minimal trajectory. A similar
statement holds true for an extremal of the functional of area: a
minimal surface has the property that for an arbitrary perturbation
1) of the minimal radius vector r with small support (i.e. perturbation
is non-zero only in a small domain) the area of the "perturbed surface"
r + 1) is not less than the area of the initial surface. The proof of
this assertion requires an additional analysis, since the vanishing
of the first variation by no means ensures that the extremal "point" is

428 7. VarialioRal Problerru in RIemannian Geom.lry
locally minimal in Lhe space of all radius vectors. Even for an ordi-
nary Iunction of one argument, the vanishing of the first derivative
does not show with certainty whether the function has a minimum,
a maximum, or an inflection point, For example, in the case of geo-
desics we had to analyse thoroughly the local properties of the func-
tionals of length and action. And this is a "second-order" analysis,
just as in the case of an ordinary function of one argument for which
we usually analyse the second-order differential to study the local
behaviour of the function near a critical point.
While investigating local minimality of a geodesic, we considered,
in fact, the so-called "second variation" of the functional of action
Figure 1.16
(or length), an analogue of second-order difterential. Hence, in the
case of extremals of the functional of area one should also study the
"second derivative" of the functional. We shall not go into details
of the second variation theory, but instead we shal! use a simpler
approach (though this approach is more special).
Let us consider a minimal surface M2 c: R8 (the reasoning is
similar for an arbitrary n), P E M2, and define M2 in a small neigh-
bourhood of P as the graph of a smooth function z = 1 (x, y), where
the Cartesian coordinates (x, y) vary in the tangent plane to M2 at
P. We now prove that any sUfficiently small perturbation (with small
support) of M2 does not reduce the area. Let in R8 there be given
the graph z = I (x, y) defined over a domain D in the plane R2 (x, y);
the surface is minimal, i.e. H = 0 and we consider the perturbation
of the graph that vanishes at the boundary of D. It is required to
prove that S (/1 .::;;;; S [/ + 111 (see Fig. 7.16).
Let us consider the space F (I) of all smooth functions / + 11
defined on D and such that TJI8D= O. This linear space depends on
the choice of the function I. The space F (I) is obtained from the linear
space C of smooth functions 11 vanishing at the boundary, i.e. "per-

7.S, Minimal Sur/ac..
429
turbations" of the function f due to shifting C by the function T]
(see Fig. 7,17). Consider the restriction of the functional S to F (I).
On this space. the functional S associates with each function f + T]
the areas of its graph. Let us prove that this functional is "convex
Fe ') , , + q
VI
o '1 C
Figure 7,t7
downward", i.e. S [etl' + ligl  a.S [I'I + liS Ig], where a. +  = 1;
r, g E F (I), i.e. r = I + 1].; g = f + '12' where llrlllD = 0, i =
1, 2, Note that a.r + g E F (I) because
(a.r + g)laD = a.rleD + (1 - a) gleD
= a.lleD + (1 - et) fleD = fleD,
i.e. a.r + g = t + '1a. where Tlal/tD = O. The definition of a convex-
downward functional copies the analogous definition for an ordinary
.::
'.,
..
'or
Figure 7,18
function, The graph of a convex-downward functional is shown con-
diti onally in Fig. 7.18. Thus, it uffices t<J verify that
'VI + (etr" + g,,)2+ ( cx/'u+ gV)2
";;a. V1+r t-r+ Y1+gi'+g;.

430 1. Varlat/ona! Probltm.t In R Itmannlan Geomtlt1I
Squaring this ineqURlity, we obtain
i + 2Gt (ro:g,,+rllgll):S:;;;Gt+ 2+ 2C1. V(i + r+ r) (1 +g+ gt),
and since i - 0: 2 - 2 = 20:,
i + r"g,,+rllgll:s:;;;V(i +r+ r) (1 + rl + g),
or (a, b):S:;;; I a 1.1 b I, where
a=(r", r ll , -1)= grad (r (:t, 1I)-:),
b=(g", gll' -i)=grad(g(.x, y)-:).
The inequality (8, b) :s:;;; I a II b I is obvious. Thus, we have proved
that S is convex downward on F (I), wbence it immediately fellows
that any extremal "point" in the space F (I) Is a minimum for the
functional S. In particular, for all points r in some neighbourhood
of an extremal point g E F (f) the inequality S (r]  S [g] is satis-
fied. Since the point f E F (f) is extremal, we have S (I] :s:;;; S (I + T\I,
'l}1D = 0, Thus, we have proved the following statement.
Theorem 4. Let M'l c: R8 be an arbitrary minimal surface, then this
surface is locally minimal, i,e, any smooth, sufficiently small pertur-
bation with small support does not diminish the area of the surface,
The theorem on the local minimality of extremal solutions for the
functional of an (n - i)-dimensional volume in R n is proved sim-
ilarly.
Vi. CALCULUS OF VARIATIONS
AND SYMPLECTIC GEOMETRY
Let T .M n be a tangent bundle of a manifold M", Le, a
2n-dimensional smooth manifold whose points are represented
by pairs {(x, a)}, where x EM", a E T.M n . Also, let P. Q EMil
and let Q (P, Q) be the space of all smooth curves connecting P
and Q, Let local coordinates xl, . . ., .x n be valid on M n in which a
trajectory is written in the form "l (t) = (Xl (t), ' . ., x n (t», x (0) =
p. x (i) = Q; ; (t) is the velocity vector. Consider the functional
t
1 Iv] = . L (.x, ) dt, where L is a smooth function of two groups or
o
variables, (x, ;) = a, i.e. L (.x, a) is a function on T .M",
Definition, The momentum p = (PI), t :s:;;; t :s:;;; n, is the covector
with the components PI = B (in a given coordinate system). The
8",1
energy E is the function E (x, ;) "" ;/P/ - L (x, ;).
The energy can be considered s ,) function on T .M R , i.e. E (x, B) =
aipi - L (x, a). An important example: L (.x, ;) = I[/;';J, in this

'/,4, Cclulus of Var!attom and Symplectl Geometrv 431
iJL ". .
case PI = "7 = 2g lJ :z.', I.e. p IS the covector dual to the velocity
iT:z I
vector a = ;; relative to the metric gl/' The energy E is of the form
E(x, ;;) = ;;12gjJ- KU;;I;J= gll;i,
i. e. it is "kinetic energy".
Let (!I be a Lie group which acts smoothly on M ft (xl, . , " x ft )
(i.e. each element gEe is represented by a diffeomorphism of M n ),
We shall focus on tile following example: IS = Rl, i.e. the real
straight line (the group with respect to addition). This action (for
d
(!I = Rl) generates on M ft a vector fIeld X (xo) = di' g, (xo) 1,=0,
where the orbit e (xo) = g, (x o ); g (xo) = .1"0' The action of <9 on
M n induces the action of <9 on T .M ft according to the mapping g.:
(x, a)...... (g (x), dt", (a».
Definition. The Lagrangian L (x, a) is said to be preserved under
the action of <9 (or is invariant with respect to (9) if the action of e
on T.M" transforms the function L (x, 8) into itself, i.e.
L (g (x), dg (8» 51 L (x, a), (x, a) E T .M n .
We now determine analytic conditions for L to be invariant, The
condition of the preservation of L is  = 0, where t is time along
the trajectory It (x), The differentiation of the composite function
yields
=.!£ hi .!£-O
dl azl dt + 8al dl - ·
' =X I (x)
(see above). Let us calculate  . Consider a and a tangent tra-
jectory CjJ ("C) = (Xl ('t'» such that 0. 1 = d d :<1 I . A small shift by 6.t
T ""0
along I' (x) transforms the functions Xl into xl + Xl 6.t (small
quantities of order higher than one are neglected). It follows that
_ +.!!!. 6.t i.e. al_a l + B a K :z : a6.t,
dT d'l: 8z dt' ft
. da l 8X' 
I.e. Cjt= 8ZR a,
and therefore gAdx l ) = {.xl + X f 6.t+ .. .}.
8: (gAc(.x I ») = 8 (Xl + X I 6.t + .. .)  (6+ :; .1.t) = Jt

432 7. Variational Problema in Riemannian Geomet11l
J = (J) is the Jacobi matrix, whence
j . . ax} II
a -+ Jf.a k = a' + 7iiii" a 6.t.
Thus,
E:..=.!.!:... X'+.!.!:... a"O,
dt azl oal a,,/'
where (x, a) are independent arguments. We have proved the follow-
ing statement.
Statement t. The analytic condition for the Lagrangian L (x, a) to
be preserved u.nder the action 01 the group e = Rl is
!..!:....X'+!..!:..... !.!:.ak=O,
azl oa' clzil
where (x, a) are independent arguments and X is the velocity vector
field due to the action 01 e = Rl.
We now derive the so-called law of conservation of momentum
projection along the extrema Is of tbe functional J (1')' Let us
1
consider S L (x. ;) de = J (y} and let Yo he an extremal of J 1,\,).
o
i.e. a solution of the system of Euler's equations !..£ -
azl
:'\" ( iJ!, ) =:: O. where 't denotes time along the extremal. Let a dot
azl
stand for differentiation with respect to the parameter 't (along
{JL '
the extremal), then Euler's equations take the form - , = PH
a%
where p is the momentum. Consider the contraction f ('t) = (p, X) ""-
X'Ph where 1 (T) is a smooth function along Yo ('t). Here 1 (T) may
be assumed to be the value of the covecLor P 011 the vector X,
i.e. fC') = p (X).
Theorem i. The identity U (T»' = 0 is l'alid, i.e. (p, X)'  O.
Prool. From Statement 1 we have
, . . . . . oL ax' oL dz"
(X PI)' =X'P,+X'PI =X. ---:--r--;:---r - d 
{Jz' {Jx n 8a .
=.!.£ Xi + !E...  a" = 0,
{jz' a:!< {Jail
which is what was required. The theorem is, proved: .
Let us consider our model example; L (x, x) = gux'x1; LIten PI =
2g1J' i.e. XIPi = 2g/JX i ;; = 2 (X, Yo) = consL along '\'0 (.),

'1.4. Ca/cutu, of Vartatlons and Sympl!ctic GeortUITY 0\33
Extremals of this Lagrangian are geodesi(s (see Sec. 7.2); in partic-
ular, I Yo I = const along ....0 (T), whence I X I cos Ct (-c) = const.
where ex (T) is the angle between the vectorS X and Yo (measured in
the metric go).
We now use this result to find geodesics on a surface of revolution
M 2 in R". Let r (z) be a smooth function which is the generator of
the surface of revolution, i.e. the curve y = r (z) rotates around the
a:-:is Oz (Fig. 7.19). If (iI = const is the speed of rotation, there arises
Figure 7.19
on M the action rwt of the group 1!3 = RI, and the field X of the
velocities of this action hss the modulus I X I = r6). In view of
what has been proved above, the identity r (z) cos (X (z) = c = const
holds true for any geodesic ....0 (T) on M. Thus. we have proved the
following statement.
Statement 2, Given a surface of revolution M2 c: R8 with the genera-
tor r(z). Then the identity r (z) CDS (X (z) = const is satisfied along
any geodesic Yo (T) on M2,
)-lD 
Figure 7,20
Let liS consider several examples. Let r (z) = const, then the
surface of revolution is a cylinder, and geodesics on the cylinder
are the images of straight lines after the Euclidean plane is rolled
into a cylinder (Fig. 1.20).

34 7. Varia/IDnal PToblmu In Riemannian a.DrnetTII
Let r (z) define 8 2 (Fig, 7.21). In this cast' geodesics are equators
(the angle is variable),
The equality r (z) cos 0; (z) = const is the necessary (but not
sufficient) condition for a trajectory to be a geodesic (give an example I).
Indeed, if l' (or) is an orbit of the action of tB. then 0; (or) = O. i.e.
Figure 7,21
(p, X) = I X I I p I = const (on the surface of revolution), but
this orbit is not always a geodesic, Let us consider, for example, a
right circulal' cone and its rotations about the axis (Fig. 7,22). As we
Figure 7,22
already know, all geodesics on a cone are the images of straight linE'S
after a piece of Euclidean plane is rolled up into a cone; cirdes are
not, apparently, geodesics on a cone.
Remark. Since ; 51 0 along the orbits of the operation of <!I =
RI (for an invariant Laglllngian). we CRn locally choos!! a coordinate
z" along the orbits  (.7"). We have then (p, X)' 51 0, which is

1.4, Calc..lu. oj Varlal/olll and Svmplect/c atOmelrv 435
equivalent to Pn = 0, where Pn is the momentum projected onto the
coordinate x n . In other words, in this case the momentum p does
not depend on xn, and the function L does not depend on this coor-
di nate either.
We now prove the so-called law of tonservation of energy along an
extremal. Let L (x, a) denote the Lagrangian (not necessarily in-
variant under the action of a group) and let Yo ('t') be an extremal
of the functional J ['I'I = ) L (x, ;) d't'. Consider the energy E =
;I p , _ L, p, =, and suppose L = g'j;9 - U (x), where the
. a",i
function U (x) is called a potential.
Theorem 2. The identity :: 51 0 holds true along extremak Yo (..)
of the functional J lyJ.
Proof, We have
dE ( 'I L) ' ( I L) '
(j.t= P,x - = p,a -
=Plal+PIi_.!.!!...;I_ai
ozl aa'
=al +I_al_1E30,
az l aa' a",1 aa f
since PI = , according to Euler's equations, and p, =  by
 w
definition. The theorem is proved,
We now turn to studying simple concepts of Hamiltonian mechan-
ics on a manifold.
Let T .M n be a tal1gent bundle of a manifold provided with coor-
dinates (x, II) and let L be the Lagrangian on T ,M n .
Definition, The Lagrangian L (x, a) is called non-singular if the
equation
oL(z, a) ( . aL("" 8) 1'''''''' )
P=----aa-- I.e. PI=--a;;I'"' _I"""n
has a unique solution a = a (x, p) for any x E M n .
Besides T ,M n , we also consider a cotangent bundle r'w, i.e.
a 2n-dimensional smooth manifold whose points are represented by
pairs (x, p), where p E TMn, i.e. p is a covector. Let us consider
the energy
E(x , a)=p,al-L(.x, a)= aL{z, 8) a'-L
Oaf
with P = P (x, a). By expressing a = a (x, p) from p = p (x, a)
and suhstiLuting into E, we obtain the function E (x, a (x, p» =

436 7. Yarlational Problems ill Rltmann/an Geometry
H (x, p) eaUed the Hamiltonian. It is of Lhe form H = Pial (.:c, p) -
L (x, a (x, p», i.e. L (x, a (x, p» = Pia! (x, p) - H (x, p). Let
l' (t) c: Mil be a smooth trajectory, then (x, .x) E T.M n , where; =
,; (t), x = x (t), Le, l' (t) generates a trajectory r (t) = (x (t), ;; (t»
on T .M 7I . Let us now consider the Lagrangian L (x, a) on alltrajec-
tories of T .M n , not only on trajectories r (t) = (x (t), ; (t» (noL
every curve on T.M 7I is of such a form).
We note that T .M" and T'" M 7I are diffeomorphic. I ndeed, let the
metric go be defined on M n , then there exists the invariant iden-
-L-
--;=r-
'" y
r(x, B) = X L""",....."">""""'"",,,,..... q{X, p);:;:;: x
M"
Figure 7.23
tification q>: T .M" __ T* Mil; q> ({Xi}, {aa.}) = ({xl}, {g"aa.}) E
T. M n . Besides the identification q>, there exists' snother identifl.
cation which arises in defining a non-singular Lagrangian L (x, a).
Indeed, the equation P = iJL :' a) has a unique solution a =
a (x, p) (for a fixed x), so that with each pair (.x, a) we can associate
the pair (x, p), where p = iJL :' 1\) . Let us consider this identifi-
cation of T.111" and T'" M", which is a diffeomorphism. In a particular
case of L (x, a) = glJa'a J we have PI = 2g lh a h , i.e. the identification
by this Lagrangian coincides with "Riemannian identificatioI\" (see
above) (Fig, 7.23).
We now turn to the variational problem of finding extremals of
s
the functional I [a.] = . L (x (t), a (t» de, where L (x, a) is assumed
R
to be defined on alJ smooth trajectories ct (t) in T .M 7I , and Rand S
are two fixed points on T.M", i.e. R (x o ' ao) and S (Yo, b o ) (Fig. 7.24).
Let us analyse those variations which do not shift the points Rand
S. Euler's equations on T .M" are of the form :I = t ( :; ). To

7.4. CalculI" of Variatto.... and Symplecttc Geomdrg 437
write them in terms of T'" M n , we make the substitution (x, p)--
(x, a) via the diffeomorphism constructed above by the Lagrangian L.
We have
L (x, a (x, p» = Pial (x, p) -H (x, p), P, =  L (x, a (x, p».
aa l
It should be noted that - a a L (x, a (x, p» = 0 since I. (x, a) does
PI
not contain the momentum p explicitly, and in the case of Euler's
Mn
---
Figure 7,201
equation!\ variat.ions are calculated only with respect to the functions
that enter explicitly into the Lagrangian. Thus,
0= - a 8 L (x, a (x, p» = a i (r, p) __ (J 8 H (:t, p),
PI PI
where
L (x, a (x, p» = {  L (x, a (x, p» ) al (:I:, p) - H (:t, p),
a1
. I ( ) aN (z, p) . Hence
),e. a x, p api
B B
 L (x, a (x, p» dt =  [Pial (x, p) - H (x, p)) dt
A A
B
=  [PI ilNiI; p) H(:I:,p)]dt=J(C.oI(t)],
A
where II) (t) E r" M n , II) (t) = (x (t), p (t», II) (0) = A, (J) (1) = B
(see Fig 7.25). Let us now write Euler's equations for the functional
J I (i) (t)] 011 T'" M n , Suppose (1)0 (t) = (x (t), p (t» is an extremal
of the functional J r (i) (t)l. It was shown above that aa = aH a (:>:, p) ,
Pa

38 '1. V..rlaUon..1 Probltnu In RI."'..nn!"n G.o",ttrll
but since a'" =.i'" (along an extremal), we have :I.'" = au(z, p)
op",
(along the extremal). Furthermore,
B B
J Iwl = ) (Pial-H) dt =  L (x, a (x, p» dt.
A A
. oL d ( aL ) 'ilL
Euler's equatIOns take the form - = - dt - ,whence PI = _ I
OZI aal oz
because .!.£=PI' Also, .!!:... =...2.... (alpl-H(x, p» = _ iJn (z, p)
a.. 1 iJzl azl az l
T'Mn
R

-
,,"
/""
Mn
Figure 7.25
because ...!...... (alp,) = 0, While proving the relation ..!- (a/pi) = 0,
ilz l iJ.t;1
we USe the fact that the coordinates x and a in the pair (x, a)
are independent and, therefore, :: = O. Moreover, we calculate the
variation with respect to xl if only this variable enters explicitly
into the sum alpl' But since it does not we have  (aipl) = O.
iJzl
which is what was required, Hence PI = _ an (z, p) , Thus, we
az l
hav!' proved the following important statement,
Theorem 3. In the coordinates (x, p) on the cotangent bundle
B
T. Mil Euler's equations for the functional J (w] =  L (x, a (x, p» dt
A
form: Xl = an (z, p) ,
api
can be written in the so-called Hamiltonian
PI =- oU(z, p) (Hamilton's equations),
az l
Let us make some comments on the implicit form of the variable
xl i]:1 the sUm alpl' For example, if L = Kualai, then a l is expressed
in terms of the momentum P alone (x does not appear explicitly):

7.4. Calcul... 01 Varlatlonr and SlImpl.ctte Geomdrll 439
PI = : = 2glja J , whence a lX = {- gMP4' Of course, the functions
gc4 = ga)!. (x) do depend on x, but this variable does not appear
explicitly in a lX , and therefore there is no variation with respect to x.
We now turn to studying Hamilton's equations on T.M n . They
define on T.M n II smooth vector field referred to the coordinates
(Xl, PI)' t :s:;;; i, 1 :s:;;; n. This field is a now of special form; not every
. 8H(z)'
vector field on T. M n can be represented as xi = a' p , PI =
PI
8HiJ; p) . This field is closely related to the field grad H which is
a covector field on M n . Indeed, consider on M" an exterior differen-
tial form of degree two (O()= dpi 1\ dx l referred to the aforementioned
local coordinates. At each point (x, p) this form can be represented
as the skew-symmetric matrix
o
o
-t 0
0--
o
-1 0 I
and, therefore, defines a skew-symmetric, non-singular scalar product
both in T(", pi (T. M") and T.("" pi (P M"). Hence, this scalar prod-
uct defines' identification of the tangent T(.., p) (T. M n ) and cotan-
gent T(, p) (T. M n ) spaces at each point. Let us consider the flow
grad H (x, p) with the coordinates ( 8H8i p) . 8H8' p) ), then
the dual flow relative to this scalar product is of the form
( 8H;;;P) , _' 8H:,p) }, i.e. coincides with the Hamiltonian
now derived in Theorem 3, Hence, the Hamiltonian field can be
defined as a flow dual, relative to an exterior 2-form, to the gradient
of the Hamiltonian H (x, p) given on T.M n . Let us now assume Mil
to be a Riemannian manifold, so that T* Mil and T .M" may be
considered as canonically identified, in particular. we shaH not
distinguish between vectors and cOvectors and use only upper indices
in coordinate expressions,
The material presented below does not presume that the reader
is familiar with the previous analysis of the functional J r (0]; it
merely stresses the importance of Hamiltonian Oows in Riemannian
geometry.
Let us consider an even-dimensional Riemannian manifold M2";
in the preceding c;xamplc M" was represented by r*M n e! T.M n .

440 7. VariationaL Problems In Riemannian Geometry
Let 00(2) = 0011 d:z;i 1\ dx' be a non-singular exterior 2-form on M 2n
which defines a skew-symmetric scalar product. If this product is
given on M2 n , for any smooth function J (x) on M2" we can define
a "skew-symmetric gradient", sgrad f (x).
Definition. The skew-symmetric gradient sgrad f of a smooth func-
tion f (relative to the scalar product 00) is a vector field uniquely de-
fined by CI) (Y, sgrad f) = Y (I), where: Y runs the set of all smooth
vector fields on M'l" and Y (I) is the value of the operator Y on the
function t.
Although the definition of sgrad I is similar, in its form, to the
definition of grad I (for a symmetric scalar product), their properties
differ. Note that the uniqueness of the definition of sgrad t follows
from non-singularity of (a),
We recall an important definition. Let (a)() = WI.. ..., I dXi, A . . .
. . , A dx iR be a differential form of degree k: the form 6)(R+I) = dw(l<)
defined by d<,)(II) =  OCl>i,.:. i" dxl, A dxl, A ... A dx l " is called an
0%"
i.
exterior differential of 6)("). For the i-form O}(I) =6), dx l the opera-
tion d can be written as
dOI(I)= o' d3;I, A dx l = ( OW! - Ow} ) dxl A tbJ, (i < j).
iJ% . OZ' iJz'
Definition. A smooth manifold M2n is called symplectic if the
exterior differential 2-form 6)(2) defined on it is such that (i) 6)(2)
is non-singular and (ii) d6)(2) = 0, Le, the form 6)(2) is closed.
It is known that symplectic manifolds admit a special atlas in
which the form (1)(2) (sometimes called "symplectic structure") be-
comes canonical: namely, for any point P E M2 n there exist a neigh-
bonrhood U (P) and coordinates {pI, . . ., p", q\ . , ., q"} in which
n
6)(2) can be written as ((1(2) =  dpi 1\ dqi. These coordinates are
1=1
called symplectic. Their existence is demonstrated by the Darboux
theorem which will not be proved here, for we confine ourselves only
to an example of a symplectic manifold. Let M2 T1 = R 2 ", let pI, ' . .,
p", q" .. " qn be Cartesian coordinates in R 2 ", and let 00(2) =
"
 dpi 1\ dg l . Since the coefficients ((IW of this form are constant,
1=1
d6)(2)  0, Le. the form is closed. Its non-singularity is obvious and,
therefore, R2 n with the form 00(1) becomes a symplectic manifold.
Another important example of a symplectic manifold is a smooth
two-dimensional orientable manifold, Le. 8 2 with a certain number
of handles. Such a manifold admits a symplectic structure if by the

7.4 Calculus of Varialtom and Symplecttc Geometry 441
2-form of interest we mean an exterior form of volume relative to the
Riemannian metric, i.e, the form defined in the local coordinates
(x, y) as w() = Vgdx 1\ dy, where g = det glJ' Then M 2 becomes
a symplectic manifold, Indeed, W(2) is non-singular because
vi *0; furthermore, 00(2) is, apparently, closed.
Remark, The Darboux theorem on the existence of a symplectic
atlas is not a trivial consequence of the possibility of reducing
a 2-form to a canonical "block" structure at each point. The Darboux
theorem asserts that such a reduction can be performed in the entire
neighbourhood. and this fact needs special demonstration,
Delinition. A smooth vector field X on a symplectic manifold M2 n
with the form (I)(Z) is called H amtltonian if this field is of the
form X = sgrad H, where H is a smooth function on MZ", The
function H is called the Hamiltonian 01 a flow,
Let us find an explicit coordinate expression for the operation
- - M
sgradf. We have; 00 (sgradf, Y)=wIJ(sgrad/)lyJ=y". whence
(sgrad f); = (I)u .!L J ' Le, the vector sgrad 1 is dual (relative to the
oz
scalar product w) to the coveclor grad f = ( ..!.L ) . Here is an
ozl
example of the Hamiltonian field. Let H be an arbitrary smooth
function on M2n referred to symplectic coordinates (p', gi). i,e,
n.
w(Z) =  dpi 1\ dql, where the coordinates (PI, gl) are Carlesian. The
1=1
operation sgrad f is then of the form: sgrad H = ( .!!!..., -  ) ,
iJql iJpl
Le. tbe fteld X= ( iJH . ' _.!!!.. ) , 1:i";;;in, is Hamiltonian. This
iJq 81"
eJ\.ample clearly shows the distinction between Hamiltonian and
potential fields; a Hamiltonian fteld is of the form ( .!!!..., - )
iJql opl
and a potential field grad H = ( 8B , 1!!.... ) . In particular, tbese
opt iJql
fields (for the same function H) are orthogonal with respect to the
Euclidean scalar product. Some fields, which are Hanlillonian for
a function H, may be potential fm' some other fUllctioll H.
A Hamiltonian !leld admits an impOt.tant description in terms of
It one-paramHeI' group of diffeomot'phisms generated by this field.
Let X = sgrad H be a Hamiltonian helrl dnd let @J, be the corre-
sponding gronp of shifts along the iHtegr.d trajectories of the field.
This group acts on the fOl'm w(), sending it into anothcr, generaHy
disti net. 2-fol'm (<!I,)* oo() which CIIII be rcp"esented as follows.
If, the mapping @J, sends a point y to x, (y) = @J, (y), the coordi-
n<1tcs (xi (y)) may be considercd as smooth functions of (yJ); WI.' Ita\'t'

442 7, V orloUon/t! Prob!tm. in R Itm4nnion Gtomelr1l
then
(@t)'" (j)(2) = (J)I} (Xt (Y» dx (y) !\ dxf (y)
IIx) (v) IIx{ (y)  d  d p
=CI)Ij(xr(Y»-ayrQyi>dy !\dyP=(AP(t,y) y !\ y,
IIz) (v) IIz) (u) .
where (J)AP(t,Y)=(J)I/(xr(y»8Ui' (see Fig. 7,26).
A vector field X on a symplectic manifold may not admit global
representation in the form X = sgrad H, but it may possess the
FIgure 7,26
following important property: for any point P E Mn there exist a
Jleighbl:mrhood U (P) and a function Hu defined on this neighbollr-
hood such that X (a) = sgrad Hu (a), where a E U. The field X
will he called locally Hamiltonian. Each neighbourhood U requires
its own Hamiltonian Bu, but these Hamiltonians cannot, in general,
be "matched" into a single Hamiltonian defined on the entire mani-
fold, An analysis of locally Hamiltonian fields is thus reduced to
n
studying Hamiltonian fields in R1n with the structure  dpi!\ dqi,
t=1
since the neighbourhood U can be mapped diffeomorphically onto
a domain in the symplectic manifold (R:", (J)(».
Theorem 4. A !Jector field X on a symplectic manifold lvP" is locally
Hamiltonian tf and only if the one-parameter group !!I, preserves the
symplectic structure (J)(2).
Proof. We prove the theorem only for the case wher!! Ml n is R2"
"
with the structure (J)(2) =  dpl !\ dg i . The general case is reduced
(;:1
to this one by the Darboux theorem (its proof is omitted).
We now demonstrate that if (J)() is preserved under the action of
Ell the field X is locally Hamiltonian, That !J)() is mapped into
itself under the action or el, means that the derivative of the fornl
along the field X vanishes (X is the velority field for the action or

442 7, VarIa Ilona I Problmll In Rlem4nnian Geometry
then
(<!f)* (0)(2) = (J),I (Xf (y») dx) (y) A ax{ (y)
aJ (1/) az{ (Y) k k A d
= CJlI} (Zt (Y» --ayiIBiiP dy A dyp = (k" (t, y) dy yP,
c1Z) (y) c1Z) (II)
where (J)llp(t,Y)=(J)II(.xdY»avr (see Fig. 7,26).
A vector field X on a symplectic manifold may not admit global
ropresentation in the form X = sgrad H, but it may posseS!! the
Figure 7.26
following important property: for any point P E M2 n there exist a
ueighbourhood U (P) and a function Hu defined on this neighbour-
hood such that X (a) = sgrad Hu (a), where a E U. The field X
will he called locally Hamiltonian. Each neighbourhood U requires
its own Hamiltonian H IJ . but these Hamiltonians cannot, in general,
be "matched" into a single Hamiltonian defined on the entire mani-
fold, An analysis of locally Hamiltonian lields is thus reduced to
n
studying Hamiltonian fields in R2 n with the structure  dpt!\ dq',
'i_I
since the neighbourhood U can be mapped diffeomorphically onto
a domain in the symplectic manifold (R2 11 , 111(2».
Theorem 4, A vector field X on a symplectic manifold M2n is locally
Hamiltonian if and only if the one-parameter group !!I, presen'es the
symplectic structure (,)(2),
Prodf. We prove the theorem only for the case wllere M2 n is R'zlI
n
with the structure 111(2) =  dpl 1\ dql. The general case is reduced
1=1
to this one by the Darboux theorem (its proof is omitted).
We now demonstrate that if (1)(2) is preserved under the action of
(!II the field X is locally Hamiltonian, That (1)(2) is mapped into
itself under the action of @II means that the derivative of the form
along the field X vanishes (X is the velocity field for the action or

7.4. C4/CU./U.. oj Varlatlollot and Sympltcttc C,ollUtry 443
<Be), i.e. 0 = ;, 00(2) (1' (t», where l' (t) is an integral trajectory of
X. We have
 00(2) (1' (t»)
dt
n
= :, 2: .dpl (t) !\ dql (t)
II
" .
=  [  (dpl (t» A dql (t) + dpl (t) A d dql (t) ]
i.:J1
"
= 2: [d ( dPt(t» ) !\ dql (t) + dp. (t) !\ d ( dq/t) ) ]
II
n
=  (d (x. (t») !\ dqi (I) + dp' (t) A dyi (t)l,
'_I
where X == (Xl, YI), 1  i n, Also,
.i.. (0 12 ) (y (t))
dt
..
=  [( .!.!!..d p +E!...drt ) A dql
 up" uq"
1=-1
+dpl1\ ( ; dp+ ::: drt)]
n
= "" [( + ) dp" 1\ dql
 8p1t 8ql
I-I
+ ( -  ) drt 1\ dql+ ( !!.:- 8Y ) dp' 1\ dJl' ] =0,
8q" aql <"<I) 8p" 8pl (1<11)
Le.
ax' aY"
- 8p" =aq;-'
i1X"__
BiI- aq" ·
8Y' BY"
8ji1i = a;;- .
n
This means that the i-form (0(1)=  (Y'dpl_X1dq') is closed.
. II
Indeed, according to the relations derived above,

10M 7. V"riallon"l Probl.".. In Riemannian Geometry
n
doo(l) = d (YI dpl- XI dpl)
1=1
n
=  l( : dl'+ : dq)!\ dpi
I-I
- ( ::: dJl+ ;: dt) !\ dqlJ
n
=  [( -) dP' !\ dpl
1=1 op" iJpl (1«1)
+ ( .!!!. + !li!:... ) dt/' A d 1+ ( aXIL _ axt ) dq" 1\ dql = 0,
oq" Bpi 1\ P aql iJqlr.
Since tho i-form lif.l) is defined on R2 n it is closed and therefore
there exists on BIn a slUooth function' H such that its gradient
(ordinary gradl) is equal to this form, Le. CI)(1) = dH. Indeed, H
can be expressed explicitly in terms of the form CI)(I), suffice it to
set
n 1 n
H(z)=  CJ)(lb   (Y1dpl_Xidg l )=   (YI dIe I _XI 'it l ) dt,
,. V(I) t=t 0 1=1
where 1 (t) is an arbitrary smooth trajectory in R2 n from a fixed
point (say, the point 0) to a variable point (pI, ql). The number H (x)
does not depend on the path from the point 0 to the point x =
(pi, qt), For n = 2 this fact has already been proved in Chapter 4.
For n > 2 it follows from Stokes' formula. Indeed, consider
two paths VI (t) and 1'3 (t), where 11 (0) = 1. (0), 11 (1) = 1'2 (i),
then
 (J)(I) = } 00(1) =  CI)(I) =  00(1) =  dW(I) = 0,
y, .... ...,U(-v,) liD' D'
where D2 is an arbitrary disk hounded by the trajectories 1'1 and 1'2
(the disk may have self-intersections). This completes the construc-
tion of H. It remains to prove the converse statement: if X is a Hamil-
tonian field in R2 n , the group @II preserves CI)(2) , i e. X (00(2» = o.
To prove this fact, it suffices to reverse all the calculations just made,
The thl!orem is proved.
Using this result, we can easily give examples of Hamiltonian
nows. For instance, let us consider a sphere with g handles provided
wiiemannian metric and symplectic structure W(2) =
V det glj dx /\ dy (see above). How can we describe all Hamiltonian
flows on this symplectic manifold? We recal! that a fIeld X 00 M;

1.4, Cal.ulu$ of VarlalloM and Symplecllc Geometry 445
has zero divergence if the area of an arbitrary domain A remains
unchanged under shifts along the integral trajectories of the field.
Corollary 1. A flow on M; is Hamiltonian if and only if it is in-
compressible, i.e. div X = O.
Proof. According to Theorem 4, we should describe all the flows
that preserve the form CI)(2). Since by this form we mean the form of
volume, Cl)t 2 ) = V det glj dx /\ dy, the Hamiltonian flow is the
fJeld that preserves the form of volume, As we know from Chapter 4,
this field represents flows of zero divergence, and only these flows. The
corollary is proved.
Let us consider. for instance, a two-dimensional sphere 8 2 repre-
sented by a completed complex straight line R2 U 00. In this case,
we can find on 8 z a wide variety of Hamiltonian flows (with respect
to the standard metric on 8 2 ): it suffices to consider the fields
grad Re f (z) and grad 1m f (z), where f (z) is a complex-analytic func-
tion of the variable z = x + iy. Hence, all incompressible (and irrot-
ational) flows considered in Chapter 4, Sec. 4.4, are Hamiltonian,
Th symplectic structure on 8 z referred to the coordinates (x, y)
(which are used for all "finite" points) is given by CI)(2) = dx /\ dy.
However, there also exist on 8 2 Hamiltonian nows which are not
irrotational (see below).
Let us consider an important example of Hamiltonian flow, the
dynamical system which describes the motion of a three-dimensional
rigid body with a fixed point. We should consider R' identified with
the linear space of all real-valued skew-symmetric matrices of order
3, suppose X E RS is such a matrix, Define in R' a self-adjoint
:i:C:a:::a ( t: <p: R81 ) R:::C;:::e<P:a :::e :::::n:l
o 0 ;'3
the matrix <pX is skew-symmetric. Let us consider the system of
differential equations !pi = [X. !pX) (written in matrix form). It
is known from mechanics that this system describes the motion of
a three-dimensional rigid body with a fixed point. Let us write the
equations in coordinate form. It is convenient to consider an n-
dimensional rigid body rotating in R" about a fixed point. This
rotation is described by the system <pX = [X, Ij)X), where Ij): so (Tl)_
so (n), so (n) = (X: XT = -X), Ij) is of the form Ij)X = Xl +
IX, I = (:!. ) .
Assuming X = (xu), xI} = -xli. we obtain
(cpX)'J= (A, + AJ) XIJ'

4-+.6 7. Variational Probltm. In RltmannllJPl GtOmltrll
(X)'J=  (:l:lk (rpXhJ-(rpX)'k :t:AJ)
k
=  (z rk (Ak + A J) Zkl - (A, .I- "',,) :r;lk:r;kl = (hi - Af)  Z,,,:r;kJ'
. J-I
ZjJ= J+),,' ZI".1:AJ'
For n=3 we have
 t-Al . )",-A1 . A,-A.
u= A'+"1 zl..:I: 3a , a:1 1 = A.+Al %ta:l:aa, Zal= )".+1... .:1: 81 .:1: 13 '
Setting x". = X, Z13 = y, Zaa = Z, we arrive at the following system:
'_ A 1 -;., . )..-1. 1 '_ ".-)"a
z- "1+)". yz, y= )".+),,1 .:1:%, z- 1.. 1 +)". yz.
This flow (in R3) admits two integrals: P (r, y, z) = x' + yB + Z3
and Q (x, y, z) = x' (hI + ha) + y' (AI + hI) + %' (h, + h.), In-
deed, to prove that these two functions are constant along the integral
trajectories of the flow, it is sufficient to calculate the derivatives
:k (P) and :k (Q). We have
X (P) = 2zyz ( l-t + AI-1 + At-' )
AI+AI 1.a+ 1 A.+.
=> (AI +1.a)("+I) ()".+".) [(i.. I - ha)(h.+ 1. 1 )(1. 1 + As)
+ (As-AJ (1..+ i..s> (1..+ A + (AB-i.. (A.+A.) (AS+"'JJ = 0,
x (Q) = 2zyz (A,-Aa+ "'a-A.+ Aa-A.) == 0,
which is what was required, If AI' h., h. > 0 and hi are pairwise
distinct, these two integrals are functionally independent (at the
points of general position) (verify!). Hence, the integral trajectories
are the lines of intersection of two families of surfaces, spheres P =
const and ellipsoids Q == const, It hi convenient to picture the field
X on the level i!lurface Q = coni!lt (see Fig, 7.27), Fix a sphere P =
const and restrict the flow i. to this sphere, We assert tbat this flow
is Hamiltonian, Indeed, it is required to prove that X preserves the
form of volume, i.e. that X has zero divergence, It suffices to demon-
strate that the flow preserves the form of volume in the surrounding
Euclidean space. But this is obvious because the expression :: +
:: + :: is identically zero (see the explicit form of ;, y, ). This
example shows that integrali!l of a 'flow playa key role in the descrip-
tion of flow behaviour, In the case of a three-dimensional rigid body

7.4. CaleuluB of VarIa/toM and Sump/eo/to Geom.,rv 40i7
the equations of motion are completely integrable, i,e. we have
found two integrals of motion, The equations of motion are also
completely integrable (in a certain exact sense) for an n-dimensional
rigid body; but the prool of this fact is far from being trivial and
is beyond the scope of this book.
Now a lew words about the integrals of Hamiltonian flows. On
the space of smooth functions on a symplectic manifold it is useful
to introduce a new operation generated by the symplectic structure.
Q " const
Figure 7.27
Definition. The Poisson bracket ()f two smooth functions f and g on
a symplectic manifold M2n is the function {f, g} defined by
{f, g} = (j)(2) (sgrad f, sgrad g) = CiJII (sgrad 1)1 (sgrad g)/,
The explicit formula for the operation I. g - {t, g} is written as
{t, g} = (j)01 41L. because Cagrad 1)1 = (j)ilS.!L,
8z az!1o 8z Ol
Proposition 1. The operation f. g - {t, g} satisfies the following con-
ditions: (1) it Is bilinear, (2) It Is skew-symmetric, t.e, {j, g} = - {g, f}
and (3) the J ac()bl identtty
{h, {I. gn + {g, {h, f}} + {j, {g, h}} = 0
holds true,
Proof. We prove this theorem lor a particular case of M2n = Rn
with a canonical symplectic 8tructure. Conditions (1) and (2) directly
follow from the definition of a symplectic structure. Let us prove (3).
Denote by Lf differentiation along the field sgrad I. i,e. L f (h) =
(sgrad I) h, We assert that the expression. on the lelt-hand side of
(3) is the sum of monomials, each containing the second partial

448 7. Variational ProbielM in Riemannian Geomdrll
derivative of one of the functions f, g, and h. Indeed. consider {j, g};
from the definition of sgrad we obtain
{I, g} = 00(2) (sgrad f, sgrad g) = - (sgrad g) f
I iJf
= -LI/(/) = - (sgradg) -.
8",1
The components of tbe field (sgred g)! are partial derivatives of
It
the function g. Really, since 00(2) =  dP!1\ dg l , we have (sgradg)i =
1...1
_.!L for 1  tni (sgradg)i= for ijn; the matrix of
!Jql iJpJ
the form 00(2) in R 2 " (pi, . . " pil, ql, . , ., ,t) is constant and has
canonical "block" structure. Hence, ft, g} is a linear combination of
monomials which are the products of first-order partial derivatives.
Thus,
{h, {I, gn = - {{I, g) h} = - L(!, 8} (h) = - (sgrad {I, g})1 :; ,
Le. the expression {h, {I, gn is the sum of monomials, each including
second-order partial derivatives either of function I or of function g.
Thus, we have demonstrated that the left-hand side of identity (3)
is the sum of monomials, each containing, as a factor, a second-
order partial derivative. Let us now fix the function h and collect
all the terms on the left-hand side of (3) that contain second-ordel'
partial derivatives of h
{f, (g, h}) - {e, {I, h}} = LfLsh - LsLfh = [Lt, LsJ h.
On the other hand. we know that the commutator of two first-order
operators, LI and Ls. is an operator of the first (but not second I)
order, i.e, the expression [L" LI/J k includes only first-order deriv-
atives of h. Thus, no second-order derivative of h appears on the
left-hand side of (3). Using the same reasoning, we can easily verify
that the left-hand side of (3) does not contain second-order deriv-
atives of 1 or g either, Le. the left-hand side of (3) is equal to zero,
The proposition is proved.
A linear space with the bilinear operation satisfying conditions
(1)-(3) is called a Lie algebra.
Corollary 2, The linear space F (M2R) of all smooth functions on a
symplectic manifold (M", (J) is a Lte algebra with respect to the Poisson
. bracket {I, g}.
This follows from Proposition 1. The Lie algebra F (M) is in-
finite-dimensional. Let us consider the mapping", of the Lie algebra
F (M) into the Lie algebra V (111) of all smooth vector fields on M
defined by the formula ct (f) = sgrad I.

'1.4. Calculus of Var/alloM and Symplectic Geomelrll 449
Corollary 3. The mapping ct: 1-+ sgrad f is a homomorphism 01 Lie
algebras, i.e. cr. {I, g} = [cr.!. cr.g].
Proof. It is required to prove that sgrad {f, g} = (sgrad f, sgrad g].
From the Jacobi identity we have
(sgrad {t. g}) 11, = CI) (sgred {I. g}. sgrad h) = - {h. {f. en
= - {{g, {f, hn+{f, {g, h}} = -L,L1k+ L/L.h.
={L/. L,lh,
sgrad {t, g}= [sgrad f. sgrad fl,
which is what was required.
Since ct is a homomorphism of Lie algebras, a subalgebra, denoted
by H (M). is the image of the Lie algebra F (M) in the Lie algebra
V (M) of all smooth fields on M. The definition of cr. implies that.
this subalgebra is the Lie algebra of all Hamiltonian fields on M.
Thus, cr.: F (M) _ H (M) is an epimorphism. But cr. is not a mono-
morphism. for it has a non-trivial kernel: this kernel is formed by
locally constant functions. If we assume that M is connected, the
kernel of cr. is one-dimensional and consists of constant functions on M.
Hence. H (M) !:'5 F (M) lB., The subalgebra H(M) c::: V(M), H(M) =
Ho> (M) depends on the choice of the symplectic structure CI) on M.
Jf we introduce on M another symplectic structure CI)', the subalgebra.
Hr.' (M) c: V (M) will differ from Hr. (M). It should be noted that.
potential fields grad f are distinct from Hamiltonian fields sgrad I.
Indeed, potential fields do not form a subalgebra in V (M), i.e.
the commutator of two potential fields need not necessarily be a
potential field, potential fields on a circle being an example. Each.
vector field on 8 1 can he defined by a smooth function P (11'), where
rp is the angular coordinate and P (II') is the tangent vector compo-
nent, Then. the commutator of two fields P (cp) and Q (<p) on 8 1 is.
the field with the component R = PQ - QP (verifyl), Set P =
cos <p, Q = sin <p, then R liS 1. Thus, we have represented the field
with constant component (equal to 1) as a commutator of two poten-
tial fields. A constant field is not potential, for it has a closed trajec-
tory without singularities,
The operation of calculating the Poisson bracket plays an impor-
tant role in the study of integrals of Hamiltonian Dows.
Proposition 2. Let v = sgrad H be a Hamiltonian flow on M....
and let f (x) be a snwoth function which commutes with the Hamiltonian
H, i.e. {j, H} 53 O. Then I is an integral of the flow v, i.e. the function
f is constant along the integral trajectories of the field v. Conversely.
any integral g of the field sgrad H commutes with H. i.e. {g, H} = O.
Proof. It suffices to calculate the derivative of f along the field v.
I.e. (sgrad H) j. By the definition of sgrad H, we have (sgrad H) f =
{B. j} = 0, which is what was required.

450 7. Varla/lonal PrqbltllU In R ltmannlan GeotM/'1/
Since {H, H} EiI 0, the function H is an integral of the flow sgrad H.
Proposition 3. Let f and g be two integrals of sgrad H, then {I, g}
.ls also an Integral of the flow sgrad H.
The proof directly follows from the Jacobi identity.
Thus, by calculating the Poisson bracket of two integrals, we
obtain a new integral and the latter may, however, functionally
depend on the two initial integrals.

Name Index
Abel N. H.. 252
Laarange . L.. 408
Lambert . H,. 10
Lam6 O. 80
Lag:ace P. 8,1. 240
J-'::: !if, IO
Lefbnlt G, W., 829, a9
Lll8aJoul . A.. 130
Lobachevak)' N, I., 10. 2
Beltraml E.. 181, 199
Bol)'al J., 10
Borelli O. A., 10
1I0nnet p, G" 408
Brouwer L E, ;J" 10
Cataldi P., 10
gt:trEO'B1 328
Clavlu. Ch.. 10
=ga tt.., a, If I It, 2&&
Newton J., 89
DarbouJ: 1. 0.. (40
Deecarlea R., 10, 12
Oreame N., 10
Oltro8:lad.ky M. V., 39
Eflmov N. V,. 191
Euclid, 10, 12, 60
Euler L., 10, II
PalC8l D., 804
PIQcker J., 10, II
POlncard J, H" 10, II
g":It Sl, D v .:1t, 11
Ptolemy C., 10
Perm at P., 10
J,rp..t.,151
GIUBB C. P., 10, 895, 40B
Genonlde L., 10
Gram J. P., 64
Gl'Hn O. 894
Riool C, 0., 8S4
Riemann C. P. 13" 10, 316
de Rhem 0., 878
Hamilton W. R.. 288, U8
Haudorll' P., 10, 79
UffC:rtHD.'i9I'
Hook. R.. 80$
Seecherl O. Q,. 10
9ard A., 138
SchwBlt H. A.. "6
Sebwetkarl P. K., 10
Staudt K, 0., 10, II
SIokea Q, G., 893
Taurfnua F . 10
Jordan C.. 10
Joukowakl N. E., 2U
tlt780hn P. 8" 88
Vitali J., IQ
K.plar 1., 16
Klein 1'., 10, 258
Wallie J" 10

Subject Index
Absolute, 63. S9, 60, 188, 347
Image 01, 9
Atlas(el), 9'
equivalent. 101, 102
maximal, 98
oriented, 249
Basea, or the &Bme orientation, 250
Bellr.ml.Laplace equ.tlon, 199
Beltraml .urrace, 181, 184
Gauaalan curvature ot, 183
mean curvature of, 191
Riemannian metric Indued on. 181
section 01, 185
BIII:':-.:e::ie43340, 165
cf.dllteomorphl.ID, 10'
c'.!Iomeomorpblsm, 104, 105
C'.manllold, 102
C" .mapp\ng, 10'
c'-tunUon, 102
C".manUold, 102
COO-manifold, 102
Cartesian product, 74, 86
r.ctofll 01, 74
C.ter.rne: or, H
Caucby.Rlemann equallon, 2'0
Cb8,;\IUeo4"rpblc 1 08
Cbrlstoltel Iymbois, 32S, 844
Circle, equation of. 12
Circular permutation, 58
Cloaure, 88, 70, 90
Cobomology Sroups, 372, 378
3m1fc 3%opertles ot, 381
Compactlncatlon, 277.287
Comp.cln..., 82
Conlormal tranotormatlon, 44
fu2 79
alrlne, 328
torsion tensor ot. 328
WIrag'3U 341, m
.ymmetrlc, 329
ContracUon , 306
Convlt!rgence, uniform, 81
coordInate linea, 20
CoOrdlna US
Cartesian I 12
contormal. 198
cylindrical, 22, 31
local, 94. 831
polar 30
aPber!cal ! 8 t
.ymplect c, 440
CoordInate ayatem
contlnuoua, I, 17
curvilinear, 17
local, B4
regular, 10, 17
Coordinate tranatormatton, 1 "1
t"j':WoI, Br!6 103
COvariant dU1erentiallon, 321, 328
along a curve, 339
COvering
open. 81
relined, 81
Cra.. cap, 2S5
Curvature, 148
01 a curve, 148. 158
01 a plane oectlon. 170
radlu., 149
..alar, 364
curv:21;:r:oJltlon 01 360
Invariant deltlon 01, 361
Riemann, 380
algebraic propertlee ot, 362
Curves
angle between, 20
f:atnfral, 234
cuapehlj' 25, 28, 30, 31, 32, 37
Darboux theorem, 440
Diffeomorphism, 353, 40', 419
Dirichlet lunctlonal, U7
Divergence, 236
EIIIg.,", equation 01. 18
rglr!d, :h 137, 141, 162
EpimorphIsm, 380, H9
igUnnlaIi'dJgi, 5& , 30

Sublecl Indez
453
Euler's equations. 407.  1. U 7, "18, '2!t.
Exterg; rfir!at3forllUJ, 311, 372, 439
closed, "8
r&':nr;.1f or, m
exact, 378
f:i:I' g1.2'8W 403
Exte:r:u\rtfi:a f;'n 8309
Extremal, H 7, 436, 437
Factori.atlon, 253, 349
Plb...., 127
Fll$t fundamental form, 160
Flow. 232
comple>: potential 01, 241
conJugate, 26.0
W::fffant, 4f1h5 334
tJ'a';t8 232
Fr8DJ.(tJ; IW' 149
coordinate, 14B
Prenet formulas, 151, 154
Function
algebraic, 271, 272
Ch"otfzN3
ana\¥tlc, 271
bounded, 211
gya\C844fol
01 coordinate tranBtormatlon, 99, 103
dll!e.rentlal ot, 18, 382
Invarlance property 01, 382
directional derivative ot, 122
gfPiIAlea of, 9
ttl,I'8" 9
SUfcPOIt or, 90
FUnCJe';f:u;:7of, 412
linear, 408
Fundamental tbeorem or algebra, 402
Gauss.Bonnet theorem, 408
Gauss.Oetrogradsky lormula, 395
Gaussian curvature, 187. 178, 367
Geode.lc(s), 68, 336, 341, 416, 22,
434
equation. 01, 32, 345, 346
1r:IltI2I, m
,",rtace, 370
Geometry
elliptic, 51, 59
Lobachevsl<!an, 16, 52
non.EuclideDn. 10
on a pJane, 3
on a sphere. 38
symplectic, 130
Gradient ot a smoolh tunctlon,
properti.s 01, 229
skew-symmetric, '40
Gram determinent, 64
Grep!1 l s formula, 39.
Group(a)
Abelian, 252
connected componen t (n, 216
factor, 216
18Ometry, 201
laomorpblam Of, 202
Lie, 216
linear, 216, 217
orthogonal, 201, 217
f.n1d, 217, 2!9, 221
aymplectlc, 225, 228
unit element of, 21B
unitary, 219
Group methods, 10
Hamilton'. equatlOll8, 238
Hamllton!an, 436
Helicoid, rlbt. 192
Homce:or;gfa:::n9l5. f6, 13. 'H, 89. 353.
Homology theory, 371
Homomorpblam. 3 B8. 449
Homotopy, 384, 402
between mapping., 3B'
amooth, 38'
Hook.'. law, 305
HypaIBurfac:e
curvature ot, '0'
GaUll8!an curvature ot. t76
DIBt fundamental form or, 161
fa:"Ytoti617i
principal direction. ot, 168
Identity matrl>:, 18
Immersion, 187. 138
rmpt function theorem, t09, 135. !Sa,
inclusion, 71
Idlces
contravariant, 297
covariant, 297
::,".rI?' an 7
raotroplc cone, 48, 47
i:aafi'X' 18, 2', ", 105, 130
JoukowBl<1 transformation, 244
Kepler's law, 14
Klein boltle, 253
Lagrangian, 08. 409. 426, 432
nDnw,Bingular. 43:». 4.36
Lame con"tents, 3U5
Laplacian. 210
Latitude, 21, 4U
Lelbnlz lormula, 329, 331
Level lines, lOB
Level manitold, 108, 112
LeveJ Gurtace.o, 108
Lie algebra, 4J.8
Lie group, 31

454
Subject 1nM3';
r.=lrJ, tJl,"'o 13B
Lo1Od,orne, 44, 48
parametric .1P......lon for. 44, 4B
MI1ty(aU' 2:
canonical aymmelr/o lorm 01, 269
closed, 28
compact, 109
compleJt.analytlc, 102, 108
conUnuoaly.dllterentlable. 101
dllteomorpblo, 100
dlmonllon or. 93
hcmeoil'apblc, 106
'::feC&9 201
real-analytic. 102
....ootb, 92, 9B, 101
mf(',. 260. 261
Map'lnuoul. 72, 74, 78
coordinate. 16
f»mng'.t 3: 1, 180
Mapping
.:rponentlal U 9
tg::eb:880
bopWe, 07. 80
lengtb.p.....rvIni". 200
linear, 182
Ipherlcal, O'
Metrl:r, *aw"YmmetrIC, 321
Mean curvature. 167, 178
Metric. 87
contormal. 42, 292
Euclidean, 36
Indef1nlte, 86
Ulbacbovlklan, AS
non-Euclidean. 3'
lCUdo.Euctidean, 35
1n0268 33, 14&
Ipberical. 2, 46
Hllbpr1hm. 250
l1t<";U:dd 1:?'I:2
:M:onomorphl'm, 137, 363, 448
Natural parameter, 147
Ne1rbbourbood
ball, 8S
die/oint, 79
01 point, 70
ot lOt,' 70
':1-g 1r.'W7' m
Operallon(e)
algebralo over tenlOl'IJ, 305
or glulnr, 206-207
Oper&wr
compleJ:, 222
Or/emt:,if.19 223
conel,tent, 2'8, au
gf;,,2i0
Parallel dllplaoemenl. 330
equation 01, 83
properties ut, 3'0
para:::..:;rU7 B'
Partition Oi unIty, 87, 90, 107, 390
Paacal'. law, 30'
Po/ncari'l lemma, 387
Polncan1'. model, 2, I BB, 3'6
Polnt!l)
boundary, a&7
brancblng, 270
coordlnatr.8 ot. IS
critical, 230, 410
eJ:tremaJ U 0
bYjlorbol!c, 177
Interior, 27
Jaolated 230
non.crltlcal 6 230
='41,
.lngular, 230
Poluon bracket, 447-U9
Pot::le92,t35
&g:, 2'0
Pratul. 268
Prlnclpel aJ:es, 168
Principal dlrectlonl, 188
7fNe nI. /8
paeUdOtlf.here, ,if. 16 I
 ::lgl:rfU8,:,d18, 47
01 sero radlul, 48
Quatemlon(o), 228
alreb.. Of. 226
dlmenalon, 228
Imarlnary. 228
Radius vector
barmonlc. 191, 428
minimal, 198
Rellnement, at
RI<:<:I tenlor, 384
Riemlllln In\earal. 31
Sard'. Iheorem, 138, 13B. 141, '00
Sca1 fdtOIB, 26
8kew..metrlc. 321
t,.'tund:m' r:. 163, 185, 387
tr(rtlon nloma. 79
c/:,dc!8B70
cloaure or, 611-1 I
=J'ir':"' j
::CeJberf) dente), 71
diameter ot, 68

Subject 1 ndez
455
Set('I&tance between, 68
Interior ot, 69, 70
rlOr;;l,OIW ot, 68, 70
point ot contact, 68, 70
{'g1s"on7 70
01 .ero measure, 388
spacf,\ompact, 82
clOJJed, 88
compact, 82, 84
properUes 01, 83
connected, 17
Imare ot, 78
disconnected. 77
dJecrete, 78, SO
Hauadodl, 79
:eg¥,bIC. 73
norma), 80, 83
patb....loe.COnnecte1l. 78
pbaoe, 129
points or. 87
IUbap_ or, 08{ 71
toil":l3to IB
toPOlol[lcally Invariant, 78
Sphere, n.dlmenuonal. 88
spJr :: rrnateB, 18
In polar coordinate.. sa
f',g,J.,Jn"'9'U'U' U,
r.::.l:ee;:re2! 393
Structure
:pl::i,23UO
SubmanWOld(8), 128, 137. 159
Surface
\1I!.1286
or conltant CW'Yature, 179
to;:;::oc, 2897 . 188
minimal, 181, 424. U8
pbyolcal model tor. 191
aV:ra"';lre, 179
or =Ive curvature. 179
JUemannlan, 272
ot uro curvature-. 179
Syllem8
autonomou..8, 23.2,
dynamical, 229
If[ !t:'80
Tangen t bundle.. 127
Tangent plane, 114
parametric equation ot. 116
U.69
Tangent Ipace, 113, 116
Tanr':.rne;i:' o1,6'IN,3'11U, 117, 121"
Tansor, 297
alternation ot, 308
=il:J;[0;9 0 \' 306
InerUa, 301
lowerlns 01 Indlcea In. 308
!all 01 Intll..1 In, 305
product, 300
properllea 01 299
railing 01 Incl.l... In 308
:l're,:rYWo'l':t'8! 30
atre.., 608, 804
IYmmelritatJon 01, 307
Tene:trie9b8 800
t: ! tf:t rJar5e, 147
Topolorlcal auUl, 79
grol::r./, Varll1tt propertlea, 78:
dlBCi'ete, 71
lundamental tbeorem 01, 108
f:.Ik 9U
set.tbeoretlc. 9
=n 1'3,a fW68 848
Tranaltlon loncuana, 89, tOS
i:::a'i1a111K: \6,89866
UrJlObn'a lemma, 88
:Ilonal problellll, 407
binormal, 167
lootropl., 88
Ir;Ci,h a''l' 88
=:!k;.i8!8m
tanrent, 26
vecE\1 80, 88
Hamiltonian, 441
linearly Independent, 283-
Velocity, area.. U
'fuy of,,:I:,t{n0:16
rJ'II: J 88