Foundations of Differential Geometry

FOUNDATIONS OF DIFFERENTIAL GEOMETRY VOLUME I

SHosmCHI KOBAYASHI UnIVersIty of CahfornIa, Berkeley, Cahfornia and

KATSUMI NOMIZU Brown UnIversIty, ProvIdence, Rhode Island

1963 INTERSCIENCE PIJBLISHERS a division of John Wiley & Sons, New York · London

Copyright © 1963 I;J;--john Wiley & Sons, Inc. AIYttights Reserved Library of Congress Catalog Card Number: 63-19209 PRINTED IN THE UNITED STATES OF AMERICA

PREFACE Differential geometry has a long history as a field of mathematics and yet its rigorous foundation in the realm of contemporary mathematics is relatively new. We have written this book, the first of the two volumes of the Foundations of Differential Geometry, with the intention of providing a systematic introduction to differential geometry which will also serve as a reference book. Our primary concern was to make it self-contained as much as possible and to give complete proofs of all standard results in the foundation. We hope that this purpose has been achieved with the following arrangements. In Chapter I we have given a brief survey of differentiable manifolds, Lie groups and fibre bundles. The readers who are unfamiliar with them may learn the subjects from the books of Chevalley, Montgomery-Zippin, Pontrjagin, and Steenrod, listed in the Bibliography, which are our standard references in Chapter I. We have also included a concise account of tensor algebras and tensor fields, the central theme of which is the notion of derivation of the algebra of tensor fields. In the Appendices, we have given some results from topology, Lie group theory and others which we need in the main text. With these preparations, the main text of the book is self-contained. Chapter II contains the connection theory of Ehresmann and its later development. Results in this chapter are applied to linear and affine connections in Chapter III and to Riemannian connections in Chapter IV. Many basic results on normal coordinates, convex neighborhoods, distance, completeness and holonomy groups are proved here completely, including the de Rham decomposition theorem for Riemannian manifolds. In Chapter V, we introduce the sectional curvature of a Riemannian manifold and the spaces of constant curvature. A more complete treatment of properties of Riemannian manifolds involving sectional curvature depends on calculus of variations and will be given in Volume II. We discuss flat affine and Riemannian connections in detail. In Chapter VI, we first discuss transformations and infinitesimal transformations which preserve a given linear connection or a Riemannian metric. We include here various results concerning Ricci tensor, holonomy and infinitesimal isometries. We then v

treat the extension of local transformations and the so-called equivalence problem for affine and Riemannian connections. The results in this chapter are closely related to differential geometry of homogeneous spaces (in particular, symmetric spaces) which are planned for Volume II. In all the chapters, we have tried to familiarize the reaaers with various techniques of computations which are currently in use in differential geometry. These are: (1) classical tensor calculus with indices; (2) exterior differential calculus of E. Cartan; and (3) formalism of covariant differentiation V xY, which is the newest among the three. We have also illustrated, as we see fit, the methods of using a suitable bundle or working directly in the base space. The Notes include some historical facts and supplementary results pertinent to the main content of the present volume. Tqe Bibliography at the end contains only those books and papers which we quote throughout the book. Theorems, propositions and corollaries are numbered for each section. For example, in each chapter, say, Chapter II, Theorem 3.1 is in Section 3. In the rest ofthe same chapter, it will be referred to simply as Theorem 3.1. For quotation in subsequent chapters, it is referred to as Theorem 3.1 of Chapter II. We originally planned to write one volume which would include the content of the present volume as well as the following topics: submanifolds; variations of the length integral; differential geometry of complex and Kahler manifolds; differential ge9metry of homogeneous spaces; symmetric spaces; characteristic classes. The considerations of time and space have made it desirable to divide the book in two volumes. The topics mentioned above will therefore be included in Volume II. In concluding the preface, we should like to thank Professor L. Bers, who invited us to undertake this project, and Interscience Publishers, a division of John Wiley and Sons, for their patience and kind cooperation. We are greatly indeeted to Dr. A.J. Lohwater, Dr. H. Ozeki, Messrs. A. Howard and E. Ruh for their kind help which resulted in many improvements of both the content and the presentation. We also acknowledge the grants of the National Science Foundation which supported part ofthe work included in this book. SHOSHICHI KOBAYASHI KATSUMI NOMIZU

CONTENTS

Interdependence of the Chapters and the Sections CHAPTER

.

Xl

I

Differentiable Manifolds

1. 2. 3. 4. 5.

1 17 26

Differentiable manifolds Tensor algebras Tensor fields Lie groups . Fibre bundles .

38 50

CHAPTER

II

Theory of Connections

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Connections in a principal fibre bundle. Existence and extension of connections . Parallelism. Holonomy groups . Curvature form and structure equation Mappings of connections Reduction theorem Holonomy theorem Flat connections Local and infinitesimal holonomy groups Invariant connections CHAPTER

63 67 68 71

75 79 83

89 92 94

103

III

Linear and Affine Connections

1. Connections in a vector bundle . 2. Linear connections 3. Affine connections' 4. Developments 5. Curvature and torsion tensors 6. Geodesics 7. Expressions in local coordinate systems Vll

113 118 125

130 132 138

140

Vll1

8. 9.

CONTENTS

Normal coordinates Linear infinitesimal holonomy groups

146 151

CHAPTER IV RielDannian Connections

1. Riemannian metrics 2. Riemannian connections . .. 3. Normal coordinates and convex neighborhoods 4. Completeness 5. Holonomy groups 6. The decomposition theorem of de Rham 7. Affine holonomy groups

,

154 158 162 172 179 187 193

CHAPTER V Curvature and Space ForlDs

1. 2. 3. 4.

Algebraic preliminaries Sectional curvature Spaces of constant curvature Flat affine and Riemannian connections

198 201 204 209

CHAPTER VI TransforlDations

1. Affine mappings and affine transformations 2. Infinitesimal affine transformations 3. Isometries and infinitesimal isometries 4. Holonomy and infinitesimal isometries 5. Ricci tensor and infinitesimal isometries 6. Extension of local isomorphisms . 7. Equivalence problem .

225 229 236 244 248 252 256

ApPENDICES

1. 2. 3. 4. 5. 6. 7.

Ordinary linear differential equations 267 A connected, locally compact metric space is separable 269 Partition of unity . 272 On an arcwise connected subgroup of a Lie group 275 Irreducible subgroups of O(n) 277 Green's theorem 281 Factorization lemma . 284

CONTENTS

IX

NOTES

1. Connections and holonomy groups 2. Complete affine and Riemannian connections . 3. Ricci tensor and scalar curvature 4. Spaces of constant positive curvature 5. Flat Riemannian manifolds 6. Parallel displacement of curvature 7. 8. 9. 10.

11.

Symmetric spaces . Linear connections with recurrent curvature The automorphism group of a geometric structure Groups of isometries and affine transformations with maximum dimensions Conformal transformations of a Riemannian manifold

287 291 292 294 297 300 300 304 306 308 309

Summary of Basic Notations . Bibliography

313 315

Index

325

Interdependence

0/ the

Chapters and the Sections

Exceptions Chapter II: Chapter III: Chapter IV: Chapter IV: Chapter Chapter Chapter Chapter Chapter

Theorem 11.8 requires Section 11-10. Proposition 6.2 requires Section 1II-4. Corollary 2.4 requires Proposition 7.4 in Chapter III. Theorem 4.1, (4) requires Section 1II-4 and Proposition 6.2 in Chapter III. V: Proposition 2.4 requires Section 111-7. VI: Theorem 3.3 requires Section V-2. VI: Corollary 5.6 requires Example 4.1 in Chapter V. VI: Corollary 6.4 requires Proposition 2.6 in Chapter IV. VI: Theorem 7.10 requires Section V-2 .

.

Xl

CHAPTER I

Differentiable Manifolds 1. Differentiable manifolds A pseudogroup of transformations on a topological space S is a set r of transformations satisfying the following axioms: (1) Each fEr is a homeomorphism of an open set (called the domain off) of S onto another open set (called the range off) of S;

(2) Iff E r, then the restriction off to an arbitrary open subset of the domain off is in r; (3) Let U = UUi where each U i is an open set of S. A homeoi

morphismf of U onto an open set of S belongs to r if the restriction off to Ui is in r for every i; (4) For every open set U of S, the identity transformation of U is in r; (5) Iff E r, thenf-l E r; (6) Iff E r is a homeomorphism of U onto V and f' E r is a homeomorphism of U' onto V' and if V n U' is non-empty, then the homeomorphism f' f of f-l(V n U') onto .['(V n U') is in r. We give a few examples of pseudogroups which are used in this book. Let Rn be the space ofn-tuples ofreal numbers (xt, x 2 , ••• ,xn ) with the usual topology. A mapping f of an open set of Rn into Rm is said to be of class r = 1, 2, ... , 00, if f is continuously r times differentiable. By class Co we mean that f is continuous. By class ew we mean that f is real analytic. The pseudogroup rr(Rn) of transformations of class Cr of Rn is the set of homeomorphisms f of an open set of R n onto an open set of Rn such that both f and f- l are of class Cr. Obviously rr(Rn) is a pseudogroup of transformations of Rn. If r < s, then rS(Rn) is a 0

cr,

1

2

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

subpseudogroup of rr(Rn). If we consider only those f E rr(Rn) whose Jacobians are positive everywhere, we obtain a subpseudogroup of rr(Rn). This subpseudogroup, denoted by r~ (R n), is called the pseudogroup of orientation-preserving transformations of class Cr of Rn. Let Cn be the space of n-tuples of complex numbers with the usual topology. The pseudogroup of holomorphic (i.e., complex analytic) transformations of Cn can be similarly defined and will be denoted by r(cn). We shall identify Cn with h necessary, b · (1 R 2n,wen y mappIng z, ... , Z n) E Cn·Into (1 x, ... , 2 xn,y!, ... ,yn) E R n, where Zi = xi + iyi. Under this identification, rlcn) is a subpseudogroup of r~(R2n) for any r. An atlas of a topological space M compatible with a pseudogroup r is a family of pairs (Ui , flJt), called charts, such that (a) Each Ui is an open set of M and UUi = M; t

(b) Each fIJi is a homeomorphism of U i onto an open set of S; (c) Whenever U i n U j is non-empty, the mapping fIJi flJi- 1 of fIJi ( U i n Uj) onto flJj( U i n Uj) is an element of r. A complete atlas of M compatible with r is an atlas of M compatible with r which is not contained in any other atlas of M compatible with r. Every atlas of M compatible with r is contained in a unique complete atlas of M compatible with r. In fact, given an atlas A = {( Ut , fIJi)} of M compatible with r, let A be the family of all pairs (U, fIJ) such that fIJ is a homeomorphisn1 of an open set U of M onto an open set of S and that 0

fIJi

0

fIJ-l: fIJ( U n Ui)

---+

flJt (U nUt)

is an element of r whenever U n U i is non-empty. Then A is the complete atlas containing A. If r' is a subpseudogroup of r, then an atlas of M compatible with r' is compatible with r. A differentiable manifold of class Cr is a Hausdorff space with a fixed complete atlas compatible with rr(Rn). The integer n is called the dimension of the manifold. Any atlas of a Hausdorff space compatible with rr(Rn), enlarged to a complete atlas, defines a differentiable structure of class Cr. Since rr(Rn) ::> rS(Rn) for r < s, a differentiable structure of class Cs defines uniquely differentiable structure of class Cr. A differentiable manifold of class CW is also called a real analytic manifold. (Throughout the book we shall mostly consider differentiable manifolds of class Coo. By

a

I.

DIFFERENTIABLE MANIFOLDS

3

a differentiable manifold or, simply, manifold, we shall mean a differentiable manifold of class COO.) A complex (analytic) manifold of complex dimension n is a Hausdorff space with a fixed complete atlas compatible with r(C n ). An oriented differentiable manifold of class Cr is a Hausdorff space with a fixed complete atlas compatible with r~(Rn). An oriented differentiable structure of class Cr gives rise to a differentiable structure of class Cr uniquely. Not every differentiable structure of class cr is thus obtained; if it is obtained from an oriented one, it is called orientable. An orientable manifold of class Cr admits exactly two orientations if it is connected. Leaving the proof of this fact to the reader, we shall only indicate how to reverse the orientation of an oriented manifold. Ifa family ofcharts ( Ui' fIJi) defines an oriented manifold, then the family of charts (Ui , "Pi) defines the manifold with the reversed orientation where "Pi is the composition of fIJi with the transformation (xl, X 2, ••• , X n ) ----+ (-xl, X 2, ••• , xn ) of Rn. Since r(Cn) c r~(R2n), every complex manifold is oriented as a manifold of class cr. For any structure under consideration (e.g., differentiable structure of class Cr ), an allowable chart is a chart which belongs to the fixed complete atlas defining the structure. From now on, by a chart we shall mean an allowable chart. Given an allowable chart (Uo fIJi) of an n-dimensional manifold M of class Cr , the system of functipns Xl flJo ••• , xn fIJi defined on U i is called a local coordinate system in Ute We say then that Ui is a coordinate neighborhood. For every point p of M, it is possible to find a chart ( Ui' fIJi) such that flJt(p) is the origin of Rn and fIJi is a homeomorphism of U i onto an open set of Rn defined by Ixll < a, ... , Ixnl < a for some positive number a. Ui is then called a cubic neighborhood of p. In a natural manner Rn is an oriented manifold of class Cr for any r; a chart consists of an elementf of r~(Rn) and the domain off Similarly, Cn is a complex manifold. Any open subset N of a manifold M ofclass Cr is a manifold of class cr in a natural manner; a chart of N is given by (Ui n N, "Pi) where (Ui , fIJi) is a chart of M and "Pi is the restriction of fIJi to U; n N. Similar-Iy, for complex manifolds. Given two manifolds M and M' of class Cr, a mapping f: M ----+ M' is said to be differentiable of class C\ k < r, if, for every chart (Ui , fIJi) of M and every chart (Vj , "Pj) of M' such that 0

0

4

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

f(Ui ) c Vj' the mapping 1.pj ofo CPi- 1 of CP~(Ui) into 1.pj(Vj) is differentiable of class Cle. If u\ ... , un is a local coordinate systemin U i and vI, ... , V m is a local coordinate system in Vj' then f may be expressed by a set of differentiable functions of class Ck: .v1 = fl (u1 ,

• • • ,

un), . . . ,

Vm

= fm (u1 ,

• • • ,

un).

By a differentiable mapping or simply, a mapping, we shall mean a mapping of class Coo. A differentiable function of class Ck on M is a nlapping of class Ck of Minto R. The definition of a holomorphic (or complex analytic) nlapping or function is similar. By a differentiable curve ofclass Ck in M, we shall mean a differentiable mapping of class Ck of a closed interval [a, bJ of R into M, namely, the restriction of a differentiable mapping of class Ck of an open interval containing [a, b] into M. We shall now define a tangent vector (or simply a vector) at a point p of M. Let 7J(p) be the algebra of differentiable functions of class Cl defined in a neighborhood of p. Let x( t) be a curve of class C\ a < t < b, such that x (to) = p. The vector tangent to the curve x(t) at p is a mapping X: 7J (p) ---+ R defined by

Xf

=

(df(x(t)) jdt) to.

In other words, Xf is the derivative of f in the direction of the curve x(t) at t = to' The vector X satisfies the following conditions:

(1) X is a linear mapping of 7J(p) into R;

(2) X(fg)

=

(Xf)g(p)

+ f(P) (Xg)

for j,g

€

7J(p).

The set of mappings X of 7J(p) into R satisfying the preceding two conditions forms a real vector space. We shall show that the set of vectors at p is a vector subspace of dimension n, where n is the dimension of M. Let u\ ... , un be a local coordinate system in a coordinate neighborhood U of p. For each j, (oj ouj)p is a mapping of 7J(p) into R which satisfies conditions (1) and (2) above. We shall show that the set of vectors at p is the vector space with basis (ojou 1 )p, ... , (ojoun)p. Given any curve x(t) with p = x (to) , let uj = xi (t), j = 1, ,n, be its equations in terms of the local coordinate system u\ , un. Then

(df(x(t))jdt)t o =

~j

(ofjouj)p' (dxj(t) jdt) to *,

* For the summation notation, see

Summary of Basic Notations.

I.

DIFFERENTIABLE MANIFOLDS

5

which proves that every vector at p is a linear cornbination of (ajaul)p' ... ' (ajaun)p. Conversely, given a linear combination ~ ~j (aj auj) P' consider the curve defined by u j = u j (p)

+ ~ t,

j = 1, . . . , n.

j

Then the vector tangent to this curve at t = 0 is ~ ~j( aj aui) p. To prove the linear independence of (aj aul) p' ... , (aj aUn) p' assume ~ ~j( aj auj)p = o. Then

o=

~ ~j(aukjaui)p

=

~k

for k = 1, ... , n.

This completes the proof of our assertion. The set of tangent vectors at p, denoted by T p (M) or T p , is called the tangent space of Mat p. The n-tuple of numbers ~\ ... , ~n will be called the com-· ponents of the vector ~ ~j(ajauj)p with respect to the local coordinate system ul , ... , un. Remark. I t is known that if a manifold M is of class Coo, then Tp(Jvf) coincides with the space of X: f!j(P) --+ R satisfying conditions (1) and (2) above, where f!j(P) now denotes the algebra of all Coo functions around p. From now on we shall consider mainly manifolds of class Coo and mappings of class Coo ._-1 A vector field X on a manifold M is an assignment of a vector X p to each point p of M. Iff is a differentiable function on M, then Xfis a function on M defined by (Xf) (P) = Xpj. A vector field X is called differentiable if Xf is differentiable for every differentiable function j. In terms of a local coordinate system u\ ... ,un, a vector field X may be expressed by X = ~ ~j(ajauj), where ~j are functions defined in the coordinate neighborhood, called the components of X with respect to u\ ... , un. X is differentiable if and only if its components ~j are differentiable. Let X(M) be the set of all differentiable vector fields on M. It is a real vector space under the natural addition and scalar multiplication. If X and Yare in X(M), define the bracket [X, Y] as a mapping from the ring of functions on M into itself by

[X, Y]f = X(Yf) - Y(Xf). We shall show that [X, Y] is a vector field. In terms of a local coordinate system u\ ... , un, we write X =

~ ~j (aj aui) ,

6

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Then

[X, Y]f =

- rl( a ~iI auk) ) (aJ! aui) .

~ j,k( ~k( ar/jauk)

This means that [X, Y] is a vector field whose components with respect to ul, ... , un are given by ~k( ~k( ar/jauk) - YJk( a~i jauk)), j = 1, . . . , n. With respect to this bracket operation, X(M) is a Lie algebra over the real number field (of infinite dimensions). In particular, we have Jacobi's identity:

[[X, Y], Z]

+ [[Y,

Z], X]

for X, Y, Z

E

+ [[Z, X],

Y] = 0

X(M).

We may also regard X(M) as a module over the algebra ~(M) of differentiable functions on M as follows. Iff is a function and X is a vector field on M, thenf X is a vector field on M defined by (f X) p = f(p)Xp for p E M. Then

[jX,gY] =fg[X, Y] +f(Xg)Y-g(Yf)X j,g

E

~(M),

X,Y E X(M).

For a point p of M, the dual vector space T; (M) of the tangent space Tp(M) is called the space of covectors at p. An assignment of a covector at each point p is called a I-Jorm (differential form of degree 1). For each function f on M, the total differential (df) p off at p is defined by where <, > denotes the value of the first entry on the second entry as a linear functional on Tp(M). If ul, ... , un is a local coordinate system in a neighborhood of p, then the total differentials (du 1 ) p, ... , (dun) p form a basis for T: (M). In fact, they form the dual basis of the basis (ajau1)p, ... , (ajaun)p for Tp(M). In a neighborhood ofp, every I-form w can be uniquely written as w

=

~j

fj du i ,

where fj are functions defined in the neighborhood of p and are called the components of w with respect to ul, ... , un. The I-form w is called differentiable if h are differentiable (this condition is independent of the choice of a local coordinate system). We shall only consider differentiable I-forms.

I.

7

DIFFERENTIABLE MANIFOLDS

A I-form W can be defined aiso as an O:(M)-linear mapping of the ~(M)-module X(M) into ~(M). The two definitions are related by (cf. Proposition 3.1) X

E

X(M),

pEM.

Let A r;(M) be the exterior algebra over r;(M). An r-form W is an assignment of an element of degree r in A r;(M) to each point p of M. In terms of a local coordinate system u\ ... , un, W can be expressed uniquely as W --

~..

..(.

"'::;"~I <~2 < •••
dU il A

• • •A A du ir \.

The r-form W is called differentiable if the components hI" 'ir are all differentiable. By an r-form we shall mean a differentiable r-form. An r-form W can be defined also as a skew-symmetric r-linear mapping over ~(M) of X( M) X X( M) X • • • X X( M) (r times) into ~(M). The two definitions are related as follows. If WI' ••• , W r are I-forms and Xl' ... , X r are vector fields, then (WI A ••. A W r ) (Xl' , X r ) is l/r! times the determinant of the matrix (wj(Xk))j,k=I, ,r of degree r. We denote by !Y = 1Y(M) the totality of (differentiable) rforms on M for each r = 0, 1, ... ,n. Then 1)O(M) = O:(M). Each 1)r(M) is a real vector space and can be also considered as an ~(M)-module: for j E O:(M) and W E 1)r(M) , jw is an r-form defined by (fw)p = j(p)w,p, P EM. We set 1) = 1)(M) = ~~=o1)r(M). With respect to the exterior product, 1)(M) forms an algebra over the real number field. Exterior differentiation d can be characterized as follows: (1) d is an R-linear mapping of 1)(M) into itself such that d( 1)r) c 1)r+l; (2) For a functionj E 1)0, djis the total differential; (3) If W E 1)r and 7T E 1)8, then

dw

A 7T

+(

1) r W A d7T;

(4) d2 = o. In terms of a local coordinate system, if W = ~il < ...
8

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

real vector space. A V-valued r-form w on M is an assignment to each point p E M a skew-symmetric r-linear mapping of T'O(M) x ... x T'O(M) (r times) into V. Ifwe take a basis el , • . • , em for V, we can write w uniquely as w ~j=IWj • ej, where wi are usual r-forms on M. w is differentiable, by definition, if wi are all differentiable. The exterior derivative dw is defined to be ~j=l dw i . ej , which is a V-valued (r + 1)-form. Given a mappingJ of a manifold M into another manifold M', the differential at p of J is the linear mapping J* of T'O(M) into Tf('O)(M') defined as follows. For each X E 'T'O(M), choose a curve x(t) in M such that X is the vector tangent to x(t) at p = x(to). Then!*(X) is the vector tangent to the curve J(x(t)) at J(P) = J(x( to) ). I t"follows immediately that if g is a function differentiable in a neighborhood of J(P) , then (J*(X))g = X(g oj). When it is necessary to specify the point p, we write (f*) 'P' When there is no danger of confusion, we may simply write J instead of J*. The transpose of (1*)'0 is a linear mapping of Tf~P)(M') into T;(M). For any r-form w' on M', we define an r-formJ*w' on M by

The exterior differentiation d commutes with f*: d(f*w') f*(dw') . A mappingf of Minto M' is said to be of rank r at p E M if the dimension of J*(T'O(M)) is r. If the rank of J at p is equal to n = dim M, (1*) '0 is injective and dim M dim M'. If the rank ofJatp is equal to n' - dim M', (J*) '0 is sur:jective and dim M > dim M'. By the implicit function theorem, we have PROPOSITION

1.1.

Let J be a mapping oj Minto M' and p a point

of M. (1) If (J*) '0 is injective, there exist a local coordinate system ul, ... un in a neighborhood U ofp and a local coordinate system VI, ••• , vn' in a neighborhood ojJ(p) such that Jor q E U and i In particular, J is a homeomorphism oj U onto f( U).

1, ... , n.

1.

9

DIFFERENTIABLE MANIFOLDS

(2) If (f*) p is surjective, there exist a local coordinate system u1 , • • • , un in a neighborhood U of P and a local coordinate system VI, ••• , vn' of f(p) such that for q € U

and i

=

1, ... , n'.

In particular, the mapping f: U -+ M' is open. (3) If (f*) p is a linear isomorphism of Tp(M) onto Tf(p/M') , thenf defines a homeomorphism of a neighborhood U ofp onto a neighborhood V off(p) and its inverse f- 1 : V -+ U is also differentiable. For the proof, see Chevalley [1, pp. 79-80J. A mapping f of Minto M' is called an immersion if (f*) p is injective for every point p of M. We say then that M is immersed in M' by f or that M is an immersed submanifold of M'. When an immersionfis injective, it is called an imbedding of Minto M'. We say then that M (or the imagef(M)) is an imbeddedsubmanifold (or, simply, a submanifold) of M'. A submanifold mayor may not be a closed subset of M'. The topology of a submanifold is in general finer than the relative topology induced from M'. An open subset M of a manifold M', considered as a submanifold of M' in a natural manner, is called an open submanifold of M'. Example 1.1. Let f be a function defined on a manifold M'. Let M be the set of points p € M' such thatf(p) = O. If (df) p i=- 0 at every point p of M, then it is possible to introduce the structure of a manifold in M so that M is a closed submanifold of M', called the hypersurface defined by the equation f = O. More generally, let M be the set of common zeros of functions fl' ... ,f, defined on M'. If the dimension, say k, of the subspace of T;(M') spanned by (dfl) p' • • • , (df,) p is independent of p € M, then M is a closed submanifold of M' of dimension dim M' - k. A diffeomorphism of a manifold M onto another manifold M' is a homeomorphism cp such that both cp and cp-l are differentiable. A diffeomorphism of M onto itself is called a differentiable transformation (or, simply, a transformation) of M. A transformation cp of M induces an automorphism cp* of the algebra 1)(M) of differential forms on M and, in particular, an automorphism of the algebra ty(M) of functions on M:

(cp*f) (p) = f( cp(p)),

f

€

ty(M),

p€M.

10

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

I t induces also an automorphism cp* of the Lie algebra X( M) of vector fields by where

cp(q) = p,

X

X(M).

E

They are related by for X

E

X(M)

and

f

E

fJ(M).

Although any mapping cp of Minto M' carries a differential form w' on M' into a differential form cp* (w') on M, cp does not send a vector field on M into a vector field on M' in general. We say that a vector field X on M is cp-related to a vector field X' on M' if (cp*) pXp = X;(p) for all p E M. If X and Yare cp-related to X' and Y' respectively, then [X, Y] is cp-related to [X', Y']. A distribution S of dimension r on a manifold M is an assignment to each pointp of Man r-dimensional subspace Sp of Tp(M). I t is called differentiable if every point p has a neighborhood U and r differentiable vector fields on U, say, Xl' ... , X r , which form a basis of Sq at every q E U. The set Xl' ... , X r is called a local basis for the distribution S in U. A vector field X is said to belong to S if X p E Sp for all p E M. Finally, S is called involutive if [X, Y] belongs to S whenever two vector fields X and Y belong to S. By a distribution we shall always mean a differentiable distribution. A connected submanifold N of M is called an integral manifold of the distribution S iff*(Tp(N)) = Sp for all pEN, wherefis the imbedding of N into M. If there is no other integral manifold of S which contains N, N is called a maximal integral manifold of S. The classical theorem of Frobenius can be formulated as follows.

1.2. Let S be an involutive distribution on a manifold M. Through every point p E M, there passes a unique maximal integral manifold N(P) of S. Any integral manifold through p is an open submanifold of N(P)· For the proof, see Chevalley [1, p. 94]. We also state PROPOSITION

.

1.3. Let S be an involutive distribution" on a manijold M. Let W be a submanifold of M whose connected components are all integral manifolds of S. Let f be a differentiable mapping of a manifold N PROPOSITION

I. DIFFERENTIABLE MANIFOLDS

11

into M such that f( N) c W. If W satiifies the second axiom of countability, then f is differentiable as a mapping of N into W. For the proof, see Chevalley [1, p. 95, Proposition 1]. Replace analyticity there by differentiability throughout and observe that W need not be connected since the differentiability olfis a local matter. We now define the product of two manifolds M and N of dimension m and n, respectively. If M is defined by an atlas A {( Ui' 9Ji)} and N is defined by an atlas B = {( Vi' 1p I)}, then the natural differentiable structure on the topological space M x N is defined by an atlas {( Ui X Vj' 9Ji X 1p I)}, where 9Ji x "Pj: Ui x Vi R m+n - R m x R n is defined in a natural manner. Note that this atlas is not complete even if A and B are complete. For every point (p, q) of M x N, the tangent space T(p,q)(M x N) can be identified with the direct sum Tp(M) Tq(N) in a natural manner. Namely, for X E Tp(M) and Y E TiN), choose x(to) and curves x(t) andy(t) such that X is tangent to x(t) at p Y is tangent to yet) at q = y(to)' Then (X, Y) E Tp(M) Tq(N) is identified with the vector Z E T(p,q)(M x N) which is tangent to the curve z(t) = (x(t),y(t)) at (P, q) = (x(to),y(to))' Let X E T(p q)(M x N) be the vector tangent to the curve (x(t), q) in M x N at (P, q). Similarly, let Y E T(p,q)(M x N) be the vector tangent to the curve (p,y(t)) in M x Nat (p, q). In other words, X is the image of X by the mapping M M x N which sends p' E M into (P', q) and Y is the image of Y by the mapping N M x N which sends q' E N into (p, q'). Then Z = X + Y, because, for any functionf on M x N, 2f = (df(x(t),y(t))/dt)t=t o is, by the chain rule, equal to --')0-

--')0-

--')0-

(df(x(t),y(to))/dt)t=to

+ (df(x(to),y(t))/dt)t=to =

Xf

+ Y1.

More generally: PROPOSITION 1.4 (Leibniz's formula). Let 9J be a mapping of the product manifold M x N into another manifold V. The differential 9J* at (P, q) EM x N can be expressed as follows. If Z E T(p,q)(M x N) corresponds to (X, Y) E T p (M) + T q (N), then

12

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

where

ffJ l: M

~

V and ffJ2: N

ffJl(P') = ffJ(p', q) for P'

E

M

---+

V are defined by

and ffJ2(q') = ffJ(p, q') for q' EN.

Proof. From the definitions of X, Y, ffJl' and ffJ2' it follows that ffJ*(X) = ffJl*(X) and ffJ*(Y) = ffJ2*(Y). Hence, ffJ*(Z) = ffJ*(X) + ffJ*(Y) = ffJl*(X) + ffJ2*(Y). QED. Note that if V = M X Nand ffJ is the identity transformation, then the preceding proposition reduces to the formula Z = X + Y. Let X be a vector field on a manifold M. A curve x( t) in M is called an integral curve of X if, for every parameter value to, the vector XX(to) is tangent to the curve x(t) at x( to). For any point Po of M, there is a unique integral curve x( t) of X, defined for It I < 8 for some 8 > 0, such that Po = x(O). In fact, let ul , ••• , un be a local coordinate system in a neighborhood U of Po and let X = ~ ~j( aj au j) in U. Then an integral curve of X is a solution of the following system of ordinary differential equations:

j

=

1, ... , n.

Our assertion follows from the fundamental theorem for systems of ordinary differential equations (see Appendix 1). A I-parameter group of (differentiable) transformations of M is a mapping of R X Minto M, (t, p) ER X M ---+ ffJt(p) EM, which satisfies the following conditions: (1) For each t ER, ffJt: P ---+ ffJt(p) is a transformation of M; (2) For all t,s ERand P E M, ffJHs(P) = ffJt(ffJs(p)). Each I-parameter group of transformations ffJt induces a vector field X as follows. For every point P EM, X p is the vector tangent to the curve x(t) = ffJt(p), called the orbit of P, at P = ffJo(P). The orbit ffJt(p) is an integral curve of X starting at p. A loeall-parameter group of local transformations can be defined in the same way, except that ffJt(P) is defined only for t in a neighborhood of 0 and P in an open set of M. More precisely, let Ie be an open interval (-8, 8) and U an open set of M. A local I-parameter group of local transformations defined on Ie X U is a mapping of Ie X U into M which satisfies the following conditions: (1') For each t E Ie' ffJt: P ---+ ffJ t(p) is a diffeomorphism of U onto the open set ffJt( U) of M;

I.

(2') If t,s,t

+s

E

13

DIFFERENTIABLE MANIFOLDS

1£ and if p, f{Js(P)

E

U, then

As in the case of a I-parameter group of transformations, f{Jt induces a vector field X defined on U. We now prove the converse.

1.5. Let X be a vector field on a manifold M. For each point Po of M, there exist a neighborhood U of Po, a positive number C and a local I-parameter group of local transformations f{J t: U ---+ M, t E Ie' which induces the given X. PROPOSITION

We shall say that X generates a local I-parameter group of local transformations f{Jt in a neighborhood of Po' If there exists a (global) I-parameter group of transformations of M which induces X, then we say that X is complete. If f{Jt(p) is defined on Ie X M for some c, then X is complete. Proof. Let u\ ... , un be a local coordinate system in a neighborhood W of Po such that ul(po) = ... = un(po) = O. Let X = ~ ~i(U\ ... , un) (oj OUi) in W. Consider the following system of ordinary linear differential equations:

i = 1, ... ,

n

with unknown functions fl(t), ... ,fn(t). By the fundamental theorem for systems of ordinary differential equations (see Appendix 1), there exists a unique set of functionsfl(t; u), ... , fn(t; u), defined for u = (u\ ... ,un) with luil < dl and for It I < Cl' which form a solution of the differential equation for each fixed u and satisfy the initial conditions:

fiCO; u) = ui • Set f{Jt(u) = (fl(t; u), ... ,fn(t; u)) for ItI < Cl and u in U l = {u; luil < dl }. If Itl, lsi and It + sl are all less than Cl and both u and f{Js(u) are in U l , then the functions gi(t) = fi(t + s; u) are easily seen to be a solution of the differential equation for the initial conditions gi(O) =fi(s; u). By the uniqueness of the solution, we havegi(t) = fi(t; f{Js(u)). This proves that f{Jt(f{Js(u)) = f{JHS(U). Since f{Jo is the identity transformation of U l , there exist d > 0 and C > 0 such that, for U = {u; luil < d}, f{Jt( U) c U 1 if

14

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

It I < e. Hence, CP-t(CPt(u)) = CPt(CP-t(u)) = CPo(u) = U for every U E U and ItI < e. This proves that CPt is a diffeomorphism of U for It I < e. Thus, CPt is a local I-parameter group of local transformations defined on Ie X U. From the construction of CPt, it is obvious that CPt induces the given vector field X in U. QED. Remark. In the course of the preceding proof, we showed also that if two local I-parameter groups of local transformations CPt and 'f/Jt defined on Ie X U induce the same vector field on U, they coincide on U. PROPOSITION

1.6.

On a compact manifold M, every vector field X is

complete. Proof. For each point p E M, let U(p) be a neighborhood of p and e(p) a positive number such that the vector field X generates a local I-parameter group of local transformations CPt on Ie
1.7. Let cP be a transformation of M. If a vector field X generates a local I-parameter group of local transformations cP t, then the vector field cP*X generates cP CPt cp-l. Proof. It is clear that cP CPt cp-l is a local I-parameter group of local transformations. To show that it induces the vector fie1d cP*X, let P be an arbitrary point of M and q = cp-l(P). Since CPt induces X, the vector Xq E Tq(M) is tangent to the curve x(t) = CPt(q) at q = x(D). It follows that the vector PROPOSITION

0

0

0

(cp*X)p

=

0

cP*(Xq)

E

Tp(M)

is tangent to the curve yet) = cP CPt(q) = cP CPt cp-l(p). 0

0

0

QED.

1.8. A vector field X is invariant by cP, that is, qJ*X = X, if and only if cP commutes with CPt. COROLLARY

I.

15

DIFFERENTIABLE MANIFOLDS

We now give a geometric interpretation of the bracket [X, Y] of two vector fields.

Let X and Y be vector fields on M./f X generates" a local I-parameter group of local transformations ffJ t, then PROPOSITION

1.9.

More precisely, [X, Y] p = lim! [Yp - (( ffJ t) * Y) p] , t-O t

P EM.

The limit on the right hand side is taken with respect to the natural topology of the tangent vector space Tp(M). We first prove two lemmas.

1. If j(t, p). is a junction on Ie X M, where Ie is an open interval (-s, s), such thatj(O,p) = Ojor allp E M, then there exists a junction g(t, P) on Ie X M such that j(t, P) = t . g(t, p). Moreover, g(O,p) =j'(O,P), wherej' = oj/ot,jor p EM. Proof. It is sufficient to define LEMMA

get, p) =

ff'

QED.

(ts, p) ds.

2. Let X generate ffJt. For any junctionj on M, there exists a junction gt(p) = g(t, p) such thatj 0 ffJt = j + t . gt and go = Xj on M. The function g(t, P) is defined, for each fixed p E M, in ItI < s LEMMA

for some s. Proof. Consider f(t, p) = j( ffJt(P)) - j(P) and apply Lemma 1. Thenf 0 ffJt = j + t . gt. We have

lim~[f(ffJt(P)) t--O

t

j(p)] = lim! j(t,P) = limgt(p) = go(P)· t--O t t--O

QED. Proof of Proposition 1.9. Given a function j on M, take a functiong t such thatjo ffJt = j + t· gt and go = Xj (Lemma 2). Set P(t) = ffJt-1(p). Then

((ffJt)*Y)p! = (Y(jo ffJt))p(t) = (Yj)p(t)

+ t·

(Ygt)p(t)

16

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

and

QED.

proving our assertion.

1.10. With the same notations as in Proposition 1.9, we have more generally COROLLARY

(cpJ*[X, Y] = lim ~ [(cpJ*Y - (CPs+t)*Y] t--O t for any value of s. Proof. For a fixed value of s, consider the vector field (CPs) *Y and apply Proposition 1.9. Then we have

= lim ~ [(CPs) * Y - (cp s+t) *Y], t--O t since CPs 0 CPt = CPs+t. On the other hand, (CPs) *X = X by Corollary 1.8. Since (CPs) * preserves the bracket, we obtain QED.

Remark.

The conclusion of Corollary 1.10 can be written as

1.11. Suppose X and Y generate local I-parameter groups CPt and "Pg, respectively. Then CPt 0 "Ps = "Ps 0 CPt for every sand t if and only if [X, Y] = O. Proof. If CPt 0 "Ps = "Ps 0 CPt for every sand t, Y is invariant by every CPt by Corollary 1.8. By Proposition 1.9, [X, Y] = O. Conversely, if [X, Y] = 0, then d((cpt) *Y)/dt = 0 for every t by Corollary 1.10 (see Remark, above). Therefore, (cpt)*Y is a constantvector at each point p so that Y is invariant by every CPt. By Corollary 1.8, every "Ps commutes with every CPt. QED. COROLLARY

I.

17

DIFFERENTIABLE MANIFOLDS

2. Tensor algebras We fix a ground field F which will be the real nUlnber field R or the complex number field C in our applications. All vector spaces we consider are finite dimensional over F unless otherwise stated. We define the tensor product U ® V of two vector spaces U and Vas follows. Let M( U, V) be the vector space which has the set U X V as a basis, i.e., the free vector space generated by the pairs (u, v) where u E U and v E V. Let N be the vector subspace of M( U, V) spanned by elements of the form

(u

+ u', v)

(u, v + v') - (u, v) - (u, v'),

- (u, v) - (u', v), ~

(ru, v) - r(u, v),

(u, rv) - r(u, v),

where u,u' E U, V,V' E V and rEF. We set U ® V = M( U, V)/ N. For every pair (u, v) considered as an element of M( U, V), its image by the natural projection M( U, V) ~ U ® V will be denoted by u ® v. Define the canonical bilinear mapping cp of U x ·V into U ® Vby cp (u, v)

=

u®v

for (u, v)

E

U

X

V.

Let W be a vector space and "p: U X V ~ W a bilinear mapping. We say that the couple (W, "p) has the universaljactorization property for U X V if for every vector space S and every bilinear mapping j: U X V ~ S there exists a unique linear mapping g: W ~ S such thatj = g 0 "p.

The couple (U ® V, cp) has the universal jactorization property jor U X v. If a couple (W, "p) has the universal jactorization property jor U X V, then (U ® V, cp) and (W, "p) are isomorphic in the sense that there exists a linear isomorphism (1: U ® V ~ W such that "P = (1 0 cp. Proof. Let S be any vector space and j: U X V ~ S any bilinear mapping. Since U X V is a basis for M( U, V), we can extend j to a unique linear mapping j': M( U, V) ~ S. Since j is bilinear,j'vanishes on N. Therefore,j'induces a linear mapping g: U ® V ~ S. Obviously, j = g 0 cpo The uniqueness of such a mapping g follows from the fact that cp( U X V) spans U ® V. Let (W, "P) be a couple having the universal factorization property for U X V. By the universal factorization property of ( U ® V, cp) PROPOSITION

2.1.

18

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(resp. of (W, "p)), there exists a unique linear mapping a: U ® V ~ W (resp. 7: W ->- U ® V) such that "p = a 0 cp (resp. cp = 7 0 cp). Hence, cp = 7 a cp and"P = a 7 "P. Using the uniqueness of g in the definition of the universal factorization property, we conclude that 7 0 a and a 0 7 are the identity transformations of U X V and W respectively. QED. 0

0

0

0

There is a unique isomorphism of U ® V onto V ® U which sends u ® v into v ® u for all u E U and v E V. Proof. Let f: U X V --+ V ® U be the bilinear mapping defined by feu, v) = v ® u. By Proposition 2.1, there is a unique linear mapping g: U ® V --+ V ® U such that g(u ® v) = v ® u. Similarly, there is a unique linear mapping g': V ® U ~ U ® V such that g'(v ® u) = u ® v. Evidently, g' 0 g and gog' are the identity transformations of U ® V and V ® U respectively. Hence, g is the desired ison10rphism. QED. The proofs of the following two propositions are similar and hence omitted. PROPOSITION

2.2.

2.3. Ifwe regard the groundfieldF as a I-dimensional vector space over F, there is a unique isomorphism of F ® U onto U which sends r ® u into rufor all r E F and u E U. Similarly,for U ® F and U. PROPOSITION 2.4. There is a unique isomorphism of (U ® V) ® W onto U ® (V ® W) which sends (u ® v) ® w into u (8) (v ® w) for PROPOSITION

all u E U,

V E

V, and w

E

W.

Therefore, it is meaningful to write U ® V ® W. Given vector spaces U 1 , . • • , Uk' the tensor product U 1 ® ... ® Uk can be defined inductively. Let cp: U 1 X . . . X Uk ~ U 1 ® ... ® Uk be the multilinear mapping which sends (u 1 , ••• , Uk) into U 1 ® ... ® Uk. Then, as in Proposition 2.1, the couple (U 1 ® ... ® Uk' cp) can be characterized by the universal factorization property for U 1 X . . . X Uk· Proposition 2.2 can be also generalized. For any permutation 7T of (1, ... , k), there is a unique isomorphism of U 1 ® ... ® Uk onto U (l) ® ... ® U (k) which sends U 1 ® ... ® Uk into u (1) ® ... ® U (k)· 71

71

71

71

2.4.1. Given linearmappingsfj: Uj~ Vj,} = 1,2, there is a unique linear mapping f: U 1 ® U2 --+ VI ® V2 such that f(u 1 ® u2 ) = fl(u 1 ) ®J;(u 2 ) for all U I E U 1 and U2 E U2 • PROPOSITION

I.

19

DIFFERENTIABLE MANIFOLDS

Proof. Consider the bilinear mapping U 1 X U2 -+ V1 @ V2 which sends (u v u2) intof1(u 1) @};(u 2) and apply Proposition 2.1. QED. The generalization of Proposition 2.4.1 to the case with more than two mappings is obvious. The mappingfjust given will be denoted by f1 @};. PROPOSITION

2.5.

If U 1 + U2 denotes the direct sum of U1 and U2,

then Similarly,

U @ (V1 + V2) = U @ V1 + U @ V2. Proof. Let i 1: U1 -+ U 1 + U2 and i2: U2 -+ U 1 + U2 be the injections. Let P1: U1 + U2 -+ U 1 and P2: U 1 + U2 -+ U2 be the projections. Then P1 0 i 1 and P2 0 i 2 are the identity transformations of U 1 and U2 respectively. Both P2 0 i 1 and P1 0 i 2 are the zero mappings. By Proposition 2.4.1, i 1 and the identity transformation of V induce a linear mapping Z1: U1 @ V -+ (U 1 + U2) @ V. Similarly, Z2' P1' and P2 are defined. It follows that P1 zl and P2 Z2 are the identity transformations of U 1 @ V and U2 @ V respectively and P2 0 Z1 and P1 0 Z2 are the zero mappings. This proves the first isomorphism. The proof for the second is similar. QED. By the induction, we obtain 0

0

(U1 +

... + Uk)

@ V

=

U1 @ V

+ ... + Uk

@ V.

2.6. Ifu 1, ... , Um is a basisfor U and V1, ... , Vn is a basis for V, then {u i @ Vj; i = 1, ... , m;j = 1 , ... , n} is a basis for U @ V. In particular, dim U @ V = dim U dim V. Proof. Let U i be the I-dimensional subspace of U spanned by U i and V j the I-dimensional subspace of V spanned by Vj' By Proposition 2.5, U @ V = ~ .. U·t @ Vi. PROPOSITION

~,J

By Proposition 2.3, each U i @ V j is a I-dimensional vector space spanned by U i @ V j ' QED. For a vector space U, we denote by U* the dual vector space of U. For u E U and U* E U*, (u, u*) denotes the value of the linear functional u* on u.

20

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

2.7. Let L( U*, V) be the space of linear mappings of u* into V. Then there is a unique isomorphism g of U ® V onto L( U*, V) such that PROPOSITION

(g(u ® v))u*

=

(u, u*)v

for all u E U, V E

Vandu~

E

U*.

Proof. Consider the bilinear mapping f: U X V ---+ L( U*, V) defined by (f(u, v))u* = (u, u*)v and apply Proposition 2.1. Then there is a unique linear mapping g: U ® V ---+ L( U*, V) such that (g(u ® v) )u* = (u, u*)v. To prove that g is an isomorphism, let u 1, . . . , Um be a basis for U, ui, ... ,u~ the dual basis for u* and v1, . . . , V n a basis for V. We shall show that {g(u i ® v j ) ; i = 1, ... , m;j = 1, ... , n} is linearly independent. If ~ aijg(u i ® Vj) = 0 where aij E F, then

o=

(~

aijg(u i ® vj))u% =

~

akjv j

and, hence, all aij vanish. Since dim U ® V = dim L( U*, V), g is an isomorphism of U X V onto L( U*, V). QED.

2.8. Given two vector spaces U and V, there is a unique isomorphism g of u* ® v* onto (U ® V) * such that PROPOSITION

(g(u* ® v*))(u ® v)

=

(u, u*)(v, v*)

for all u E U, u*

E

U*, V E V, v*

E

V*.

Proof. Apply Proposition 2.1 to the bilinear mapping f: U* X V* ---+ (U ® V) * defined by (f(u*, v*)) (u ® v) = (u, u*)(v, v*). To prove that g is an isomorphism, take bases for U, V, U*, and V* and proceed as in the proof of Proposition 2.7. QED. We now define various tensor spaces over a fixed vector space V. For a positive integer r, we shall call Tr = V ® ... ® V (r times tensor product) the contravariant tensor space of degree r. An element of Tr will be called a contravariant tensor of degree r. If r = 1, Tl is nothing but V. By convention, we agree that TO is the ground field F itself. Similarly, T s = V* ® ... ® V* (s times tensor product) is called the covariant tensor space ofdegree s and its elements covariant tensors of degree s. Then T 1 = V* and, by convention, To = F. We shall give the expressions for these tensors with respect to a basis of V. Let e1, . • • , en be a basis for V and el, ... , en the dual

I.

21

DIFFERENTIABLE MANIFOLDS

basis for V*. By Proposition 2.6, {e t1 ® ... ® eir ; 1 < iI' ... , i r < n} is a basis for T r • Every contravariant tensor K of degree r can be expressed uniquely as a linear combination

K =

~.

. Ki 1 ••• ir e·1,1 \[Y tX". • . \[Y tX" e·1 ' r

1,1' ••• ,?'r

where Ki 1 ••• ir are the components of K with respect to the basis el , . . . , en of V. Similarly, every covariant tensor L of c;legree s can be expressed uniquely as a linear combination L =

~.h,· .. ,Jr . L·h ... Jr ell \[Y tX" .•• \tX"[ Y eir,

where L i1 ... i r are the components of L. For a change of basis of V, the components of tensors are subject to the following transformations. Let el , . . . , en and el , . . . , en be two bases of V related by a linear transformation

i = 1, ... , n. The corresponding change of the dual bases in

v*

is given by

i = 1, ... , n,

where B = (Bj) is the inverse matrix of the matrix A = (Aj) so that ~j AjBt = bt· If K is a contravariant tensor of degree r, its components Ki 1 ••• ir and Ki 1 . . . i r with respect to {e i } and {e i } respectively are related by Similarly, the components of a covariant tensor L of degree s are related by The verification of these formulas is left to the reader. We define the (mixed) tensor space of type (r, s), or tensor space of contravariant degree r and covariant degree s, as the tensor product T~ = Tr ® T s = V ® ... ® V ® v* ® ... ® v.* (V: r times, V*: s times). In particular, T~ = Tr, T~ = T sand Tg = F. An element of T~ is called a tensor of type (r, s), or tensor of contravariant degree r and covariant degree s. In terms of a basis el , . • . , en of V and

22

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

the dual basis el , . . . , en of V*, every tensor K of type (r, s) can be expressed uniquely as K = ~."'1" •• ,"'r' ..h . ... ,J. Khh ... ~r e· t)(\ . • • t)(\ e· t)(\ eh t)(\ • • • t)(\ ej .. · J 11 'CJ 'CJ "'r 'CJ 'CJ 'CJ, 8

8

8

J:

where K1::: are called the components of K with respect to the basis el , . . . , en' For a change of basis ei = ~j Aie j , we have the following transformations of components :

(2.1 ) Set T = ~r~ =0T~, so that an element of T is of the form ~r~=oK~, where K~ E T~ are zero except for a finite number of them. We shall now make T into an associative algebra over F by defining the product of two tensors K E T~ and LET: as follows. From the universal factorization property of the tensor product, it follows that there exists a unique bilinear mapping of T~ X T~ into T~t~ which sends (VI ® ... ® Vr ® vi ® ... ® v:, WI ® ... ® W p ® wi ® ... ® wi) E T~ x T~ into VI ® ... @ Vr

®

WI

® ... ®

Wp

®

vi

® ... ® vi ® wi ® ... ®

wi

E

T~t~.

The image of (K, L) E T~ X T~ by this bilinear mapping will be denoted by K ® L. In terms of components, if K is given by K~1 •.• ~r and L is given by Lk 1 •• · k p then J1·· ·J m ... m q, 1

8

(K

=

t)(\ L) ~1 ••• ~r+P h .. ·J 8 +Q

'CJ

K~1'" ~rL~r+l ..• ~r+p • h·· ·J 8 J8+1· .. J8+Q

We now define the notion of contraction. Let r,s > 1. To each ordered pair of integers (i,j) such that 1 < i < rand 1 <j < S, we associate a linear mapping, called the contraction and denoted by C, ofT~ into T~=l which maps VI ® ... ® Vr ® vi ® ... ® v: into (Vi' vj)V I ® ... ®

® Vi + 1 ® ... ® Vr ® vi ® ® V!-l ® Vj+1 ® ... ®

Vi - I

vi,

where VI, ••• , Vr E V and Vi, , vi E V*. The uniqueness and the existence of C follow from th~ universal factorization property of the tensor product. In terms of components, the contraction C maps a tensor K E T~ with components Kj~:: into a tensor CK E T~=i whose components are given by

:J:

.. ~r-1 ( CK)~1' J1 ••• J8 -1

-

~

k

Ki1 ··· k ... i r h .. • k ••. j 8

'

where the superscript k appears at the i-th position and the subscript k appears at the j-th position.

1.

DIFFERENTIABLE MANIFOLDS

23

We shall now interpret tensors as multilinear mappings.

T r is isomorphic, in a natural way, to the vector space of all r-linear mappings of V X . . . X V into F. PROPOSITION

2.9.

Tr is isomorphic, in a natural way, to the vector space of all r-linear mappings of V* X . . . X V* into F. PROPOSITION

2.10.

Proof. We prove only Proposition 2.9. By generalizing Proposition 2.8, we see that T r = V* ® ... ® V* is the dual vector space of Tr = V ® ... ® V. On the other hand, it follows from the universal factorization property of the tensor product that the space of linear mappings of Tr = V @ ... ® V into F is isomorphic to the space of r-linear mappings of V X . . . X V into F. QED. Following the interpretation in Proposition 2.9, we consider a tensor K E T r as an r-linear mapping V X . . . X V ---+ F and write K(v 1 , . • . , Vr) E F for v1 , ••• , Vr E V. PROPOSITION

2.11.

T; is isomorphic, in a natural way, to the vector

space of all r-linear mappings of V

X . . . X

V into V.

Proof. T; is, by definition, V @ T r which is canonically isomorphic with T r @ V by Proposition 2.2. By Proposition 2.7, T r @ V is isomorphic to the space of linear mappings of the dual space of Tn that is Tr, into V. Again, by the universal factorization property of the tensor product, the space of linear mappings of Tr into V can be identified with the space of r-linear mappings of V X . . . X V into V. QED. With this interpretation, any tensor K of type (1, r) is an r-linear mapping of V X . . . X V into V which maps (v 1 , . . . ,vr) into K(v 1 , ••• , Vr) E IV. If e1 , • . • ,en is a basis for V, then K = ~ Kjp .. j/i @ ej1 @ ... @ ejr corresponds to an r-linear mapping of V X ... X V into V such that K(ej1 , ... ,ejJ = ~i KJ1 ... j/i. Similar interpretation can be made for tensors of type (r, s) in general, but we shall not go into it. Example 2.1. If v E V and V* E V*, then v ® v* is a tensor of type (1, 1). The contraction C: Tt ---+ F maps v @ v* into (v, v*). In general, a tensor K of type (1, 1) can be regarded as a linear endomorphism of V and the contraction CK of K is then the trace of the corresponding endomorphism. In fact, if el' . . . , en is a

24

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

basis for V and K has components KJ with respect to this basis, then the endomorphism corresponding to K sends ej into ~i Kje i . Clearly, the trace of K and the contraction CK of K are both equal to ~i K1· Example 2.2. An inner product g on a real vector space V is a covariant tensor of degree 2 which satisfies (1) g( V, v) > 0 and g(v, v) = 0 if and only if v = 0 (positive definite) and (2) g(v, v') = g(v', v) (symmetric). Let T( U) and T( V) be the tensor algebras over vector spaces U and V. If A is a linear isomorphism of U onto V, then its transpose A * is a linear isomorphism of V* onto U* and A *-1 is a linear isomorphism of U* onto V*. By Proposition 2.4, we obtain a linear isomorphism A ® A *-1: U ® U* ~ V ® V*. In general, we obtain a linear isomorphism of T( U) onto T( V) which maps T~( U) onto T~( V). This isomorphism, called the extension of A and denoted by the same letter A, is the unique algebra isomorphism T( U) ~ T( V) which extends A: U ~ V; the uniqueness follows from the fact that T( U) is generated by F, U and U*. It is also easy to see that the extension of A commutes with every contraction C. 2.12. There is a natural 1 : 1 correspondence between the linear isomorphisms of a vector space U onto another vector space V and the algebra isomorphisms of T( U) onto T( V) which preserve type and commute with contractions. In particular, the group ofautomorphisms of V is isomorphic, in a natural way, with the group of automorphisms of the tensor algebra T( V) which presserve type and commute with contractions. Proof. The only thing which has to be proved now is that every algebra isomorphism, say 1, of T( U) onto T( V) is induced from an isomorphism A of U onto V, provided that j preserves type and commutes with contractions. Since j is type-preserving, it maps T6( U) = U isomorphically onto T6( V) = V. Denote the restriction ofj to U by A. Since j maps every element of the field F = Tg into itself and commutes with every contraction C, we have, for all u E U and u* E U*, PROPOSITION

(Au,ju*)

=

(ju,ju*)

=

C(ju @ju*)

=

C(f(u ® u*))

= f(C(u ® u*)) = f(u, u*») = (u, u*).

I.

DIFFERENTIABLE MANIFOLDS

25

Hence, ju* = A*-1U *. The extension of A and j agrees on F, U and U*. Since the tensor algebra T( U) is generated by F, U and U*,j coincides with the extension of A. QED. Let T be the tensor algebra over a vector space V. A linear endomorphism D ofT is called a derivation if it satisfies the following conditions: (a) D is type-preserving, i.e., D maps T~ into itself; (b) D(K ® L) = DK (8) L + K (8) DL for all tensors K and L; (c) D commutes with every contraction C. The set of derivations of T forms a vector space. It forms a Lie algebra if we set [D, D'] = DD' - D'D for derivations D and D'. Similarly, the set of linear endomorphisms of V forms a Lie algebra. Since a derivation D maps T6 = V into itself by (a), it induces an endomorphism, say B, of V.

The Lie algebra of derivations of T( V) is isomorphic with the Lie algebra of endomorphisms of V. The isomorphism is given by assigning to each derivation its restriction to V. PROPOSITION

2.13.

Proof. It is clear that D -» B is a Lie algebra homomorphism. From (b) it follows easily that D maps every element of F into o. Hence, for v € V and v* € V*, we have

o=

D«v, v*») = D(C(v (8) v*)) = C(D(v (8) v*)) = C(Dv (8) v*

+ v (8) Dv*)

= (Dv, v*)

+ (v, Dv*).

Since Dv = Bv, Dv* = -B*v* where B* is the transpose of B. Since T is generated by F, Vand V*, D is uniquely determined its restriction to F, V and V*. It follows that D -» B is injective. Conversely, given an endomorphism B of V, we define Da = 0 for a € F, Dv = Bv for v € V and Dv* = -B*v* for v* € V* and, then, extend D to a derivation of T by (b). The existence of D follows from the universal factorization property of the tensor QED. product.

Example 2.3.

Let K be a tensor of type (1, 1) and consider it as an endomorphism of V. Then the automorphism of T(V) induced by an automorphism A of V maps the tensor K into the tensor AKA -1. On the other hand, the derivation of T (V) induced by an endomorphism B of V maps K into [B, K] = BK - KB.

26

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

3. Tensor fields Let T x = Tx(M) be the tangent space to a manifold M at a point x and T(x) the tensor algebra over T x : T(x) = l: T~(x), where T~(x) is the tensor space of type (r, s) over T x • A tensor field of type (r, s) on a subset N of M is an assignment of a tensor K x E T~(x) to each point x of N. In a coordinate neighborhood U with a local coordinate system xl, . . . , xn , we take Xi = a/a Xi, i = 1, ... ,n, as a basis for each tangent space T x , x E U, and i Wi = dx , i = 1, ... , n, as the dual basis of T x*. A tensor field K of type (r, s) defined on U is then expressed by K x = l:

K~l·" ~r X· t)(\ ••• \C:J t)(\ 11 ... J s ~l \C:J

X·~r

t)(\

\C:J

w jl

t)(\ ••• t)(\

\C:J

\C:J

w is ,

where KJ~::: j: are functions on U, called the components of K with respect to the local coordinate system xl, ... ,xn • We say that K is of class Ck if its components Kj::: are functions of class Ck; of course, it has to be verified that this notion is independent of a local coordinate system. This is easily done by means of the formula (2.1) where the matrix (Aj) is to be replaced by the Jacobian matrix between two local coordinate systems. From now on, we shall mean by a tensor field that of class Coo unless otherwise stated. In section 5, we shall interpret a tensor field as a differentiable cross section of a certain fibre bundle over M. We shall give here another interpretation of tensor fields of type (0, r) and (1, r) from the viewpoint of Propositions 2.9 and 2.11. Let ty be the algebra of functions (of class COO) on M and X the (Y-module of vector fields on M.

J:

3.1. A tensor field K if type (0, r) (resp. type (1, r)) on M can be considered as an r-linear mapping of X X ... X X into ty (resp. X) such that PROPOSITION

K(fl X l' ... ,irXr)

=

11 .. 'irK(X1 , ••• , X r) for h (Y E

and Xi

E

X.

Conversely, any such mapping can be considered as a tensor field if type (0, r) (resp. type (1, r)). Proof. Given a tensor field K of type (0, r) (resp. type (1, r)), K x is an r-linear mapping of T x X . . . X T x into R (resp. T x)

I.

DIFFERENTIABLE MANIFOLDS

27

by Proposition 2.9 (resp. Proposition 2.11) and hence (Xl' ... , X r) --* (K(X I, ... , Xr))x = KX((XI)x, ... , (Xr)x) is an r-linear mapping of X X ... X X into ty (resp. X) satisfying the preceding condition. Conversely, let K: X X . . . X X --* ty (resp. X) be an r-linear mapping over ty. The essential point of the proof is to show that the value of the function (resp. the vector field) K(X I, ... ,Xr) at a point x depends only on the values of Xi at x. This will imply that K induces an r-linear mapping of Tx(M) X ... X Tx(M) into R (resp. Tx(M)) for each x. We first observe that the mapping K can be localized. Namely, we have LEMMA.

If Xi =

then we have

Y i in a neighborhood U of x for i = 1, ... , r,

Proof of Lemma. It is sufficient to show that if Xl = a in U, then K(X I, ... ,Xr) = a in U. For any y E U, letfbe a differentiable function on M such that f(y) = a and f = 1 outside U. Then Xl = fX I and K(XI, ... ,Xr) = f K(XI, ... , X r), which vanishes at y. This proves the lemma. To complete the proof of Proposition 3.1, it is sufficient to show that if Xl vanishes at a point x, so does K(X I, ... , X r). Let xl, ... ,xn be a coordinate system around x so that Xl = ~ifi (oj ox i ). We may take vector fields Y i and differentiable functions gi on M such that gi = fi and Y i = (oj ox i ) for i = 1, ... ,n in some neighborhood U of x. Then Xl = ~i gi Y i in U. By the lemma, K(X I , ••• , X r) = ~igi. K(Yi, X 2 , ••• , X r) in U. Since gi(X) = fi(X) = a for i = 1, ... , n, K(XI, ... , X r) vanishes at x.

QED.

Example 3.1.

A (positive definite) Riemannian metric on M is a covariant tensor field g of degree 2 which satisfies (1) g(X, X) > a for all X E X, and g(X, X) = a if and only if X = a and (2) g(Y, X) = g(X, Y) for all X, Y E X. In other words, g assigns an inner product in each tangent space Tx(M), x E M (cf. Example 2.2). In terms of a local coordinate system xl, ... ,xn , the comj ponents of g are given by gii = g( oj ox i, oj ox ). It has been customary to write ds 2 = ~ gij dx i dx j for g.

28

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Example 3.2. A differential form OJ of degree r is nothing but a covariant tensor field of degree r which is skew-symmetric: OJ(

X (l), ... ,X (f») 7T

7T

=

e( 7T) OJ (Xb

••• ,

X r),

where 7T is an arbitrary permutation of (1, 2, ... , r) and e( 7T) is its sign. For any covariant tensor K at x or any covariant tensor field K on M, we define the alternation A as follows: 1 (AK) (Xl' ... ,Xr ) = --, ~7T e(7T) • K(X (l), ... , X (f»)' 7T

r.

7T

where the summation is taken over all permutations 7T of (1, 2, ... , r). It is easy to verify that AK is skew-symmetric for any K and that K is skew-symmetric if and only if AK = K. If OJ and OJ' are differential forms of degree rand s respectively, then OJ ® OJ' is a covariant tensor field of degree r + s and OJ A OJ' = A( OJ ® OJ'). Example 3.3. The symmetrization S can be defined as follows. If K is a covariant tensor or tensor field of degree r, then

(SK)(Xb

· •• ,

1 X r ) = r! ~7T K(X (l), ... , !7T(f»)· 7T

For any K, SK is symmetric and SK = K if and only if K is symmetric. We now proceed to define the notion of Lie differentiation. Let ~;(M) be the set of tensor fields of type (r, s) defined on M and set X(M) = ~~=o~~(M). Then :!(M) is an algebra over the real number field R, the multiplication ® being defined pointwise, i.e., if K,L E ~(M) then (K ® L)x = K x ® Lx for all x E M. If cP is a transformation of M, its differential cp* gives a linear isomorphism of the tangent space Tcp-l(x) (M) onto the tangent space Tx(M). By Proposition 2.12, this linear isomorphism can be extended to an isomorphism ofthe tensor algebra T( cp-I(X)) onto the tensor algebra T(x), which we denote by ip. Given a tensor field K, we define a tensor field ipK by

(ipK) x

ip(Kcp-l(x) ' x E M. In this way, every transformation cp of M induces an algebra =

automorphism of :!:(M) which preserves type and commutes with contractions. Let X be a vector field on M and CPt a local I-parameter group of local transformations generated by X (cf. Proposition 1.5). We

I.

29

DIFFERENTIABLE MANIFOLDS

shall define the Lie derivative LxK of a tensor field K with respect to a vector /;,'ld X as follows. For the sake of simplicity, we assume that CPt is ;l global I-parameter group of transformations of M; the reader will have no difficulty in modifying the definition when X is not :"'omplete. For each t, rpt is an automorphism of the algebra X(M). For any tensor field K on M, we set lim ~ [K x - (¢itK ) x] • t-+o t The TIJapping Lx of X( M) into itself which sends K into L xK is called the Lie diJTerentiation with respect to X. We have

(LxK) x

PROPOSITION

-

3.2. Lie diJTerentiation Lx with respect to a vector

field X satisfies the following conditions: (a) Lx is a derivation ofX(M), that is, it is linear and satisfies Lx(K @ K') (b) (c) (d)

(e)

K @ (LxK') for all K, K' L x is type-preserving: L x (:t~ (M)) c X~ (M) ; L x commutes with every contraction of a tensor field; Lx f = Xf for every function f; Lx Y = [X, Y] for every vector field Y. =

(LxK) @ K'

€

X( M) ;

Proo£ It is clear that L x is linear. Let CPt be a local I-parameter group of local transformations generated by X. Then

Lx(K @ K') = lim ~ [K ® K' - fPt(K @ K')] t-+O t 1 - lim - [K @ K' - (fPtK ) @ (fPtK')] t-+o t

= lim ~ [K t-+O

t

K' - (¢itK) @ K']

@

lim~ [(¢i~) ® K' f-+O

t

(ipt K ) @ (¢it K ')]

( lim ~ [K - ()?itK)]) @ K' t-+O t

/1 ) lim (ip tK) @ ,- [K' - (ip tK') ] t-+O \ t = (LxK) @ K' + K @ (LxK').

30

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Since rf;t preserves type and commutes with contractions, so does Lx. Iffis a function on M, then

(Lxf) (x) Ifwe observe that f{Jt-t = f{J-t is a local1-parameter group of local ( X)f transformations generated by - X, we see that L xf = Xf. Finally (e) is a restatement of Proposition 1.9. QED. By a derivation of X(M), we shall mean a mapping of X(M) into itself which satisfies conditions (a), (b) and (c) of Proposition 3.2. Let S be a tensor field of type (1, 1). For each x € M, Sf[) is a linear endomorphism of the tangent space Tf[)(M). By Proposition 2.13, Sf[) can be uniquely extended to a derivation of the tensor algebra T(x) over Tf[)(M). For every tensor field K, define SK by (SK)f[) = Sf[)Kf[)' x € M. Then S is a derivation of X(M). We have 3.3. uniquely as follows: PROPOSITION

Every derivation D of X(M) can be decomposed D

=

Lx

S,

where X is a vector field and S is a tensor field of type (1, 1). Proo£ Since D is type-preserving, it maps ~(M) into itself and satisfies D(fg) = Df· g + f· Dg for 1,g € ~(M). It follows that there is a vector field X such that Df = Xf for every f € ~(M). Clearly, D L x is a derivation of X(M) which is zero on tJ(i1v1). We shall show that any derivation D which is zero on ~(M) is induced by a tensor field of type (1, 1). For any vector field Y, D Y is a vector field and, for any function 1, D(fY) Df· Y + f· DY f· DY since Df = 0 by assumption. By Proposition 3.1, there is a unique tensor field S of type (1, 1) such that DY = SY for every vector field Y. To show that D coincides with the derivation induced by S, it is sufficient to prove the following Two derivations D 1 and D 2 of X(M) coincide if thl!J coincide on ~(M) and X( M). Proo£ We first observe that a derivation D can be localized, that is, if a tensor field K vanishes on an open set U, then DK vanishes on U. In fact, for each x € U, letfbe a function such that f(x) = 0 and f = I outside U. Then K = f· K and hence DK = Df· K + f· DK. Since K and f vanish at x, so does DK. LEMMA.

I.

31

DIFFERENTIABLE MANIFOLDS

It follows that if two tensor fields K and K' coincide on an open set U, then DK and DK' coincide on U. Set D = D I - D 2 • Our problem is now to prove that if a • derivation D is zero on 6(M) and X(M), then it is zero on 1:(M). Let K be a tensor field of type (r, s) and x an arbitrary point of M. To show that DK vanishes at x, let Vbe a coordinate neighborhood of x with a local coordinate system Xl, ••• , xn and let

K =

~ K~l'

.. ~r X.21

J1 ••• J 8

t)(\... t)(\

'CJ

'CJ

X

2r

t)(\

'CJ

wi1

t)(\ • • • t)(\

'CJ

wi8

'CJ,

where X.2 = oj oxi and w j = dx j • We may extend K11i 1·.· i r X.2 and .•. J 8 ' w j to M and assume that the equality holds in a smaller neighborhood U of x. Since D can be localized, it suffices to show that D(K~l'" irX.21 'CJ t)(\ ••• t)(\ J1 •• • J 8 'CJ

X. r 2

t)(\

'CJ

wl1

t)(\ ••• t)(\

'CJ

'CJ

o.

wi8 ) =

But this will follow at once if we show that Dw = 0 for every I-form w on M. Let Y be any vector field and C: 1:i(M) ---+ tJ(M) the obvious contraction so that C( Y ® w) = w( Y) is a function (cf. Example 2.1). Then we have

o=

D(C(Y ® w)) =

=

C(DY ® w)

C(D(Y ® w))

+ C(Y ® Dw)

=

C(Y ® Dw)

Since this holds for every vector field Y, we have Dw

= =

(Dw)(Y).

o.

QED.

The set of all derivations of 1:(M) forms a Lie algebra over R (of infinite dimensions) with respect to the natural addition and multiplication and the bracket operation defined by [D, D']K = D(D'K) - D'(DK). From Proposition 2.13, it follows that the set of all tensor fields S of type (1, 1) forms a subalgebra of the Lie algebra of derivations of 1:(M). In the proof of Proposition 3.3, we showed that a derivation of 1:( M) is induced by a tensor field oftype (1, 1) ifand only ifit is zero on tJ(M). It follows immediately that if D is a derivation of 1:(M) and S is a tensor field of type (1, 1), then [D, S] is zero on tJ(M) and, hence, is induced by a tensor field of type (1, 1). In other words, the set of tensor fields of type (1, 1) is an ideal of the Lie algebra of derivations of 1:(M). On the other hand, the set of Lie differentiations Lx, X € X( M), forms a subalgebra of the Lie algebra of derivations of 1: (M). This follows from the following

32

FOUNDATIONS OF DIFFERENTIAL GEOMETRY ~

PROPOSITION

For any vector fields X and Y, we have

3.4.

L[x, Y]

[Lx, L y ].

=

Proof. By virtue of Lemma above, it is sufficient to show that [L x, L y ] has the same effect as L[x, Y] on tJ(M) and I(M). For f € tJ(M), we have

[Lx, L y Jf For Z

=

XYf- YXf = [X, Y]f

=

L[x, y]j.

I(M), we have

€

[Lx, Ly]Z = [X, [Y, Z]] - [Y, [X, Z]] = [[X, Y], Z] by the Jacobi identity.

QED.

3.5. Let K be a tensor field of type (1, r) which we interpret as in Proposition 3.1. For any vector field X, we have then PROPOSITION

(LxK)(Y 1 ,

••• ,

Yr)

=

[X, K(Y1 ,

••• ,

Yr)]

- l:i=l K(Y1 , · · · , [X, Y i ] , · · · , Yr). Proof.

We have

K(Y 1 ,

••• ,

Yr)

=

C1

•..

Cr(Y l @ ... @ Y r @ K),

where C1 , ••• , Cr are obvious contractions. Using conditions (a) and (c) of Proposition 3.2, we have, for any derivation D of :!(M),

D(K( Y 1 ,

••• ,

Yr)) = (DK) (Y 1 ,

••• ,

Y r)

+ l:i K(Y If D

=

1, · · · ,

DYi , · · · , Yr)·

Lx, then (e) of Proposition 3.2 implies Proposition 3.5. QED.

Generalizing Corollary 1.10, we obtain

3.6. Let CPt be a local I-parameter group of local transformations generated by a vector field X. For any tensor field K, we have PROPOSITION

Proof.

By definition,

LxK =

Iim~ l--O

t

[K - )OtK].

I.

33

DIFFERENTIABLE MANIFOLDS

Replacing K by fPsK, we obtain

•

Our problem is therefore to prove that fP s (L x K) = Lx (fP sK) , i.e., LxK=fPs-1oL}{ofPs(K) for all tensor fields K. It is a straightforward verification to see that fPs- 1 ° Lx ° fP sis a derivation of:.t(M). By Lemma in the proof of Proposition 3.3, it is sufficient to prove that Lx and fPfJ- 1 ° Lx ° fPs coincide on ~(M) and X(M). We already noted in the proofof Corollary 1.10 that they coincide on X( M). The fact that they coincide on ~(M) follows from the following formulas (cf. §l, Chapter I) :

cp* ((cp*X)f) - X( cp*f) , fP-y = cp*j,

which hold for any transformation cp of M and from (CPs) *X = X (cf. Corollary 1.8). QED. 3.7. O.

COROLLARY

only if LxK

=

A tensor field K is invariant by cptfOr every t if and

Let :Dr(M) be the space of differential forms of degree r defined on M, i.e., skew-symmetric covariant tensor fields of degree r. With respect to the exterior product, :D(M) - ~~=o :Dr(M) forms an algebra over R. A derivation (resp. skew-derivation) of :D(M) is a linear mapping D of :D(M) into itself which satisfies

D(OJ (resp.

=

A

OJ')

DOJ

A

=

OJ'

DOJ

A

OJ'

+ OJ A DOJ'

( -1) rOJ

A

DOJ'

for OJ

for OJ, OJ' €

€

:D(M)

:Dr(M) , OJ'

€

:D(Al)).

A derivation or a skew-derivation D of :D(M) is said to be of degree k if it maps :D r(M) into :D r+k ( M) for every r. The exterior differentiation d is a skew-derivation of degree 1. As a general result on derivations and skew-derivations of :D(M), we have

3.8. (a) If D and D' are derivations of degree k and k' respectively, then [D, D'] is a derivation of degree k + k'. (b) If D is a derivation of degree k and D' is a skew-derivation of degree k', then [D, D'] is a skew-derivation of degree k + k'. PROPOSITION

34

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

..

(c) If D and D' are skew-derivations of degree k and k' respectively, then DD' + D'D is a derivation of degree k + k'. (d) A derivation or a skew-derivation is completely determined by its effect on 1)O(M) = ~(M) and 1)l(M). Proof. The verification of (a), (b), and (c) is straightforward. The proof of (d) is similar to that of Lemma for Proposition 3.3.

QED. 3.9. For every vector field X, Lx is a derivation of degree 0 of 1) (M) which commutes with the extericr differentiation d. Conversely, every derivation of degree 0 of 1)(M) which commutes with d is equal to L x for some vector field X. Proof. Observe first that Lx commutes with the alternation A defined in Example 3.2. This follows immediately from the following formula: PROPOSITION

(LXw)(Y b

• •• ,

Yr )

=

X(w(Y 1 , -

~i

••• ,

Yr ))

w(Yr, ... , [X, Yi ],

••• ,

Yr ),

whose proof is the same as that of Proposition 3.5. Hence, L x(1)(M)) C 1)(M) and, for any w, w' E 1)(M), we have

Lx(w

A

w')

=

Lx(A(w @ w'))

= A(Lxw @ w')

=

A(Lx(w @ w'))

+ A(w

@

Lxw')

=LXWAW' +wALxw'. To prove that Lx commutes with d, we first observe that, for any transformation cP of M, rpw = (cp-l) *w and, hence, rp commutes with d. Let CPt be a local I-parameter group oflocal transformations generated by X. From rpt( dw) = d( fPtw) and the definition of Lxw it follows that Lx(dw) = d(Lxw) for every w E 1)(M). Conversely, let D be a derivation of degree 0 of 1)(M) which commutes with d. Since D maps 1)O(M) = ~(M) into itself, D is a derivation of ~(M) and there is a vector field X such that Df = Xf for every f E ~(M). Set D' = D - Lx. Then D' is a derivation of 1)(M) such that D'j = 0 for every f E ~(M). By virtue of (d) of Proposition 3.8, in order to prove D' = 0, it is sufficient to prove D'w = 0 for every I-form w. Just as in Lemma for Proposition 3.3, D' can be localized and it is sufficient to show that D'w = 0 when w is of the formfdg wherej,g E ~(M) (because

I.

35

DIFFERENTIABLE MANIFOLDS

OJ is locally of the form ~h dx i with respect to a local coordinate system xl, ... , xn ). Let OJ = fdg. From D'f = 0 and D' (dg) = d(D'g) = 0, we obtain D'(OJ) = (D'f) dg

+ f·

D'(dg) = O.

QED. For each vector field X, we define a skew-derivation (, x, called the interior product with respect to X, of degree -1 of!) (M) such that (a) (,xf = 0 for every f E !)O(M); (b) (, xOJ = OJ(X) for every OJ E !)l(M). By (d) of Proposition 3.8, such a skew-derivation is unique if it exists. To prove its existence, we consider, for each r, the contraction C: :!;(M) ---* :!~-l(M) associated with the pair (1, 1). Consider every r-form OJ as an element of :!~(M) and define (,xOJ = C( X (8) OJ). In other words,

((,xOJ)(Y I , · · · , Yr -

I)

=r·OJ(X,YI , · · · , Y r -

forYiEX(M).

I)

The verification that (, x thus defined is a skew-derivation of !)(M) is left to the reader; (,x(OJ A OJ') = {,xOJ A OJ' + (-It OJ A (,xOJ', where OJ E !)r(M) and OJ' E !)s(M), follows easily from the following formula:

(OJ

A

OJ') (YI , Y 2 , · · · , Yr + s ) (

1) , ~ e(j; k) w(Y;, ... , Y; )w'(Yk , r+s. 1 r 1

... ,

Yk

), 8

where the summation is taken over all possible partitions of (1, ... , r + s) into (j1} ... ,jr) and (k I , , k s) and s(j; k) stands for the sign of the permutation (1, , r + s) ---* (jI' ... ,jr, kI , • • • , k s). Since ({,iOJ) ( Y l' . • . , Yr - 2 ) = r(r - 1) . OJ (X, X, Y l' • . . , Y r - 2 ) = 0, we have

(,i

=

o.

As relations among d, Lx, and (, x, we have

= do (,x + (,x dfor every vectorfield X. (b) [Lx, (,y] = (,[X, Y] for any vector fields X and Y. Proof. By (c) of Proposition 3.8, d x + (, x d is a derivation PROPOSITION

3.10.

(a) Lx

0

0

(,

0

36

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

of degree O. It commutes with d because d2 = O. By Proposition 3.9, it is equal to the Lie differentiation with respect to some vector field. To prove that it is actually equal to Lx, we have only to show that Lxf = (d "x + tx d)ffor every function): But this is obvious since Lxf - Xfand (d tx + "x d)f = "x(dj) = (df) (X) = X): To prove the second assertion (b), observe first that [Lx, t y ] is a skew-derivation of degree -1 and that both [Lx, ty] and t[X, Yl are zero on ~(M). By (d) of Proposition 3.8, it is sufficient to show that they have the same effect on every I-form w. As we noted in the proof of Proposition 3.9, we have (Lxw)(Y) = X(w(Y)) w([X, Y]) which can be proved in the same way as Proposition 3.5. Hence, 0

0

0

[Lx, ty]w

Lx(w(Y))

=

ty(Lyw)

0

=

X(w(Y)) - (Lxw)(Y)

= w([ X, Y]) = t[X, y,(1). QED. As an application of Proposition 3.10 we shall prove

If w is an r-form,

3.11.

PROPOSITION

then

(dw) (Xo, Xl' ... , X r ) ~

1

r + 1 ~i=o (-I)iXz (w(Xo,···, Xi"·" X r )) 1

+ r+

1 ~o

<j~r(

l)i+i W ([Xi, Xi]' Xo,···, Xi"'" Xi'··" X r ), 1 and 2

where the symbol means that the term is omitted. (The cases r are particularly useful.) If w is a I-form, then A

(dw) (X, Y) - !{X(w(Y)) - Y(w(X))

w([X, Y])}, X, Y € X (.A1) .

If w

is a 2-form, then

(dw) (X, Y, Z)

=

!{X(w(Y, Z))

+ Y(w(Z, X)) + Z(w(X, Y))

- w([X, Y], Z) - w([Y, Z], X)

w([Z, X], Y)}, ~Y,Z

€

X(M).

Proof. The proof is by induction on r. If r = 0, then w is a Xow, which shows that the formula above function and dw(Xo)

37

DIFFERENTIABLE MANIFOLDS

1.

is true for r = O. Assume that the formula is true for r - 1. Let w be an r-form and, to simplify the notation, set X = X o• By (a) of Proposition 3.10,

(r

+ ~)

dw(X, X b

••• ,

X r ) = (t x

(LxW) (Xl' ... ,Xr )

=

dw)(XI ,

0

-

••• ,

Xr)

(d txW) (Xl' ... , X r ). 0

As we noted in the proof of Proposition 3.9,

(Lxw) (Xl' ... ,Xr ) = X(W(XI ,

Since

t XW

~i =

0

=

1

1

"

1

X i (t X W(Xl' ... , Xi' ... , X r )) ~

"+ °

+-;'~I~i<j~r(-I)~

1

"

=-~i=I(-I)~-

1

r

1

w(Xl' . . . , [X, Xi], . . . , X r ).

-~i=1(-I)~r X

-;.

X r ))

is an (r - 1)-form, we have, by induction assumption,

(d txw)(Xl' ... , X r )

-

1

••• ,

~l ~i <j ~r

~

J(tx w )([Xi' Xj], Xl' ... ' Xi' ... ' X j, ... , X r) ~

Xi(W(X, Xl'···' Xi'···' X r )) 0+"

(-1) ~ X

J

w([Xi, Xj], X, Xl' ... , Xi' ... ,Xj, ... , X r ).

Our Proposition follows immediately from these three formulas.

QED. Remark. Formulas in Proposition 3.11 are valid also for vector-space valued forms. Various derivations allow us to construct new tensor fields from a given tensor field. We shall conclude this section by giving another way of constructing new tensor fields. PROPOSITION

SeX, Y)

=

3.12.

Let A and B tensor fields

[AX, BY]

of type

(1, 1). Set

+ [BX, AY] + AB[X, Y] + BA[X, Y]

-A [X, BY] - A [BX, Y] - B[X, A:f] - B[AX, Y], X,Y € :£(M).

38

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Then the mapping S: :£(M) X :£(M) -4- :£(M) is a tensor field of type (1, 2) and S(X, Y) = -S(Y, X). Proof. By a straightforward calculation, we see that S is a bilinear mapping of the ~(M)-module :£(M) X :£(M) into the ~(M)-module :£(M). By Proposition 3.1, S is a tensor field of type (1,2). The verification of S(X, Y) = -S(Y, X) is easy.

QED. We call S the torsion of A and B. The construction of S was discovered by Nij enhuis [1].

4. Lie groups A Lie group G is a group which is at the same time a differentiable manifold such that the group operation (a, b) E G X G -4- ab- 1 E G is a differentiable mapping of G X G into G. Since G is locally connected, the connected component of the identity, denoted by GO, is an open subgroup of G. GO is generated by any neighborhood of the identity e. In particular, it is the sum of at most countably many compact sets and satisfies the second axiom of countability. I t follows that G satisfies the second axiom of countability if and only if the factor group GIGo consists of at most countably many elements. We denote by La (resp. R a ) the left (resp. right) translation of G by an element a E G: Lax = ax (resp. Rax = xa) for every x E G. For a E G, ad a is the inner automorphism of G defined by (ad a)x = axa- 1 for every x E G. A vector field X on G is called left invariant (resp. right invariant) if it is invariant by all left translations La (resp. right translations Ra ), a E G. A left or right invariant vector field is always differentiable. We define the Lie algebra 9 of G to be the set of all left invariant vector fields on G with the usual addition, scalar multiplication and bracket operation. As a vector space, 9 is isomorphic with the tangent space Te(G) at the identity, the isomorphism being given by the mapping which sends X E 9 into X e , the value of X at e. Thus 9 is a Lie subalgebra of dimension n (n = dim G) of the Lie algebra of vector fields :£( G). Every A E 9 generates a (global) I-parameter group of transformations ofG. Indeed, if
I.

DIFFERENTIABLE MANIFOLDS

39

transformations generated by A and f(Jte is defined for It I < c, then f(Jta can be defined for It I < c for every a E G and is equal to La(f(Jte) as f(Jt commutes with every La by Corollary 1.8. Since f(Jta is defined for It I < s for every a E G, f(Jta is defined for It I < 00 for every a E G. Set at = f(Jte. Then at+ s = ata s for all t,s E R. We call at the I-parameter subgroup olG generated by A. Another characterization of at is that it is a unique curve in G such that its tangent vector at at at is equal to LatA e and that ao = e. In other words, it is a unique solution of the differential equation at-lat = A e with initial condition ao = e. Denote al = f(Jle by exp A. It follows that exp tA = at for all t. The mapping A ~ exp A of g into G is called the exponential mapping. Example 4.1. GL(n; R) and gl(n; R). Let GL(n; R) be the group of all real n X n non-singular matrices A = (al) (the matrix whose i-th row and j-th column entry is aJ); the multiplication is given by

(AB)j = ~k=l a1 bJ

for A = (aJ)

and

B = (bJ).

GL(n; R) can be considered as an open subset and, hence, as an 2

open submanifold of Rn • With respect to this differentiable structure, GL(n; R) is a Lie group. Its identity component consists of matrices of positive determinant. The set gl(n; R) of all n X n real matrices forms an n 2 -dimensional Lie algebra with bracket operation defined by [A, B] = AB - BA. It is known that the Lie algebra of GL(n; R) can be identified with gl(n; R) and the exponential mapping gl(n; R) ~ GL(n; R) coincides with the usual exponential mapping exp A = ~k=O Ak/k! Example 4.2. D(n) and o(n). The group D(n) of all n X n orthogonal matrices is a compact Lie group. Its identity component, consisting of elements of determinant 1, is denoted by SD(n). The Lie algebra o(n) of all skew-symmetric n X n matrices can be identified with the Lie algebra of D(n) and the exponential mapping o(n) ~ D(n) is the usual one. The dimension of D(n) is equal to n(n - 1) /2. By a Lie subgroup of a Lie group G, we shall mean a subgroup H which is at the same time a submanifold of G such that H itself is a Lie group with respect to this differentiable structure. A left invariant vector field on His tletermined by its value at e and this tangent vector at e of H determines a left invariant vector field on

40

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

G. It follows that the Lie algebra 1) of H can be identified with a subalgebra of g. Conversely, every subalgebra 1) of 9 is the Lie algebra of a unique connected Lie subgroup H of G. This is proved roughly as follows. To each, point x of G, we assign the space of all Ax, A E 1). Then this is an involutive distribution and the maximal integral submanifold through e of this distribution is the desired group H (cf. Chevalley [1; p. 109, Theorem 1]). Thus there is a 1: 1 correspondence between connected Lie subgroups of G and Lie subalgebras of the Lie algebra g. We make a few remarks about nonconnected Lie subgroups. Let H be an arbitrary subgroup of a Lie group G. By providing H with the discrete topology, we may regard H as a O-dimensional Lie subgroup of G. This also means that a subgroup H of G can be regarded as a Lie subgroup of G possibly in many different ways (that is, with respect to different differentiable structures). To remedy this situation, we impose the condition that HIHO, where HO is the identity component of H with respect to its own topology, is countable, or in other words, H satisfies the second axiom ofcountability. (A subgroup, with a discrete topology, of G is a Lie subgroup only if it is countable.) Under this condition, we have the uniqueness of Lie subgroup structure in the following sense. Let H be a subgroup of a Lie group G. Assume that H has two differentiable structures, denoted by HI and H 2 , so that it is a Lie subgroup of G. Ifboth HI and H 2 satisfy the second axiom of countability, the identity mapping of H onto itself is a diffeomorphism of HI onto H 2 • Consider the identity mapping f: HI ~ H 2 • Since the identity component of H 2 is a maximal integral submanifold of the distribution defined by the Lie algebra of H 2 ,f: HI ~ H 2 is differentiable by Proposition 1.3. Similary f- I : H 2 ~ HI is differentiable. Every automorphism ffJ of a Lie group G induces an automorphism ffJ* of its Lie algebra g; in fact, if A E g, ffJ*A is again a left invariant vector field and ffJ* [A, B] = [ffJ*A, ffJ*B] for A,B E g. In particular, for every a E G, ad a which maps x into axa- I induces an automorphism of g, denoted also by ad a. The representation a ~ ad a of G is called the adjoint representation of G in g. For every a E G and A E g, we have (ad a)A = (Ra-l) *A, because axa- I = LaRa-1X = Ra-1Lax and A is left invariant. Let A,B E 9 and ffJt the I-parameter group of transformations of G generated by A. Set at = exp tA = ffJt(e). Then ffJt(x) = xa t for

X

E

41

DIFFERENTIABLE MANIFOLDS

1.

G. By Proposition 1.9, we have

[B, A] = lim! [(
t

t-+o

t

= lim ~ [ad (at-1)B - B]. t-+O

t

It follows that if H is an invariant Lie subgroup of G, its Lie algebra 1) is an ideal of g, that is, A E 9 and B E 1) imply [B, A] E 1). Conversely, the connected Lie subgroup H generated by an ideal 1) of 9 is an invariant subgroup of G. A differential form OJ on G is called left invariant if (La) *OJ = OJ for every a E G. The vector space g* formed by all left invariant I-forms is the dual space of the Lie algebra g: if A E 9 and OJ E g*, then the function OJ(A) is constant on G. If OJ is a left invariant form, then so is dOJ, because the exterior differentiation commutes with
dOJ(A, B) = -!OJ([A, B])

for OJ

E

g*

and

A,B E g.

The canonical I-form 0 on G is the left invariant g-valued I-form uniquely determined by

O(A) Let E 1 ,

••• ,

=

A

for A

E

g.

E r be a basis for 9 and set

o=

l:i=l OiEi . Then 0\ ... , Or form a basis for the space of left invariant real I-forms on G. We set [E j , E k ] = l:i=l CJkEi' where the Cjk'S are called the structure constants of 9 with respect to the basis E 1, ••• , E r • It can be easily verified that the equation of Maurer-Cartan is given by dO i = -!l:j,k=l Cjk Oj A Ok, i = 1, ... , r. We now consider Lie transformation groups. We say that a Lie group G is a Lie transformation group on a manifold M or that G acts (differentiably) on M if the following conditions are satisfied: (1) Every element a of G induces a transformation of M, denoted by x ~ xa where x E M;

42

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(2) (a, x) E G X M ---+ xa E M is a differentiable mapping; (3) x(ab) = (xa)b for all a,b E G and x E M. We also write Rax for xa and say that G acts on M on the right. If we write ax and assume (ab)x = a(bx) instead of (3), we say that G acts on M on the left and we use the notation Lax for ax also. Note that R ab = R b R a and Lab = La Lb. From (3) and from the fact that each R a or La is one-to-one on M, it follows that R e and L e are the identity transformation of M. We say that G acts effectively (resp. freely) on M if Rax = x for all x E M (resp. for some x E M) implies that a = e. If G acts on M on the right, we assign to each element A Ega vector field A* on M as follows. The action of the I-parameter subgroup at = exp tA on M induces a vector field on M, which will be denoted by A * (cf. §I) . 0

0

Let a Lie group G act on M on the right. The mapping a: 9 ---+ X( M) which sends A into A * is a Lie algebra homomorphism. If G acts effectively on M, then a is an isomorphism of 9 into X(M). IfG actsfreely on M, then,for each non-zero A E g, a(A) never vanishes on M. Proof. First we observe that a can be defined also in the following manner. For every x E M, let ax be the mapping a E G ---+ xa E M. Then (a x) *A e = (aA) x. It follows that a is a linear mapping of 9 into X( M). To prove that a commutes with the bracket, let A,B E g, A* = aA, B* = aB and at = exp tAo By Proposition 1.9, we have PROPOSITION

4.1.

[A*, B*]

lim It [B* - RatB*].

=

t-O

From the fact that Rat axat-l(C) = xat-1ca t for c EG, we obtain (denoting the differential of a mapping by the same letter) 0

(RatB*)x

=

Rat axat-lBe = ax(ad (at-1)B e) 0

and hence

[A*, B*]

=

lim! [axBe - ax(ad (at-1)B e)] t-O

t

= ax (lim! [Be - ad (at-1)B e]) t-O t

I.

DIFFERENTIABLE MANIFOLDS

43

by virtue of the formula for [A, B] in 9 in terms of ad G. We have thus proved that a is a homomorphisn1 of the Lie algebra 9 into the Lie algebra X(M). Suppose that aA = 0 everywhere on M. This means that the I-parameter group of transformations Rat is trivial, that is, Rat is the identity transformation of M for every t. If G is effective on M, this implies that at = e for every t and hence A = O. To prove the last assertion of our proposition, assume aA vanishes at some point x of M. Then Rat leaves x fixed for every t. If G acts freely on M, this implies that at = e for every t and hence A = O. QED. Although we defined a Lie group as a group which is a differentiable manifold such that the group operation (a, b) ~ ab- 1 is differentiable, we may replace differentiability by real analyticity without loss of generality for the following reason. The exponential mapping is one-to-one near the origin of g, that is, there is an open neighborhood N of 0 in 9 such that exp is a diffeomorphism of N onto an open neighborhood U of e in G (cf. Chevalley [1; p. 118] or Pontrjagin [1; §39]). Consider the atlas of G which consists of charts (Ua, CPa), a E G, where CPa: Ua ~ N is the inverse mapping of R a exp: N ~ Ua. (Here, Ua means R a ( U) and N is considered as an open set ofRn by an identification of 9 with Rn.) With respect to this atlas, G is a real analytic manifold and the group operation (a, b) ~ ab-1 is real analytic (cf. Pontrjagin [1; p. 257]). We shall need later the following 0

4.2. Let G be a Lie group and H a closed subgroup of G. Then the quotient space GjH admits a structure of real analytic manifold in such a way that the action of G on GjH is real analytic, that is, the mapping G X GjH ~ GjH which maps (a, bH) into abH is real analytic. In particular, the projection G ~ GjH is real analytic. For the proof, see Chevalley [1; pp. 109-111]. PROPOSITION

There is another important class of quotient spaces. Let G be an abstract group acting on a topological space M on the right as a group of homeomorphisms. The action of G is called properly discontinuous if it satisfies the following conditions: (1) If two points x and x' of M are not congruent modulo G (i.e., Rax x' for every a E G), then x and x' have neighborhoods U and U ' respectively such that R a ( U) n U ' is empty for all a E G·,

"*

44

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(2) For each x E G, the isotropy group Gx = {a E G; Rax = x} is finite; (3) Each x E M has a neighborhood U, stable by Gx , such that U n R a (U) is empty for every a E G not contained in Gx. Condition (1) implies that the quotient space MIG is Hausdorff. If the action of G is free, then condition (2) is automatically satisfied. 4.3. 'Let G be a properly discontinuous group of differentiable (resp. real analytic) transformations acting freely on a differentiable (resp. real analytic) manifold M. Then the quotient space MIG has a structure of differentiable (resp. real analytic) manifold such that the projection 1T: M ~ MIG is differentiable (resp. real analytic). Proof. Condition (3) implies that every point of MIG has a neighborhood V such that 1T is a homeomorphism ofeach connected component of 1T- 1 (V) onto V. Let U be a connected component of 1T- 1 ( V). Choosing V sufficiently small, we may assume that there is an admissible chart (U, cp), where cp: U ~ Rn, for the manifold M. Introduce a differentiable (resp. real analytic) structure in MIG by taking (V, "P), where "P is the composite of 1T- 1 : V ~ U and cp, as an admissible chart. The verification of details is left to the reader. QED. PROPOSITION

Remark. A complex analytic analogue of Proposition 4.3 can be proved in the same way. To give useful criteria for properly discontinuous groups, we define a weaker notion of discontinuous groups. The action of an abstract group G on a topological space M is called discontinuous if, for every x E }lVf and every sequence of elements {an} of G (where an are all mutually distinct), the sequence {Ranx} does not converge to a point in M. 4.4. Every discontinuous group G of isometries of a metric space M is properly discontinuous. Proof. Observe first that, for each x E M, the orbit xG = {Rax; a E G} is closed in M. Given a point x' outside the orbit xG, let r be a positive number such that 2r is less than the distance between x' and the orbit xG. Let U and U' be the open spheres of radius r and centers x and x' respectively. Then R a ( U) n U ' is empty for all a E G, thus proving condition (1). Condition (2) PROPOSITION

1.

DIFFERENTIABLE MANIFOLDS

45

is always satisfied by a discontinuous action. To prove (3), for each x E M, let r be a positive number such that 2r is less than the distance between x and the closed set xG - {x}. It suffices to take the open sphere of radius r and center x as U. QED. Let G be a topological group and H a closed subgroup of G. Then G, hence, any subgroup of G acts on the quotient space GjH on the left. 4.5. Let G be a topological group and H a compact subgroup of G. Then the action of every discrete subgroup D of G on GjH (on the left) is discontinuous. Proof. Assuming that the action of D is not discontinuous, let x andy be points ofGjH and {dn} a sequence of distinct elements ofD such that dnx converges toy. Letp: G ~ GjHbe the projection and write x = p(a) and y = p(b) where a,b € G. Let V be a neighborhood of the identity e of G such that bVVV-l V-1b- 1 contains no element of D other than e. Since p( bV) is a neighborhood ofy, there is an integer N such that dnx E p(b V) for all n > N. Hence, dnaH = p-l( dnx) c p-l(P(bV)) = bVH for n > N. For each n > N, there exist Vn E V and hn E H such that dna = bvnhn. Since H is compact, we may assume (by taking a subsequence if necessary) that hn converges to an element h E H and hence hn = unh for n > N, where Un E V. We have therefore dn = bvnunha- 1 for n > N. Consequently, d'/,dj- 1 is in bVVV-l V-1b- 1 if i,} > N. This means d'/, = d j if i,} > N, contradicting our assumption. QED. PROPOSITION

In applying the theory of Lie transformation groups to differential geometry, it is important to show that a certain given group of differentiable transformations of a manifold can be made into a Lie transformation group by introducing a suitable differentiable structure in it. For the proof of the following theorem, we refer the reader to Montgomery-Zippin [1; p. 208 and p. 212]. 4.6 Let G be a locally compact effective transformation group of a connected manifold M of class C\ 1 < k < W, and let each transformation of G be of class Cl. Then G is a Lie group and the mapping G X M ~ M is of class Ck. THEOREM

We shall prove the following result, essentially due to van Dantzig and van der Waerden [1].

46

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

4.7. The group G of isometries of a connected, locally com.. pact metric space M is locally compact with respect to the compact-open topology. Proof. We recall that the compact-open topology of G is defined as follows. For any finite number of pairs (K i , Ui) of compact subsets K i and open subsets U i of M, let W = W(Ki) ... , K s ; U1 , ••• , Us) = {gJ € G; gJ(Ki ) C Ui for i = 1, ... ,s}. Then the sets W of this form are taken as a base for the open sets of G. Since M is regular and locally compact, the group multiplication G X G -+ G and the group action G x M -+ M are continuous (cf. Steenrod [1; p. 19]). The continuity of the mapping G -+ G which sends gJ into gJ-l will be proved using the assumption in Theorem 4.7, although it follows from a weaker assumption (cf. Arens [1]). Every connected, locally compact metric space satisfies the second axiom of countability (see Appendix 2). Since M is locally compact and satisfies the second axiom of countability, G satisfies the second axiom of countability. Thisjustifies the use ofsequences in proving the local compactness of G (cf. Kelley [1; p. 138]). The proof is divided into several lemmas. THEOREM

1. Let a € M and let 13 > 0 be such that U(a; B) = {x € M; dCa, x) < B} has compact closure (where d is the distance). Denote by Va the open neighborhood U(a; 13/4) of a. Let gJn be a sequence of isometries such that gJn(b) converges for some point b € Va. Then there exist a compact set K and an integer N such that gJn(Va) C K for every n > N. Proof. Choose N such that n > N implies d(gJn(b), gJn(b)) < 13/4. If x E Va and n > N, then we have LEMMA

d(gJn(x) , gJN(a))

d(gJn(x), gJn(b))

d(gJn(b), gJN(b))

+ d(gJN(b), gJN(a)) = d(x, b)

d(gJn(b), gJN(b))

+ deb, a)

< 13,

using the fact that gJn and gJN are isometries. This means that gJn(Va) is contained in U(gJN(a); B). But U(gJN(a); B) = gJN(U(a; B)) since gJN is an isometry. Thus the closure K of U(gJN(a); B) = gJN(U(a; B)) is compact and gJn(Va) c K for n > N. LEMMA

2. In the notation of Lemma 1, assume again that gJn(b)

I.

47

DIFFERENTIABLE MANIFOLDS

converges for some b E Va. Then there is a subsequence cpnk of cp n such that cpnk(X) converges for each x EVa. Proof. Let {b n } be a countable set which is dense in Va. (Such a {b n } exists since .tv! is separable.) By Lemma 1, there is an N such that CPn( Va) is in K for n > N. In particular, CPn(b I ) is in K. Choose a subsequence cpI,k such that cpI,k( bI ) converges. From this subsequence, we choose a subsequence cp2,k such that cp2,k(b 2) converges, and so on. The diagonal sequence cplc,k(b n) converges for every n = 1, 2, . . .. To prove that cpk,k(X) converges for every x E Va, we change the notation and may assume that CPn(b i ) converges for each i = 1, 2, .... Let x E Va and b > o. Choose bi such that d(x, bi) < b/4. There is an N I such that d(CPn(b i ), CPm(b i )) < b/4 for n,m > N I. Then we have d(CPn(x), CPm(x)) < d(CPn(x), CPn(b i )) =

2d(x, bi)

+ d(CPn(b i), CPm(b i )) + d(CPm(b i), CPm(x))

+ d(CPn(b i), CPm(b i))

<

b.

Thus CPn(x) is a Cauchy sequence. On the other hand, Lemma 1 says that CPn(x) is in a compact set K for all n > N. Thus CPn(x) converges.

3. Let CPn be a sequence of isometries such that CPn(a) converges for some point a E M. Then there is a subsequence CPnk such that CPnk(X) convergesfor each x E M. (The connectedness of M is essentially used here.) Proof. For each x E M, let Va: = U(x; e/4) such that U(x; e) has compact closure (this e may vary from point to point, but we choose one such e for each x). We define a chain as a finite sequence of open sets Vi such that (1) each Vi is of the form Va: for some x; (2) VI contains a; (3) Vi and Vi+ I have a common point. We assert that every point y of M is in the last term of some chain. In fact, it is easy to see that the set of such points y is open and closed. .tv! being connected, the set coincides with M. This being said, choose a countabfe set {bi} which is dense in M. For bI , let VI' V2, ... , Vs be a chain with bi E Vs. By assumption CPn(a) converges. By Lemma 2, we may choose a subsequence (which we may still denote by CPn by changing the notation) such that CPn(x) converges for each x E VI. Since VI n V2 is non-empty, Lemma 2 allows us to choose a subsequence which converges for LEMMA

48

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

each x E V2 , and so on. Thus the original sequence cP n has a subsequence CPI,k such that CPI,k( bI ) converges. From this subsequence, we may further choose a subsequence CP2,k such that CP2,k( b2) converges. As in the proof of Lemma 2, we obtain the diagonal subsequence CPk,k such that CPk,k(b n) converges for each n. Denote this diagonal subsequence by CPn' by changing the notation. Thus cP n ( bi) converges for each bi· We now want to show that CPn(x) converges for each x E M. In V:v, there is some bi so that there exist an N and a compact set K such that CPn( VaJ c K for n > N by Lemma 1. Proceeding as in the second half of the prooffor Lemma 2, we can prove that CPn(x) is a Cauchy sequence. Since CPn(x) E K for n > N, we conclude that CPn(x) converges.

4. Assume that CPn is a sequence of isometries such that CPn(x) converges for each x E M. Define cp(x) = lim CPn(x) for each x. Then . . n-oo cP zs an zsometry. Proof. Clearly, d( cp(x), cp(y)) = d(x,y) for any XJ' E M. For any a EM, let a' = cp(a). From d(cp:;! cp(a), a) = d(cp(a), CPn(a)), it follows that cp:;I(a') converges to a. By Lemma 3, there is a subsequence CPnk such that cp~I(y) converges for every y E M. Define a mapping 1p by 1p(y) = lim cp:;I(y). Then 1p preserves k-oo k distance, that is, d(1p(x), 1p(y)) = d(x,y) for any x,y E M. From LEMMA

0

d(1p (cP (x) ), x) = d(lim lim d(cP:;kI ( cP (x) ), x) k-oo cP:;kI ( cP (x) ), x) = k-oo = k-oo lim d( cp(x), CPn k(x)) = d( cp(x), cp(x)) = 0, it follows that 1p(cp(x)) = x for each x E M. This means that cP maps M onto M. Since 1p preserves distance and maps M onto M, 1p-I exists and is obviously equal to cpo Thus cP is an isometry.

5. Let CPn be a sequence of isometries and cP an isometry. if cP n(x) converges to cP (x) for every x E M, then the convergence is uniform on every compact subset K of M. Proof. Let d > 0 be given. For each point a E K, choose an integer N a such that n > N a implies d(CPn(a), cp(a)) < d/4. Let W a = U(a; d/4). Then for any x E W a and n > N a , we have LEMMA

d(CPn(x), cp(x)) ~ d(CPn(x), CPn(a))

< 2d(x, a)

+

+ d/4 <

d(CPn(a), cp(a)) d.

+ d(cp(a),

cp(x))

I.

49

DIFFERENTIABLE MANIFOLDS

Now K can be covered by a finite number of W/s, say Wi = Wao , i = 1, ... , s. It follows that if n > maxi {Na }, then ~

~

d( CPn(x), cp(x)) < b

for each x E K.

6. If CPn(x) converges to cp(x) as in Lemma 5, then cp;;I(X) converges to cp-l(X) for every x E M. Proof. For any x E M, lety = cp-l(X). Then LEMMA

d(cp;;I(X), cp-l(X)) = d(cp;;l(cp(y)),y) = d(cp(y), CPn(Y))

--+

O.

We shall now complete the proof of Theorem 4.7. First, observe that CPn --+ cP with respect to the compact-open topology is equivalent to the uniform convergence of CPn to cP on every compact subset of M. If CPn --+ cP in G (with respect to the compact-open topology), then Lemma 6 implies that cp;;I(X) --+ cp-l(X) for every x E M, and the convergence is uniform on every compact subset by Lemma 5. Thus cp;;1 --+ cp-l in G. This means that the mapping G --+ G which maps cP into cp-l is continuous. To prove that G is locally compact, let a E M and U an open neighborhood of a with compact closure. We shall show that the neighborhood W = W(a; U) = {cp E G; cp(a) E U} of the identity of G has compact closure. Let CPn be a sequence of elements in W. Since CPn(a) is contained in the compact set U, closure of U, we can choose, by Lemma 3, a subsequence CPnk such that CPnk(X) converges for every x E M. The mapping cP defined by cp(x) = lim CPnk(X) is an isometry of M by Lemma 4. By Lemma 5, CPnk --+ cP uniformly on every compact subset of M, that is, CPnk --+ cP in G, proving that W has compact closure. QED. 4.8. Under the assumption of Theorem 4.7, the isotropy subgroup Ga = {cp E G; cP (a) = a} of G at a is compact for every a E M. Proof. Let CPn be a sequence of elements of Ga. Since CPn(a) = a for every n, there is a subsequence CPnk which converges to an element cP of Ga by Lemmas 3, 4, and 5. QED. COROLLARY

4.9. If M. is a locally compact metric space with a finite number of connected components, the group G of isometries of M is locally compact with respect to the compact-open topology. Proof. Decompose M into its connected components M i , M = =1 Mi. Choose a point ai in each M i and an open COROLLARY

U:

50

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

neighborhood Ui of ai in M i with compact closure. Then W(a b · · · , as; U b . · · , Us) = {cp € G; cp(a i ) € Uifori = 1, ... , s} is a neighborhood of the identity of G with compact closure.

QED. 4.10. If M is compact in addition to the assumption of Corollary 4.9, then G is compact. Proof. Let G* = {cp € G; cp(Mi ) = M i for i = 1, ... , s}. Then G* is a subgroup of G of finite index. In the proof of Corollary 4.9, let Ui = Mi. Then G* is compact. Hence, G is compact. QED. COROLLARY

5. Fibre bundles Let M be a manifold and G a Lie group. A (differentiable) principal fibre bundle over M with group G consists of a manifold P and an action of G on P satisfying the following conditions: (1) G acts freely on P on the right: (u, a) € P X G ~ ua =

Rau € P; (2) M is the quotient space of P by the equivalence relation induced by G, M = PIG, and the canonical prqjection 7T: P ~ M is differentiable; (3) P is locally trivial, that is, every point x of M has a neighborhood U such that 7T- 1 ( U) is isomorphic with U X G in the sense that there is a diffeomorphism "P: 7T- 1 ( U) ~ U X G such that "P(u) = (7T(U), cp(u)) where cp is a mapping of 7T- 1(U) into G satisfying cp(ua) = (cp(u))a for all u € 7T- 1 ( U) and a € G. A principal fibre bundle will be denoted by P(M, G, 7T), P(M, G) or simply P. We call P the total space or the bundle space, M the base space, G the structure group and 7T the projection. For each x € M, 7T- 1 (X) is a closed submanifold of P, called the fibre over x. If u is a point of 7T- 1(X), then 7T- 1(X) is the set of points ua, a € G, and is called the fibre through u. Every fibre is diffeomorphic to G. Given a Lie group G and a manifold M, G acts freely on P = M X G on the right as follows. For each b € G, R b maps (x, a) € M X G into (x, ab) € M X G. The principal fibre bundle P(M, G) thus obtained is called trivial. From local triviality of P( M, G) we see that if W is a submanifold of M then 7T- 1( W) (W, G) is a principal fibre bundle.

I.

DIFFERENTIABLE MANIFOLDS

51

We call it the portion of P over Wor the restriction of P to Wand denote it by W. Given a principal fibre bundle P( M, G), the action of G on P induces a homomorphism G of the Lie algebra 9 of G into the Lie algebra X(P) of vector fields on P by Proposition 4.1. For each A E g, A* = G(A) is called the fundamental vector field corresponding to A. Since the action of G sends each fibre into itself, Au * is tangent to the fibre at each u E P. As G acts freely on P, A * never vanishes on P (if A =1= 0) by Proposition 4.1. The dimension of each fibre being equal to that of g, the mapping A .....-+ (A*)u of 9 into T u (P) is a linear isomorphism of 9 onto the tangent space at u of the fibre through u. We prove

PI

Let A * be the fundamental vector field corresponding to A E g. For each a E G, (Ra)*A* is the fundamental vector field corresponding to (ad (a-I)) A E g. Proof. Since A* is induced by the I-parameter group of transformations Rat where at = exp tA, the vector field (R a ) *A * is induced by the I-parameter group of transformations RaRatRa-l = Ra-lata by Proposition 1.7. Our assertion follows from the fact that a-lata is the I-parameter group generated by (ad (a-I))A E g. QED. PROPOSITION

5.1.

The concept of fundamental vector fields will prove to be useful in the theory of connections. In order to relate our intrinsic definition of a principal fibre bundle to the definition and the construction by means of an open covering, we need the concept of transition functions. By (3) for a principal fibre bundle P(M, G), it is possible to choose an open covering {Uex } of M, each 1T-I ( UJ provided with a diffeomorphism U""'-+ (1T(U), tpex(u)) of7T- I(Uex ) onto Uex X G such that tpex(ua) (tpex(u))a. If UE1T- I(Uex n Up), then tpp(ua)(tpex(ua))-l = tpp(u)(tpex(u))-l, which shows that tpp(u) (tpex(U))-1 depends only on 1T(U) not on u. We can define a mapping 1jJpex: Uex n Up .....-+ G by 1jJpex( 7T( u)) - tpp(u) (tpex(u)) -1. The family of mappings 1jJpex are called transition functions of the bundle P( M, G) corresponding to the open covering {Uex } of M. It is easy to verify that

(*)

1jJyex(x) = 1jJyp(x) . 1jJpex(x)

Conversely, we have

for x

E

Uex n Up n Uy.

52

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Let M be a manifold, {Uct } an open covering of M and G a Lie group. Given a mapping "PPct: Uct n U{3 ~ G for every nonempty Uct n U{3' in such a way that the relations (*) are satisfied, we can construct a (differentiable) principal fibre bundle P( M, G) with transition functions "P{3ct. Proof. We first observe that the relations (*) imply "Pctct(x) = e for every x E Uct and "Pct{3(x)"Ppct(x) = e for every x E Uct n Up. Let Xct = Uct X G for each index rx and let X = UctXct be the topological sunl of X ct ; each element of X is a triple (rx, x, a) where rx is some index, x E Uct and a E G. Since each Xct is a differentiable manifold and X is a disjoint union of Xct' X is a differentiable manifold in a natural way. We introduce an equivalence relation p in X as follows. We say that (rx, x, a) E {rx} X Xct is equivalent to ({J,y, b) E {{J} X X{3 if and only if x = y E U/X n Up and b = "P{3ct(x)a. We remark that (rx, x, a) and (rx,y, b) are equivalent if and only if x = y and a = b. Let P be the quotient space of X by this equivalence relation p. We first show that G acts freely on P on the right and that PIG = M. By definition, each c E G maps the p-eq uivalence class of (rx, x, a) into the p-eq uivalence class of (rx, x, ac). It is easy to see that this definition is independent of the choice of representative (rx, x, a) and that G acts freely on P on the right. The projection 7T: P ~ M maps, by definition, the p-equivalence class of (rx, x, a) into x; the definition of 7T is independent of the choice of representative (rx, x, a). For u,v E P, 7T(U) = 7T(V) if and only if v = uc for some c E G. In fact, let (rx, x, a) and ({J,y, b) be representatives for u and v respectively. If v = uc for some c E G, then y = x and hence 7T(V) = 7T(U). Conversely, if 7T(U) = x = y = 7T(V) E Uct n U{3' then v = uc where c = a- 1"P{3ct(x) -lb E G. In order to make P into a differentiable manifold, we first note that, by the natural mapping X ~ P = XI p, each Xct = Uct X G is mapped 1: 1 onto 7T-1 ( Uct ). We introduce a differentiable structure in P by requiring that 7T- 1 ( Uct ) is an open submanifold of P and that the mapping X ~ P induces a diffeomorphism of Xct = Uct X G onto 7T-1 ( Uct ). This is possible since every point of P is contained in 7T-1 ( Uct ) for some rx and the identification of (rx, x, a) with ({J, x, "P{3ct(x)a) is made by means of differentiable mappings. It is easy to check that the action of G on P is differentiable and P(M, G, 7T) is a differentiable principal fibre bundle. Finally, the transition functions of P PROPOSITION

5.2.

I.

DIFFERENTIABLE MANIFOLDS

53

corresponding to the covering {Ua} are precisely the given "Ppa if we define "Pa: 7T- 1 ( Ua) ---+ Ua X G by "Pa(u) = (x, a), where u € 7T-1 ( U) is the p-equivalence class of (iX, x, a). QED. A homomorphism f of a principal fibre bundle P' (M', G') into another principal fibre bundle P(M, G) consists of a mapping f': P' ---+ P and a homomorphismf": G' ---+ G such thatf'(u'a') = f'(u')f"(a) for all u' € P' and a' € G'. For the sake of simplicity, we shall denotef' andf" by the same letterf Every homomorphism f: P' ---+ P maps each fibre of P' into a fibre of P and hence induces a mapping of M' into lvI, which will be also denoted byf A homomorphismf: P'(M', G') ---+ P(M, G) is called an imbedding or injection if f: P' ---+ P is an imbedding and if f: G' ---+ G is a monomorphism. Iff: P' ---+ P is an imbedding, then the induced mappingf: M' ---+ M is also an imbedding. By identifying P' with f(P'), G' withf(G') and M' withf(M'), we say that P'(M', G') is a subbundle of P(M, G). If, moreover, M' = M and the induced mapping f: M' ---+ M is the identity transformation of M, f: P'(M', G') ---+ P(M, G) is called a reduction of the structure group G of P(M, G) to G'. The subbundle P'(M, G') is called a reduced bundle. Given P(M, G) and a Lie subgroup G' of G, we say that the structure group G is reducible to G' if there is a reduced bundle P'(M, G'). Note that we do not require in general that G' is a closed subgroup of G.. This generality is needed in the theory of connections.

5.3. The structure group G of a principal fibre bundle P(M, G) is reducible to a Lie subgroup G' if and only if there is an open covering {Ua} of M with a set of transition functions "Ppa which take their values in G'. Proof. Suppose first that the structure group G is reducible to G' and let P'(M, G') be a reduced bundle. Consider P' as a submanifold of P. Let {Ua } be an open covering of M such that each 7T'-1( Ua ) (7T': the projection of P' onto M) is provided with an isomorphism u ---+ (7T'(U), cp~(u)) of 7T'-1( Ua) onto Ua X G'. The corresponding transition functions take their values in G'. Now, for the same covering {Ua }, we define an isomorphism of 7T- 1( Ua) (7T: the projection of Ponto M) onto Ua X G by extending cP~ as follows. Every v € 7T- 1 ( Ua ) may be represented in the form v = ua for some u € 7T'-1( Ua) and a € G and we set CPa(v) = cp~(u)a. PROPOSITION

54

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

It is easy to see that cplX(V) is independent of the choice of representationv = ua. We see then that v ~ (7T(V), cplX(V)) is an isomorphism of 7T-1 ( UJ onto UIX X G. The corresponding transition functions VJ{3IX(X) = CP{3(v) (cplX(V))-l = cpp(u)(cp~tU))-l take their values in G'. Conversely, assume that there is a covering {UJ of M with a set of transition functions VJ{31X all taking values in a Lie subgroup G' of G. For UIX n U{3 =1= 1>, VJ{31X is a differentiable mapping of UIX n U{3 into a-Lie group G such that VJ{3IX( UIX n U{3) c G'. The crucial point is that VJ{31X is a differentiable mapping of UIX n U{3 into G' with respect to the differentiable structure of G'. This follows from Proposition 1.3; note that a Lie subgroup satisfies the second axiom of countability by definition, cf. §4. By Proposition 5.2, we can construct a principal fibre bundle P' (M, G') from {UIX } and {VJ{3IX}' Finally, we imbed P' into P as follows. Let fIX: 7T'-1( UIX ) ~ 7T-1 ( UIX ) be the composite of the following three . mappIngs: 7T'-1(

UJ

~ >

UIX

X

G' ~ UIX

X

G~

7T- 1 (

UIX )'

I t is easy to see thatflX = h on 7T'-1( UIX n U{3) and that the mapping f: P' -+ P thus defined by {fJ is an injection. QED. Let P( M, G) be a principal fibre bundle and F a manifold on which G acts on the left: (a,~) E G X F ~ a~ E F. We shall construct a fibre bundle E( M, F, G, P) associated with P with standard fibre F. On the' product manifold P X F, we let G act on the right as follows: an element a E G maps (u, ~) E P X F into (ua, a-l~) E P X F. The quotient space of P X F by this group action is denoted by E = P X G F. A differentiable structure will be introduced in E later and at this moment E is only a set. The mapping P X F -+ M which maps (u,~) into 7T(U) induces a mapping 7TE' called the projection, of E onto M. For each x E M, the set 7Ti1(x) is called the fibre of E over x. Every point x of M has a neighborhood U such that 7T-1 ( U) is isomorphic to U X G. Identifying 7T-1 ( U) with U X G, we see that the action of G on 7T- 1 (U) X F on the right is given by

(x, a,

~) -+

(x, ab,

b-l~)

for (x, a,

~)

E

U

X

G

X

F

and

bEG.

It follows that the isomorphism 7T-1 ( U) ~ U X G induces an isomorphism 7Ti1( U) ~ U X F. We can therefore introduce a

I.

55

DIFFERENTIABLE MANIFOLDS

differentiable structure in E by the requirement that 1Ti1( U) is an open submanifold of E which is diffeomorphic with U X F under the isomorphism 1TE 1( U) R::;j U X F. The projection 1TE is then a differentitble mapping of E onto M. We call E or more precisely E(M, F, \G, P) thefibre bundle over the base M, with (standard) fibre F and (structure) group G, which is associated with the principal fibre bundle P. 5.4. Let P(M, G) be a principal fibre bundle and F a manifold on which G acts on the left. Let E(M, F, G, P) be the fibre bundle associated with P. For each u € P and each ~ € F, denote by u~ the image of (u, ~) € P X F ~y the natural projection P X F ~ E. Then each u € P is a mapping of F onto F x = 1TE 1(X) where x = 1T(U) and PROPOSITION

(ua)~

=

u(a~)

for u € P, a € G,

~

€

F.

The proof is trivial and is left to the reader. By an isomorphism of a fibre F x = 1Ti1(x), X € M, onto another fibre F 1I , y € M, we mean a diffeomorphism which can be represented in the form v u-\ where u € 1T- 1(X) and v € 1T- 1(y) are considered as mappings of F onto F x and F1I respectively. In particular, an automorphism of the fibre F x is a mapping of the form v u- 1 with U,V € 1T-1 (X). In this case, v = ua for some a € G so that any automorphism of F x can be expressed in the form u a u-1 where u is an arbitrarily fixed point of 1T-1 (x). The group of automorphisms of F x is hence isomorphic with the structure group G. Example 5.1. G(GjH, H): Let G be a Lie group and H a closed subgroup of G. We let H act on G on the right as follows. Every a € H maps u € G into ua. We then obtain a differentiable principal fibre bundle G(GjH, H) over the base manifold GjH with structure group H; the local triviality follows from the existence of a local cross section. It is proved in Chevalley [1; p. 110] that if 1T is the projection of G onto GjH and e is the identity of G, then there is a mapping (] of a neighborhood of 1T(e) in GjH into G such that 1T is the identity transformation of the neighborhood. See also Steenrod [1; pp. 28-33]. Example 5.2. Bundle of linear frames: Let M be a manifold of dimension n. A linear frame u at a point x € M is an ordered basis Xl> ... , X n of the tangent space Tx(M). Let L(M) be the set of 0

0

0

0

0

(]

56

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

all linear frames u at all points of M and let 7T be the mapping of L(M) onto M which maps a linear frame u at x into x. The general linear group GL(n; R) acts on L(M) on the right as follows. If a = (aJ) E GL(n; R) and u = (Xl' ... , X n) is a linear frame at x, then ua is, by definition, the linear frame (Y l' . . . , Y n) at x defined by Y i = ~i a{Xi · It is clear that GL(n; R) acts freely on L(M) and 7T(U) = 7T(V) if and only if v = ua for some a E GL(n; R). Now in order to introduce a differentiable structure in L(M), let (xl, ... , xn ) be a local coordinate system in a coordinate neighborhood U in M. Every frame u at x E U can be expressed uniquely in the form u = (Xl' ... , X n) with X t = ~k Xf(ojox k ), where (Xf) is a non-singular matrix. This shows that 7T- I ( U) is in 1: 1 correspondence with U X GL(n; R). We can make L(M) into a differentiable manifold by taking (Xi) and (Xf) as a local coordinate system in 7T- I ( U). It is now easy to verify that L(M) (M, GL(n; R)) is a principal fLbre bundle. We call L(M) the bundle of linear frames over M. In view of Proposition 5.4, a linear frame u at x E M can be defined as a non-singular linear mapping of Rn onto Tx(M). The two definitions are related to each other as follows. Let el , , en be the natural basis for Rn: el = (1, 0, ... , 0), ... , en = (0, ,0,1). A linear frame u = (Xl' ... ' X n) at x can be given as a linear mapping u: Rn -»- Tx(M) such that ue i = Xi for i = 1, ... , n. The action of GL(n; R) on L(M) can be accordingly interpreted as follows. Consider a = (aj) E GL(n; R) as a linear transformation of Rn which maps ei into ~t ajei . Then ua: Rn -»- Tx(M) is the composite of the following two mappings: a

u

R n ------+- R n ------+- T x (M) . Example 5.3. Tangent bundle: Let GL(n; R) act on Rn as above. The tangent bundle T(M) over M is the bundle associated with L(M) with standard fibre Rn. It can be easily shown that the fibre of T(M) over x E M may be considered as Tx(M). Example 5.4. Tensor bundles: Let T~ be the tensor space of type (r, s) over the vector space Rn as defined in §2. The group GL(n: R) can be regarded as a group of linear transformations of the space T; by Proposition 2.12. With this standard fibre T;, we obtain the tensor bundle T;(M) oftype (r, s) over M which is associated with L(M). It is easy to see that the fLbre of T;(M) over x E M may be considered as the tensor space over Tx(M) of type (r, s).

I.

DIFFERENTIABLE MANIFOLDS

57

Returning to the general case, let P(M, G) be a principal fLbre bundle and H a closed subgroup of G. In a natural way, G acts on the quotient space GjH on the left. Let E(M, GjH, G, P) be the associated bundle with standard fibre GjH. On the other hand, being a subgroup of G, H acts on P on the right. Let PjH be the quotient space of P by this action of H. Then we have

5.5. The bundle E = P X G (GjH) associated with P with standard fibre GjH can be identified with PjH as follows. An element ofE represented by (u, a~o) E P X GjH is mapped into the element ofPjH represented by ua E P, where a E G and ~o is the origin ofGjH, i.e., the coset H. Consequently, P(E, H) is a principalfibre bundle over the base E = PjH with structure group H. The projection P ~ E maps u E P into u~o E E, where u is considered as a mapping of the standardfibre GjH into afibre of PROPOSITION

E.

Proof. The proof is straightforward, except the local triviality of the bundle P(E, H). This follows from local triviality of E(M, GjH, G, P) and G(GjH, H) as follows. Let U be an open set of M such that 'TTi 1 ( U) ~ U X GjH and let V be an open set of GjH such that P-l(V) ~ V X H, where p: G ~ GjH is the projection. Let W be the open set of 'TTi 1 ( U) c E which corresponds to U X V under the identification 'TTi 1 ( U) ~ U X GjH. If fl: P ~ E = PjH is the projection, then fl-l( W) ~ W X H.

QED. A cross section of a bundle E(M, F, G, P) is a mapping a: M ~ E such that 'TTE a is the identity transformation of M. For P(M, G) itself, a cross section a: M ~ P exists if and only if P is the trivial bundle M X G (cf. Steenrod [1 ; p. 36]). More generally, we have 0

5.6. The structure group G of P(M, G) is reducible to a closed subgroup H if and only if the associated bundle E(M, GjH, G, P) admits a cross section a: M ~ E = P jH. Proof. Suppose G is reducible to a closed subgroup H and let Q(M, H) be a reduced bundle with injection f: Q ~ P. Let fl: P ~ E = PjH be the projection. If u and v are in the same fibre of Q, then v = ua for some a E H and hence fl(f(v)) = fl(f(u)a) = fl(f(u)). This means that fl of is constant on each fibre of Q and induces a mapping a: M ~ E, a(x) = fl(f(u)) PROPOSITION

58

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

where x = 7T(f(u)). It is clear that a is a section of E. Conversely, given a cross section a: M -+ E, let Q be the set of points u E P such that fl(U) = a(7T(u)). In other words, Q is the inverse image of a(M) by the projection fl: P -+ E = PfH. For every x E M, there is u E Q such that 7T(U) = x because fl-l( a(x)) is non-empty. Given u and v in the same fibre of P, if U E Q then v E Q when and only when v = ua for some a E H. This follows from the fact that fl(u) = fl(v) if and only if v = ua for some a E H. It is now easy to verify that Q is a closed submanifold of P and that Q is a principal fibre bundle Q(M, H) imbedded in P(M, G). QED.

Remark. The correspondence between the sections a: M -+ E = PfH and the submanifolds Q is 1:1. We shall now consider the question of extending a cross section defined on a subset of the base manifold. A mappingf of a subset A of a manifold M into another manifold is called dijJerentiable on A if for each point x E A, there is a differentiable mappingfx of an open neighborhood U x of x in Minto M' such that fx = f on U x n A. Iff is the restriction of a differentiable mapping of an open set W containing A into M', then f is clearly differentiable on A. Given a fibre bundle E(M, F, G, P) and a subset A of M, by a cross section on A we mean a differentiable mapping a of A into E such that TTE a is the identity transformation of A. 0

5.7. Let E(M, F, G, P) be a fibre bundle such that the base manifold M is paracompact and the fibre F is dijJeomorphic with a Euclidean space R m. Let A be a closed subset (possibly empty) of M. Then every cross section a: A -+ E defined on A can be extended to a cross section defined on M. In the particular case where A is empty, there exists a cross section of E defined on M. Proof. By the very definition of a paracompact space, every open covering of M has a locally finite open refinement. Since M is normal, every locally finite open covering {Ui } of M has an open refinement {Vi} jluch that Pi c Ui for all i (see Appendix 3). THEOREM

A differentiable function defined on a closed set of Rn can be extended to a differentiable function on Rn (cf. Appendix 3). LEMMA

1.

2.

Every point of M has a neighborhood U such that every section of E defined on a closed subset contained in U can be extended to U. Proof. Given a point of M, it suffices to take a coordinate LEMMA

I.

DIFFERENTIABLE MANIFOLDS

59

neighborhood U such that 7Til(U) is trivial: 7Til(U) ~ U X F. Since Fis diffeomorphic with R m, a section on U can be identified with a set of m functions!l' ... ,!m defined on U. By Lemma 1, these functions can be extended to U. Using Lemma 2, we shall prove Theorem 5.7. Let {Ui}iEI be a locally finite open covering of M such that each U i has the property stated in Lemma 2. Let {Vi} be an open refinement of {Ut } such that Vi c Ui for all i E I. For each subset J of the index set I, set SJ = U Vi. Let T be the set of pairs (T, J) where J c I

iEJ and T is a section of E defined on S J such that T = (] on A n S JO The set T is non-empty; take U i which meets A and extend the restriction of (] to A n Vi to a section on Vi' which is possible by the property possessed by Ui. Introduce an order in T as follows: (T', J') < (T", J") if J' c J" and T' = T" on Sr. Let (T, J) be a maximal element (by using Zorn's Lemma). Assume J *- I and let i E I - J. On the closed set (A U S J) n Vi contained in Ui' we have a well defined section (]i: (]i = (] on A n Vi and (]i = T on SJ n Vi. Extend (]i to a section T i on Vi' which is possible by the property possessed by Ui. Let J' = J U {i} and T' be the section on SJ' defined by T' = T on SJ and T' = T i on Then (T, J) < (T', J'), which contradicts the maximality of (T, J). Hence, I = J and T is the desired section. QED.

rio

The proof given here was taken from Godement [1, p. 151]. Example 5.5. Let L( M) be the bundle of linear frames over an n-dimensional manifold M. The homogeneous space GL(n; R)j O(n) is known to be diffeomorphic with a Euclidean space of dimension in(n + 1) by an argument similar to Chevalley [1, p. 16]. The fibre bundle E = L(M)jO(n) with fLbre GL(n; R)jO(n), associated with L(M), admits a cross section if M is paracompact (by Theorem 5.7). By Proposition 5.6, we see that the structure group of L(M) can be reduced to the orthogonal group O(n), provided that Mis paracompact. Example 5.6. More generally, let P(M, G) be a principal fibre bundle over a paracompact manifold M with group G which is a connected Lie group. It is known that G is diffeomorphic with a direct product of any of its maximal compact subgroups H and a Euclidean space (cf. Iwasawa [1]). By the same reasoning as above, the structure group G can be reduced to H.

60

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Example 5.7. Let L( M) be the bundle of linear frames over a manifold M of dimension n. Let ( , ) be the natural inner product in Rn for which el = (1, 0, ... , 0), ... , en = (0, ... , 0, 1) are orthonormal and which is invariant by O(n) by the very definition of O(n). We shall show that each reduction of the structure group GL(n; R) to O(n) gives rise to a Riemannian metric g on M. Let Q(M,O(n)) be a reduced subbundle of L(M). When we regard each u E L(M) as a linear isomorphism of Rn onto Tx(M) where x = 7T(U), each u E Q defines an inner product g in Tx(M) by g(X, Y) = (u-IX, u- l Y) for X, Y E Tx(M). The invariance of ( , ) by O(n) implies that g(X, Y) is independent of the choice of u E Q. Conversely, if M is given a Riemannian metric g, let Q be the subset of L(M) consisting of linear frames u = (Xl' ... , X n) which are orthonormal with respect to g. If we regard u E L( M) as a linear isomorphism of Rn onto T x ( M), then u belongs to Q if and only if (~, ~') = g(u~, uf) for all ~,f ERn. It is easy to verify that Qforms a reduced subbundle of L(M) over M with structure group O(n). The bundle Q will be called the bundle of orthonormal frames over M and will be denoted by 0 (M) . An element of O(M) is an orthonormal frame. The result here combined with Example 5.5 implies that every paracompact manifold M admits a Riemannian metric. We shall see later that every Riemannian manifold is a metric space and hence paracompact. To introduce the notion of induced bundle, we prove

5.8. Given a principal fibre bundle P(M, G) and a mappingf of a manifold N into M, there is a unique (of course, unique up to an isomorphism) principal fibre bundle Q(N, G) with a homomorphism f: Q ~ P which induces f: N ~ M and which corresponds to the identity automorphism of G. PROPOSITION

The bundle Q(N, G) is called the bundle induced byffrom P(M, G) or simply the induced bundle; it is sometimes denoted by f-IP. Proof. In the direct product N X P, consider the subset Q 'consisting of (y, u) E N X P such thatf(y) = 7T(U). The group G acts on Q by (y, u) ~ (y, u)a = (y, ua) for (y, u) E Q and a E G. I t is easy to see that G acts freely on Q and that Q is a principal fibre bundle over N with group G and with projection 7T Q given

I.

DIFFERENTIABLE MANIFOLDS

61

by 1TQ(Y, u) = y. Let Q' be another principal fibre bundle over N with group G and f': Q' -+ P a homomorphism which induces f: N -+ M and which corresponds to the identity automorphism of G. Then it is easy to show that the mapping of Q' onto Q defined by u' -+ (1TQ,(u'),f'(u')), u' E Q', is an isomorphism of the bundle Q' onto Q which induces the identity transformation of Nand which corresponds to the identity automorphism of G. QED. We recall here some results on covering spaces which will be used later. Given a connected, locally arcwise connected topological space M, a connected space E is called a covering space over M with projection p: E -+ M if every point x of M has a connected open neighborhood U such that each connected component of p-l( U) is open in E and is mapped homeomorphically onto U by p. Two covering spaces p: E -+ M and p': E' -+ Mare isomorphic if there exists a homeomorphism f: E -+ E' such that p' of = p. A covering space p: E -+ M is a universal covering space if E is simply connected. If M is a manifold, every covering space has a (unique) structure of manifold such that p is differentiable. From now on we shall only consider covering manifolds.

5.9. (1) Given a connected manifold M, there is a unique (unique up to an isomorphism) universal covering manifold, which will be denoted by Nt. (2) The universal covering manifold Nt is a principal fibre bundle over M with group 1T 1 (M) and projection p: Nt -+ M, where 1T 1 (M) is the first homotopy group of M. (3) The isomorphism classes of the covering spaces over M are in a 1:1 correspondence with the conjugate classes of the subgroups of 1Tl(M). The correspondence is given as follows. To each subgroup H of 1Tl(M), we associate E = Nt/H. Then the covering manifold E corresponding to H is a fibre bundle over M with fibre 1Tl(M) /H associated with the principal fibre bundle Nt (M, 1T 1 ( M) ). If H is a normal subgroup of 1T 1 ( M) , E = Nt/H is a principal fibre bundle with group 1T 1 ( M) / H and is called a regular covering manifold of M. For the proof, see Steenrod [1, pp. 67-71] or Hu [1, pp. 89-97]. The action of 1T 1 ( M) / H on a regular covering manifold E = Nt/H is properly discontinuous. Conversely, if E is a connected manifold and G is a properly discontinuous group of transforma.. tions acting freely on E, then E is a regular covering manifold of PROPOSITION

62

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

M = EjG as follows immediately from the condition (3) in the definition of properly discontinuous action in §4. Example 5.8. Consider Rn as an n-dimensional vector space and let ~1' ••• , ~n be any basis of Rn. Let G be the subgroup of Rn generated by ~1' • . . ' ~n: G = {l; mi ~i; mi integers}. The action of G on Rn is properly discontinuous and Rn is the universal covering manifold of RnjG. The quotient manifold RnjG is called an n-dimensional torus. Example 5.9. Let Sn be the unit sphere in Rn+l with center at the origin: sn = {(xl, ... ,xn+1) € Rn+l; l;i(X i )2 = I}. Let G be the group consisting of the identity transformation of Sn and the !". • 1 ... , xn+l)·Into ( -x, 1 ... , translormation 0 f Sn w h·IC h maps (x, _xn+1 ). Then Sn,n > 2, is the universal covering manifold of snjG. The quotient manifold snjG is called the n-dimensional real projective space.

CHAPTER II

Theory of Connections 1. Connections in a principal fibre bundle Let P(M, G) be a principal fibre bundle over a manifold M with group G. For each u E P, let Tu(P) be the tangent space of P at u and Gu the subspace of Tu(P) consisting of vectors tangent to the fibre through u. A connection r in P is an assignment of a subspace Qu of Tu(P) to each u E P such that (a) Tu(P) = Gu + Qu (direct sum); (b) Qua = (R a)*Qu for every u E P and a E G, where R a is the transformation of P induced by a E G, Rau = ua; (c) Qu depends differentiably on u. Condition (b) means that the distribution u -)- Qu is invariant by G. We call Gu the vertical subspace and Qu the horizontal subspace of Tu(P). A vector X E Tu(P) is called vertical (resp. horizontal) ifit lies in Gu (resp. Qu). By (a), every vector X E Tu(P) can be uniquely written as I

where Y E Gu and

Z

E

Qu.

We call Y (resp. Z) the vertical (resp. horizontal) component of X and denote it by vX (resp. hX). Condition (c) means, by definition, that if X is a differentiable vector field on P so are vX and hX. (It can be easily verified that this is equivalent to saying that the distribution u -)- Qu is differentiable.) Given a connection r in P, we define a I-form w on P with values in the Lie algebra 9 of G as follows. In §5 of Chapter I, we showed that every A E 9 induces a vector field A* on P, called the fundamental vector field corresponding to A, and that A -)- (A*) u is a linear isomorphism of 9 onto Gu for each u E P. For each X E Tu(P), we define w(X) to be the unique A E 9 such that 63

64

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(A*)u is equal to the vertical component of X. It is clear that w(X) = 0 if and only if X is horizontal. The form w is called the connection form of the given connection r. 1.1. The connection form w of a connection satisfies the following conditions: (a') w(A*) = A for every A E g; (b') (Ra)*w = ad (a-I)w, that is, w((Ra)*X) = ad (a-i) . w(X) for every a E G and every vector field X on P, where ad denotes the adjoint representation of G in g. Conversely, given a g-valued 1-form w on P satisfying conditions (a') and (b'), there is a unique connection r in P whose connection form is w. Proof. Let w be the connection form of a connection. The condition (a') follows immediately from the definition of w. Since every vector field of P can be decomposed into a horizontal vector field and a vertical vector field, it is sufficient to verify (b') in the following two special cases: (1) X is horizontal and (2) X is vertical. If X is horizontal, so is (R a)*X for every a E G by the condition (b) for a connection. Thus, both w( (R a ) *X) and ad (a-I) . w(X) vanish. In the case when X is vertical, we may further assume that X is a fundamental vector field A *. Then (R a)*X is the fundamental vector field corresponding to ad (a-I)A by Proposition 5.1 of Chapter 1. Thus we have PROPOSITION

Conversely, given a form w satisfying (a') and (b'), we define

The verification that u -» Q u defines a connection whose connection form is w is easy and is left to the reader. QED. The projection 7T: P -» M induces a linear mapping 7T: T u (P) -» Tx(M) for each u E P, where x = 7T(U). When a connection is given, 7T maps the horizontal subspace Qu isomorphically onto

Tx(M). The horizontal lift (or simply, the lift) of a vector field X on M is a unique vectqr field X* on P which is horizontal and which projects onto X, that is, 7T(X:) = X (u) for every u E P. 1T

II.

65

THEORY OF CONNECTIONS

1.2. Given a connection in P and a vector field X on M, there is a unique horizontal lift x* of X. The lift X* is invariant by R a for every a € G. Conversely, every horizontal vector field x* on P invariant by G is the lift of a vector field X on M~ Proof. The existence and uniqueness of X* is clear from the fact that 7T gives a linear isomorphism of Qu onto T (u)(M). To prove that x* is differentiable if X is differentiable, we take a neighborhood U.of any given point x of M such that 7T-1 ( U) ~ U x G. Using this isomorphism, we first obtain a differentiable vector field Y on 7T- 1 (U) such that 7T Y = X. Then X* is the horizontal component of Y and hence is differentiable. The invariance of x* by G is clear from the invariance of the horizontal subspaces by G. Finally, let X* be a horizontal vector field on P invariant by G. For every x € M, take a point u € P such that 7T(U) - x and define Xx = 7T(X:). The vector Xx is independent of the choice of u such that 7T(U) = x, since if u' - ua, then 7T(X:,) = 7T(R a • X:) = 7T(X:). It is obvious that X* is then the QED. lift of the vector field X. PROPOSITION

7T

PROPOSITION

1.3.

Let X* and y* be the horizoritallifts of X and Y

respectively. Then (1) X* y* is the horizontal lift of X + Y; (2) For everyfunctionf on M, f* . X* is the horizontal lift offX where f* is the function on P difined by f* = f 7T ~. (3) The horizontal component of [X*, Y*] is the horizontal lift of [X,YJ. Proof. The first two assertions are trivial. As for the third, we have 7T(h[X*, Y*]) = 7T([X*, Y*]) = [X, Y]. QED. 0

Let xl, ... ,xn be a local coordinate system in a coordinate neighborhood U in M. Let Xi be the horizontal lift in 7T- 1 ( U) of the vector field Xi = a/ax i in Ufor each i. Then Xi, ... , X: form a local basis for the distribution u -+ Qu in 7T-1 ( U). We shall now express a connection form OJ on P by a family of forms each defined in an open subset of the base manifold M. Let {UIY.} be an open covering of M with a family of isomorphisms "PlY.: 7T- 1 ( UIY.) -+ UIY. x G and the corresponding family of transition functions "Pa.f3: UIY. n Up -+ G. For each 0::, let (JIY.: UIY. -+ P be the

66

FOUNDATIONS OF DIIl'FERENTIAL GEOMETRY

cross section on Ua. defined by O'a.(x) = "P;l(X, e), x E Ua., where e is the identity of G. Let () be the (left invariant g-valued) canonical I-form on G defined in §4 of Chapter I (p. 41). For each non-empty Ua. n Uf3 , define a g-valued I-form ()a.f3 on Ua. n Uf3 by ()a.f3 = "p:f3(). For each

C't,

define a g-valued I-form Wa. on Ua. by

PROPOSITION

1.4.

The forms ()a.f3 and Wa. are subject to the conditions:

Conversely, for every family of g-valued l-Jorms {wa.} each defined on Ua. and satisfying the preceding conditions, there is a unique connection form W on P which gives rise to {wa.} in the described manner. Proof. If Ua. n Uf3 is non-empty, O'f3(x) = O'a.(x)"Pa.f3(x) for all x E Ua. n Uf3. Denote the differentials of O'a., 0'f3' and "Pa.f3 by the same letters. Then for every vector X E T x ( Ua. n U(3 ), the vector O'f3(X) E Tu(P), where u = O'f3(x), is the image of (O'a.(X), "P1X{3(X)) E Tu'(P) + Ta(G), where u' = O'a.(x) and a = "p1X{3(x), under the mapping P X G ~ P. By Proposition 1.4 (Leibniz's formula) of Chapter I, we have f3 (X) =

0'

0'a. (

X) "pa.f3 (x)

+ 0'a. ( x) "pa.f3 (X) ,

where O'a.( X) "Pa.f3(x) means R a(O'a.(X)) and O'a.(x) "P1X{3(X) is the image of"Pa.f3(X) by the differential of O'a.(X) , O'a.(x) being considered as a mapping of G into P which maps bEG into O'a.(x)b. Taking the values of w on both sides of the equality, we obtain

Indeed, if A Egis the left invariant vector field on G which is equal to "Pa.{3(X) at a = "Pa.f3(x) so that ()( "Pa.f3(X)) = A, then (1a.(x) "Pa.f3 (X) is the value of the fundamental vector field A* at u = O'a.(x)"Pa.{3(x) and hence w(O'a.(x)"PIX{3(X)) = A. The converse can be verified by following back the process of obtaining {wa.} from w. QED.

II.

THEORY OF CONNECTIONS

67

2. Existence and extension of connections Let P(M, G) be a principal fibre bundle and A a subset of M. We say that a connection is defined over A if, at every point u E P with 7T(U) E A, a subspace Qu of Tu(P) is given in such a way that conditions (a) and (b) for connection (see §I) are satisfied and Qu depends differentiably on u in the following sense. For every point x E A, there exist an open neighborhood U and a connection in P U = 7T- I ( U) such that the horizontal subspace at every u E 7T- I(A) is the given space Qu.

I

2.1. Let P( M, G) be a principal fibre bundle and A a closed subset of M (A may be empty). If M is paracompact, every connection defined over A can be extended to a connection in P. In particular, P admits a connection if M is paracompact. Proof. The proof is a replica of that of Theorem 5.7 in Chapter I. THEOREM

A differentiable function defined on a closed subset of Rn can be always extended to a differentiablefunction on Rn (cf. Appendix 3). LEMMA

1.

LEMMA

2. Every point of M has a neighborhood U such that every

connection defined on a closed subset contained in U can be extended to a connection defined over U. Proof. Given a point of M, it suffices to take a coordinate neighborhood U such that 7T- I ( U) is trivial: 7T- I ( U) ~ U X G. On the trivial bundle U X G, a connection form w is completely determined by its behavior at the points of U X {e} (e: the identity of G) because of the property R; (w) = ad (a-I) w. Furthermore, if 0': U -+ U X G is the natural cross section, that is, O'(x) = (x, e) for x E U, then w is completely determined by the g-valued I-form O'*w on U. Indeed, every vector X E Ta(x) ( U X G) can be written uniquely in the form X = Y where Y is tangent to U O'*(7T*X). Hence we have

X

+ Z,

{e} and Z is vertical so that Y =

w(X) = W(O'*(7T*X)) + w(Z) = (O'*w) (7T*X) + A, where A is a unique element of 9 such that the corresponding fundamental vector field A* is equal to Z at O'(x). Since A depends

68

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

only on Z, not on the connection, w is completely determined by a*w. The equation above shows that, conversely, every gvalued I-form on U determines uniquely a connection form on U X G. Thus Lemma 2 is reduced to the extension problem for g-valued I-forms on U. If {A i} is a basis for g, then w = ~ wi Ai where each wi is a usual I-form. Thus it is sufficient to consider the extension problem of usual I-forms on U. Let xl, ... , x n be a local coordinate system in U. Then every I-form on U is of the fornl ~h dx i where eachfi is a function on U. Thus our problem is reduced to the extension problem of functions on U. Lemma 2 now follows from Lemma 1. By means of Lemma 2, Theorem 2.1 can be proved exactly in the same way as Theorem 5.7 of Chapter 1. Let {Uth E I be a locally finite open covering of M such that each U i has the property stated in Lemma 2. Let {Vt } be an open refinement of {Ui} such that Vi cUt. For each subset J of I, set SJ = U Vi. i€J

Let T be the set of pairs (T, J) where J c I and T is a connection defined over SJ which coincides with the given connection over A n S J. Introduce an order in T as follows: (T', J') < (T", J") if J' c J" and T' = T" on SJ'. Let (T, J) be a maximal element of T. Then J = I as in the proof of Theorem 5.7 of Chapter I and T is a desired connection. QED. Remark. It is possible to prove Theorem 2.1 using Lemma 2 and a partition of unity {J:} subordinate to {Vt } (cf. Appendix 3). Let Wi be a connection form on 7T-1 ( U i ) which extends the given connection over A n Vi. Then w = ~i gtWi is a desired connection form on P, where each gt is the function on P defined by

gt =fi

0

7T.

3. Parallelism Given a connection r in a principal fibre bundle P(M, G), we shall define the concept of parallel displacement offibres along any given curve T in the base manifold M. Let T = x t , a < t < b, be a piecewise differentiable curve of class Cl in M. A horizontal lift or simply a lift of T is a horizontal curve T* = U t , a < t s: b, in P such that 7T( ut ) = X t for a < t < b. Here a horizontal curve in P means a piecewise differentiable curve of class Cl whose tangent vectors are all horizontal.

II.

THEORY OF CONNECTIONS

69

The notion of lift of a curve corresponds to the notion of lift of a vector field. Indeed, if X* is the lift of a vector field X on M, then the integral curve of X* through a point Uo E P is a lift of the integral curve of X through the point Xo = 7T(UO) E M. We now prove

3.1. Let T = x t, 0 < t < 1, be a curve of class Cl in M. For an arbitrary point Uo of P with 7T(U O) = X o, there exists a unique lift T* = U t of T which starts from UO• Proof. By local triviality of the bundle, there is a curve Vt of class Cl in P such that Vo = Uo and 7T( vt ) = X t for 0 < t < 1. A lift of T, if it exists, must be of the form U t = vta t, where at is a curve in the structure group G such that ao = e. We shall now look for a curve at in G which makes U t = vta t a horizontal curve. Just as in the proof of Proposition 1.4, we apply Leibniz's formula (Proposition 1.4 of Chapter I) to the mapping P X G -+ P which maps (v, a) into va and obtain PROPOSITION

where each dotted italic letter denotes the tangent vector at that point (e.g., itt is the vector tangent to the curve T* = U t at the point ut ). Let w be the connection form of r. Then, as in the proof of Proposition 1.4, we have . w(it t) = ad(at-1)w(v t ) + at-1d t , where at-1d t is now a curve in the Lie algebra 9 = Te(G) of G. The curve U t is horizontal if and only if dta t- 1 = -w(v t ) for every t. The construction of U t is thus reduced to the following

Let G be a Lie group and 9 its Lie algebra identified with T e(G). Let Y t, 0 < t :::;;: 1, be a continuous curve in T e(G). Then there exists in G a unique curve at of class Cl such that ao = e and dta t- 1 = Y t for 0 < t < 1. LEMMA.

Remark. In the case where Y t = A for all t, the curve at is nothing but the I-parameter subgroup of G generated by A. Our differential equation dta t- 1 = Y t is hence a generalization of the differential equation for I-parameter subgroups. Proof of Lemma. We may assume that Y t is defined and continuous for all t, - 00 < t < 00. We define a vector field X on

70

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

G X R as follows. The value of X at (a, t) € G X R is, by definition, equal to (Yta, (dldz) t) € Ta(G) X Tt(R), where z is the natural coordinate system in R. It is clear that the integral curve of X starting from (e, 0) is of the form (at, t) and at is the desired curve in G. The only thing we have to verify is that at is defined for all t, 0 < t < 1. Let f{Jt = exp tX be a local I-parameter group of local transformations of G X R generated by X. For each (e, s) € G x R, there is a positive number d s such that ({Jt(e, r) is defined for Ir - sl < d s and It I < d s (Proposition 1.5 of Chapter I). Since the subset {e} X [0, 1] of G X R is compact, we may choose d > 0 such that, for each r € [0, 1], f{Jt(e, r) is defined for It I < d (cf. Proof of Proposition 1.6 of Chapter I). Choose So, Sl' ... , Sk such that 0 = So < Sl < ... < Sk = 1 and Si - Si-l < d for every i. Then f{Jt(e, 0) = (at, t) is defined for o < t < Sl; f{Ju(e, Sl) = (b u, u + Sl) is defined for 0 ::;;: u < S2 - Sl' where bub;;l = Y U + S1 ' and we define at = bt-slaSl fors l < t ::;;: S2; ... ; f{Ju(e, Sk-l) = (c u, Sk-l + u) is defined for 0 ::;;: u < Sk - Sk-l' where cuC;;1 = Yu + sk-l , and we define at = Ct- sk-l ask-l , thus completing the construction of at, 0 < t < 1. QED. Now using Proposition 3.1, we define the parallel displacement of fibres as follows. Let T = X t , 0 < t < 1, be a differentiable curve of class CIon M. Let U o be an arbitrary point of P with 7T(U O) = X O• The unique lift T* of T through Uo has the end point U I such that 7T(U I ) = Xl. By varying Uo in the fibre 7T- I (XO)' we obtain a mapping of the fibre 7T- I (XO) onto the fibre 7T- I (X I) which maps U o into Up We denote this mapping by the same letter T and call it the parallel displacement along the curve T. The fact that I I T: 7T- (XO) -+ 7T- (X I) is actually an isomorphism comes from the following

3.2. The parallel displacement along any curve commutes with the action of G on P: T Ra = Ra T for every a € G. PROPOSITION

0

T

0

Proof. This follows from the fact that every horizontal curve is mapped into a horizontal curve by R a • QED. The parallel displacement along any piecewise differentiable curve of class CI can be defined in an obvious manner. It should be remarked that the parallel displacement along a curve T is

II.

THEORY OF CONNECTIONS

71

independent of a specific parametrization X t used in the following sense. Consider two parametrized curves X t , a < t ~ b, and Ys, c ~ s < d, in. M. The parallel displacement along X t and the one along Y s coincide if there is a homeomorphism cp of the interval [a, b] onto [c, d] such that (1) cp(a) = c and cp(b) = d, (2) both cp and cp-l are differentiable of class Cl except at a finite number of parameter values, and (3) Yrp(t) = X t for all t, a < t < b. If 7 is the curve X t , a < t ~ b, we denote by 7 -1 the curve Yt, a < t < b, defined by Yt = xa + b - t • The following proposition is evident. 3.3. (a) If 7 is a piecewise differentiable curve of class Cl in M, then the parallel displacement along 7- 1 is the inverse of the parallel displacement along 7. (b) If 7 is a curve from x to Y in M and fl is a curve from Y to z in M, the parallel displacement along the composite curve fl . 7 is the composite of the parallel displacements 7 and fl. PROPOSITION

4. Holonomy groups U sing the notion of parallel displacement, we now define the holonomy group of a given connection r in a principal fibre bundle P(M, G). For the sake of simplicity we shall mean by a curve a piecewise differentiable curve of class Ck, 1 < k ~ 00 (k will be fixed throughout §4). For each point x of M we denote by C(x) the loop space at x, that is, the set of all closed curves starting anq. ending at x. If 7 and fl are elements of C(x), the composite curve fl . 7 (7 followed by fl) is also an element of C(x). As we proved in §3, for each 7 € C(x), the parallel displacement along 7 is an isomorphism of the fibre 7T- 1 (X) onto itself. The set of all such isomorphisms of 1 7T- (X) onto itself forms a group by virtue of Proposition 3.3. This group is called the holonomy group of r with reference point x. Let CO(x) be the subset ofC(x) consisting of loops which are homotopic to zero. The subgroup of the holonomy group consisting of the parallel displacements arising from all 7 € CO(x) is called the restricted holonomy group of r with reference point x. The holonomy group and the restricted holonomy group of r with reference point x will be denoted by (x) and °(x) respectively.

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FOUNDATIONS OF DIFFERENTIAL GEOMETRY

I t is convenient to realize these groups as subgroups of the structure group G in the following way. Let u be an arbitrarily fixed point ofthe fibre 1T-1(X). Each T E C(x) determines an element, say, a, of G such that T(U) = ua. If a loop fl E C(x) determines bEG, then the composite fl . T determines ba because (fl . T) (u) fl(ua) = (fl(u))a uba by virtue of Proposition 3.2. The set of elements a E G determined by all T E C(x) forms a subgroup of G by Proposition 3.3. This subgroup, denoted by
~


t'..I

t'..I

t".I

4.1. (a) Ifv ua, a EG, then
t'..I

t'..I

t'..I

t'..I

t'..I

t'..I

t'..I

t'..I

II.

THEORY OF CONNECTIONS

73

(R b #*) . T* • #*-1 is a horizontal curve in P from v to vb and its projection into M is a loop at 1T(V) homotopic to zero. Thus b E °(v). Similarly, if b E °(v), then b E °(u). QED. If M is connected, then for every pair of points u and v of P, there is an element a E G such that v ~ ua. It follows from Proposition 4.1 that if M is connected, the holonomy groups (u), u E P, are all conjugate to each other in G and hence isoll1orphic with each other. The rest of this section is devoted to the proof of the fact that the holonomy group is a Lie group.

4.2. Let P(M, G) be a principal fibre bundle whose base manifold M is connected and paracompact. Let (u) and °(u), u E P, be the holonomy group and the restricted holonomy group of a connection r with r:eference point u. Then (a) °(u) is a connected Lie subgroup of G; (b) °(u) is a normal subgroup of (u) and (u)j°(u) is countable. By virtue of this theorem, (u) is a Lie subgroup of G whose identity component is °(u). Proof. We shall show that every element of °(u) can be joined to the identity element by a piecewise differentiable curve of class Ck in G which lies in °(u). By the theorem in Appendix 4, it follows then that °(u) is a connected Lie subgroup of G. Let a E °(u) be obtained by the parallel displacement along a piecewise differentiable loop T of class Ck which is homotopic to O. By the factorization lemma (Appendix 7), T is (equivalent to) a product of small lassos of the form TIl. # • Tb where T1 is a piecewise differentiable curve of class Ck from x = 1T(U) to a point, say,y, and # is a differentiable loop aty which lies in a coordinate neighborhood of y. It is sufficient to show that the element of °(u) defined by each lasso TIl. # . T1 can bejoined to the identity element. This element is obviously equal to the element of <1>0 (v) defined by the loop #, where v is the point 0 btained by the parallel displacement of u along T1. It is therefore sufficient to show that the element b E <1>0 ( v) defined by the differentiable loop # can be joined to the identity element in °(v) by a differentiable curve of G which lies in °(v). Let xl, ... , xn be a local coordinate system with origin at y THEOREM

74

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

and let ft be defined by Xi = xi(t), i = 1, ... , n. Setfi(t, s) = s + (1 - s)xi(t) for i = 1, ... , nand 0 < t, s < 1. Thenf(t, s) = (f1(t, s), ... ,fn(t, s)) is a differentiable mapping of class Ck of I X I into M (where I = [0, 1]) such thatf(t, 0) is the curve fl and f (t, 1) is the trivial curve y. For each fixed s, let b(s) be the element of O(v) obtained from the loop f (t, s), 0 < t s 1, so that b(0) = band b( 1) = identity. The fact that b(s) is of class Ck in s (as a mapping of I into G) follows from the following

Let f: I x I ~ M be a dijJerentiable mapping of class Ck and uo(s), 0 < S < 1, a dijJerentiable curve of class Ck in P such that 7T(U O(S)) = f (0, s). For each fixed s, let U1(s) be the point of P obtained by the parallel displacement of Uo(s) along the curve f (t, s), where o < t s 1 and s isfixed. Then the curve u1(s), 0 < s s 1, is dijferentiable of class Ck. Proof of Lemma. Let F: I X I ~ P be a differentiable mapping of class Ck such that 7T(F( t, s)) = f (t, s) for all (t, s) E I X I and that F (0, s) = uo(s). The existence of such an F follows from local triviality of the bundle P. Set vt(s) = F (t, s). In the proof of Proposition 3.1, we saw that, for each fixed s, there is a curve at(s), 0 S t s 1, in G such that ao(s) = e and that the curve vt(s)at(s), 0 < t < 1, is horizontal. Set ut(s) = vt(s)at(s). To prove that u1 (s), 0 < s < 1, is a differentiable curve of class Ck, it is sufficient to show that a1 (s), 0 S s s 1, is a differentiable curve of class Ck in G. Let w be the connection form of r. Set Yt(s) = -w(vt(s)), where vt(s) is the vector tangent to the curve described by vt(s), 0 < t < 1, when s is fixed. Then as in the proof of Proposition 3.1, at(s) is a solution of the equation dt(s)at(s) -1 = Yt(s). As in the proof of the lemma for Proposition 3.1, we define, for each fixed s, a vector field X(s) on G X R so that (at(s), t) is the integral curve of the vector field X(s) through the point (e, 0) E G X R. The differentiability of at(s) in s follows from the fact that each solution of an ordinary linear differential equation with parameter s is differentiable in s as many times as the equation is (cf. Appendix 1). This completes the proof of the lemma and hence the proof of (a) of Theorem 4.2. We now prove (b). If 7 and fl are two loops at x and if fl is homotopic to zero, the composite curve 7 • fl . 7 -1 is homotopic to zero. This implies that °(u) is a normal subgroup of (u). LEMMA.

II.

THEORY OF CONNECTIONS

75

Let 7Tl(M) be the first homotopy group of M with reference point x. We define a homomorphismf: 7Tl(M) ~
Remark. In §3, we defined the parallel displacement along any piecewise differentiable curve of class Cl. In this section, we defined the holonomy group ••• :::>
5. Curvature form and structure equation Let P(M, G) be a principal fibre bundle and p a representation of G on a finite dimensional vector space V; p(a) is a linear transformation of V for each a E G a,nd p(ab) = p(a)p(b) for a,b E G. A pseudotensorial form of degree ·r on P of type (p, V) is a Vvalued r-form rp on P such that for a E G. Such a form rp is called a tensorialform ifit is horizontal in the sense that rp(X1 , ••• ,Xr ) = 0 whenever at least one of the tangent vectors Xi of P is vertical, i.e., tangent to a fibre. Example 5.1. If Po is the trivial representation of G on V, that is, po(a) is the identity transformation of V for each a E G, then a tensorial form of degree r of type (Po, V) is nothing but a form rp on P which can be expressed as rp = 7T*rpM where rpM is a V-valued r-form on the base M.

76

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Example 5.2. Let p be a representation of G on V and E the bundle associated with P with standard fibre Von which G acts through p. A tensorial form ep of degree r of type (p, V) can be regarded as an assignment to each x E M a multilinear skewsymmetric mapping CPa; of Ta;(M) X ... X Ta;(M) (r times) into the vector space 7Ti1(x) which is the fibre of E over x. Namely, we define

where u is any point of P with 7T(U) = x and Xf is any vector at u such that 7T(X!) = Xi for each i. ep(Xf, ... , Xi) is then an element of the standard fibre V and u is a linear mapping of V onto 7Ti1(x) so that u( ep(Xf, ... , Xi)) is an element of 7Ti1(x). It can be easily verified that this element is independent of the choice of u and Xl. Conversely, given a skew-symmetric multilinear mapping CPa;: Ta;(M) X ... X Ta;(M) --+ 7Ti1(x) for each x E M, a tensorial form ep of degree r of type (p, V) on P can be defined by

where x = 7T(U). In particular, a tensorial O-form of type (p, V), that is, a functionf: P --+ V such thatf (ua) = p(a-1)f (u), can be identified with a cross section M --+ E. A few special cases of Example 5.2 will be used in Chapter III. Let r be a connection in P(M, G). Let G u and Qu be the vertical and the horizontal subspaces of Tu(P), respectively. Let h: T u (P) --+ Qu be the projection. PROPOSITION

If

5.1.

ep is a pseudotensorial r-form on P of type

(p, V), then (a) -The form eph defined by (eph) (X b . . . ,Xr ) = ep(hXb . . • , hXr ), Xi E T u (P), is a tensorial form of type (p, V); (b) dep is a pseudotensorial (r + 1) :form of type (p, V); (c) The (r + 1)-form Dep defined by Dep = (dep)h is a tensorialform of type (p, V). Proof. From R a h = h R a , a E G, it follows that eph is a pseudotensorial form of type (p, V), It is evident that 0

0

II.

77

THEORY OF CONNECTIONS

if one of X/s is vertical. (b) follows from R; (c) follows from (a) and (b).

0

d = d R;, a E G. 0

QED.

The form Dcp = (dcp) h is called the exterior covariant derivative of cp and D is called exterior covariant differentiation. If p is the adjoint representation of G in the Lie algebra g, a (pseudo) tensorial form of type (p, g) is said to be of type ad G. The connection form w is a pseudotensorial I-form of type ad G. By Proposition 5.1, Dw is a tensorial 2-form of type ad G and is called the curvature form of w. THEOREM

5.2 (Structure equation). Let

o its curvature form. dw(X, Y)

=

w

be a connection form and

Then

-i[w(X), w(Y)]

+ O(X,

Y) for X,Y

E

Tu(P),

u E P.

Proof. Every vector of P is a sum of a vertical vector and a horizontal vector. Since both sides of the above equality are bilinear and skew-symmetric in X and Y, it is sufficient to verify the equality in the following three special cases. (1) X and Yare horizontal. In this case, w (X) = w (Y) = 0 and the equality reduces to the definition of O. (2) X and Yare vertical. Let X = A * and Y = B* at u, where A,B E g. Here A* and B* are the fundamental vector fields corresponding to A and B respectively. By Proposition 3.11 of Chapter I, we have

2dw(A*, B*)

A*(w(B*)) - B*(w(A*)) - w([A*, B*]) = -[A, B] = -[w(A*), w(B*)],

=

since w(A*) = A, w(B*) = Band [A*, B*] = [A, B]*. On the other hand, O(A*, B*) = O. (3) X is horizontal and Y is vertical. We extend X to a horizontal vector field on P, which will be also denoted by X. Let Y = A * at u, where A E g. Since the right hand side of the equality vanishes, it is sufficient to show that dw(X, A*) = O. By Proposition 3.11 of Chapter I, we have

2dw (X, A *)

= =

X( w(A *)) - A * (w(X)) - w([X, A *]) - w ([X, A *]) .

Now it is sufficient to prove the following

78

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

If A *

is the fundamental vector field corresponding to an element A € 9 and X is a horizontal vector field, then [X, A *] is horizontal. Proof of Lemma. The fundamental vector field A * is induced by Rat' where at is the I-parameter subgroup of G generated by A € g. By Proposition 1.9 of Chapter I, we have LEMMA.

[X, A*] = lim ~ [R a (X) - X]. t--+O t t If X is horizontal, so is Rat(X). Thus [X, A*] is horizontal. COROLLARY

5.3.

If both

QED.

X and Yare horizontal vector fields on P,

then w([X, Y]) = -20(X, Y). Proof. Apply Proposition 1.9 of Chapter I to the left hand side of the structure equation just proved. QED. The structure equation (often called "the structure equation of E. Cartan") is sometimes written, for the sake of simplicity, as follows: dw = -lew, w] + O. Let el' . . . , er be a basis for the Lie algebra 9 and Cjb i, j, k = 1, ... , r, the structure constants of 9 with respect to e1 , • . • , en that is, j, k = 1, ... , r. Let w = ~i wie i and 0 = ~i Oi ei . Then the structure equation can be expressed as follows:

i

=

1, ... , r.

5.4 (Bianchi's identity). DO = O. Proof. By the definition of D, it suffices to prove that dO( X, Y, Z) = 0 whenever X, Y, and Z are all horizontal vectors. We apply the exterior differentiation d to the structure equation. Then o = ddw i = -i~ Cjk dw i A wk + !~ Cjkwi A dw k + dOi. THEOREM

Since wi(X) = 0 whenever X is horizontal, we have dOi(X, Y, Z) = 0 whenever X, Y, and Z are all horizontal.

QED.

II.

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THEORY OF CONNECTIONS

5.5. Let w be a connection form and cp a tensorial 1form of type ad G. Then PROPOSITION

Dcp(X, Y)

=

dcp(X, Y)

+ t[cp(X), w(Y)] + t[w(X), cp(Y)] for X,Y E Tu(P),

u E P.

Proof. As in the proof of Theorem 5.2, it suffices to consider the three special cases. The only non-trivial case is the case where X is vertical and Y is horizontal. Let X = A * at u, where A E g. We extend Y to a horizontal vector field on P, denoted also by Y, which is invariant by R a , a E G. (We first extend the vector 'TT Y to a vector field on M and then lift it to a horizontal vector field on P.) Then [A*, Y] = O. As A* is vertical, Dcp(A*, Y) = O. We shall show that the right hand side of the equality vanishes. By Proposition 3.11 of Chapter I, we have dcp(A*, Y) = t(A*(cp(Y)) - Y(cp(A*)) - cp([A*, Y]) = tA*(cp(Y)), so that it suffices to show A * (cp( Y)) + [w(A *), cp( Y)] = 0 or A*(cp(Y)) = -[A, cp(Y)]. If at denotes the I-parameter subgroup of G generated by A, then I

A~(cp( Y))

= lim! [CPuat( Y) - CPu( Y)] t--+O t

=

lim! [(Rdtcp) u( Y) - CPu( Y)] t--+O

t

= lim! [ad (a t- 1 ) ( cp u(Y)) - cp u(Y)] = - [A, cpu(Y) ] , t--+O t since Y is invariant by Rat. QED.

6. Mappings of connections In §5 of Chapter I, we considered certain mappings of one principal fibre bundle into another such as a homomorphism, an injection, and a bundle map. We now study the effects of these mappings on connections.

Let f: P' (M', G') -4- P (M, G) be a homomorphism with the corresponding homomorphism f: G' -4- G such that the induced mapping f: M' -4- M is a diffeomorphism of M' onto M. Let r' be a connection in P', w' the connectionform and 0' the curvature form of r'. Then (a) There is a unique connection r in P such that the horizontal subspaces qf r' are mapped into horizontal subspaces of r by f PROPOSITION

6.1.

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FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(b) If wand 0 are the connection form and the curvature form of r respectively, then f*w = f· w' and f*O = f· 0', where f· w' or f· 0' means the g' -valuedform on P' defined by (f· w') (X') = f (w' (X')) or (f· 0') (X', Y') = f (0' (X', Y')), where f on the right hand side is the homomorphism g' ~ 9 induced by f: G' ~ G. (c) If u' E P' and u = f (u') E P, then f: G' ~ G maps
the mappings 7T': Qu' ~ T~/(M') and f: Tx,(M') -+- Tx(M) are linear isomorphisms and hence the remaining two mappings must be also linear isomorphisms. The uniqueness of r is evident from its construction. (b) The equality f* w = f . w' can be rewritten as follows:

w(fX')

=

f(w'(X'))

for X'

E

Tu'(P'),

u' E P'.

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81

THEORY OF CONNECTIONS

It is sufficient to verify the above equality in the two special cases: (1) X' is horizontal, and (2) X' is vertical. Since f: P' ~ P maps every horizontal vector into a horizontal vector, both sides of the equality vanish if X' is horizontal. If X' is vertical, X' = A'* at u', where A' E g'. Set A = f (A') E g. Since f (u' a') = f (u')f (a') for every a' E G', we havef (X') = A* atf (u'). Thus

w(fX') = w(A*) = A =f(A') =f(w'(A'*)) =f(w'(X')). From f*w = f· w', we obtain d(f*w) = d(f· w') and f*dw f· dw'. By the structure equation (Theorem 5.2) : -if*([w, wJ) + f*O = -!f([w', w'J) + f· 0 /,

=

we have

-![f*w,f*wJ + f*O = -![f· w',f·

w'J

+ f· 0'.

This implies thatf*O = f· 0'. (c) Let T be a loop at x = 7T(U). Set T' =f-1(T) so that T' is a loop at x' = 7T' (u'). Let T'* be the horizontal lift of T' starting from u'. Thenf (T'*) is the horizontal lift of T starting from u. The statement (c) is now evident. QED. In the situation as in Proposition 6.1, we say that f maps the connection r' into the connection r. In particular, in the case where P'(M', G') is a reduced subbundle of P(M, G) with .injection f so that M' = M and f: M' ~ M is the identity transformation, we say that the connection r in P is reducible to the connection r' in P'. An automorphismf of the bundle P(M, G) is called an automorphism of a connection r in P ifit maps r into r, and in this case, r is said to be invariant by j.

6.2. Let f: P'(M', G') ~ P(M, G) be a homomorphism such that the corresponding homomorphism f: G' ~ G maps G' isomorphically onto G. Let r be a connection in P, w the connection form and 0 the curvature form of r. Then (a) There is a unique connection r' in P' such that the horizontal subspaces of r' are mapped into horizontal subspaces of r by f (b) If w' and 0' are the connection form and the curvature form of r' respectively, thenf*w = f· w' andf*O = f· 0'. (c) If u' E P' and u = f (u') E P, then the isomorphism f: G' ~ G maps (u') into (u) and <1>0 (u') into <1>0 (u) . PROPOSITION

82

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. We define r' by defining its connection form w'. Set w' = f- 1 ·f*w, wheref-1: 9 ---+ g' is the inverse of the isomorphism f: g' ---+ 9 induced from f: G' ---+ G. Let X' € Tu'(P') and a' € G' and set X = fX' and a = f (a'). Then we have

w'(Ra,X')

f-1(W(f(R a,X'))) = f-1(W(R aX)) = f-1( ad (a- 1)(w(X))) = ad (a'-1) (f-1( w(X))) = ad (a'-1)(w(X')).

=

Let A' € g' and set A = f(A'). Let A* and A'* denote the fundamental vector fields corresponding to A and A' respectively. Then we have w'(A'*) =f-1(W(A*)) =f-1(A) = A'. This proves that the form w' defines a connection (Proposition 1.1). The verification of other statements is similar to the proof of Proposition 6.1 and is left to the reader. QED. In the situation as in Proposition 6.2, we say that r' is induced by ffrom r. Iffis a bundle map, that is, G' = G andf: G' ---+ Gis the identity automorphism, then w' = f*w. In particular, given a bundle P(M, G) and a mapping f: M' ---+ M, every connection in P induces a connection in the induced bundlef-1P. For any principal fibre bundles P (M, G) and Q (M, H), P X Qis a principal fibre bundle over M X M with group G X H. Let P + Q be the restriction of P X Q to the diagonal ~M of M X M. Since ~M and M are diffeomorphic with each other in a natural way, we consider P + Q as a principal fibre bundle over M with group G X H. The restriction of the projection P X Q ---+ P to P + Q, denoted by fp, is a homomorphism with the corresponding natural homomorphism fa: G X H ---+ G. Similarly, for fQ: P + Q ---+ Q andfH: G X H ---+ H.

Let r p and r Q be connections in P(M, G) and Q(M, H) respectively. Then (a) There is a unique connection r in P + Q such that the homomorphisms fp: P + Q ---+ P and f Q: P + Q ---+ Q maps r into r p and r Q respectively. (b) If w, wp and wQ are the connectionforms and 0, Op, and OQ are the curvature forms of r, r p, and r Q respectively, then PROPOSITION

6.3.

w =f~wp

+ fQwQ'

°

=f~Op

+ f QO Q.

II.

THEORY OF CONNECTIONS

83

(c) Let u E P, V E Q, and (u, v) E P + Q. Then the holonomy group <1>(u, v) of r (resp. the restricted holonomy group <1>°(u, v) of r) is a subgroup of <1>(u) X <1>(v) (resp. <1>°(u) X <1>°(v)). The homomorphism fa: G X H --+ G (resp. fH: G X H --+ H) maps <1>(u, v) onto <1>(u) (resp. onto <1> (v)) and <1>0 (u, v) onto <1>0 (u) (resp. onto <1>0 (v) ), where (u)and O(u) (resp. <1>(v) and O(v))are the holonomy group and the restricted holonomy group of r p (resp. r Q)' The proof is similar to those of Propositions 6.1 and 6.2 and is left to the reader. 6.4. Let Q(M, H) be a subbundle of P(M, G), where H is a Lie subgroup of G. Assume that the Lie algebra 9 of G admits a subspace m such that 9 = m + ~ (direct sum) and ad (H) (m) m, where ~ is the Lie algebra of H. For every connection form w in P, the ~­ component w' of w restricted to Q is a connection form in Q. Proof. Let A E ~ and A * the fundamental vector field corresponding to A. Then w' (A *) is the ~-component of w(A *) = A. Hence, w'(A*) A. Let cp be the m-component of w restricted to Q. Let X E Tv(Q) and a E H. Then w(RaX) - W' (RaX) + cp(RaX) , ad (a-1)(w(X)) ad (a-1)(w'(X)) + ad (a-I) (cp(X)). The left-hand sides of the preceding two equalities coincide. Comparing the ~-components of the right hand sides, we obtain w'(RaX) - ad (a-I) (w'(X)). Observe that we used the fact that ad (a-I) (cp(X)) is in m. QED. PROPOSITION

Remark. The connection defined by w in P is reducible to a connection in the subbundle Q if and only if the restriction of w to Q is ~-valued. Under the assumption in Proposition 6.4, this means w' = w on Q.

7. Reduction theorem Unless otherwise stated, a curve will mean a piecewise differentiable curve of class Coo. The holonomy group oo(uo) will be denoted by (uo) . We first establish

7.1 (Reduction theorem). Let P( M, G) be a principal fibre bundle with a connection r, where M is connected and paracompact. Let Uo be an arbitrary point of P. Denote by P(uo) the set of points in P THEOREM

84

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

which can be joined to Uo by a horizontal curve. Then ( 1) P(uo) is a reduced bundle with structure group
1. Let Q be a subset of P( M, G) and H a Lie subgroup of G. Assume: (1) the projection 1T: P --+ M maps Q onto M,o (2) Q is stable by H, i.e., R a ( Q) = Q for each a E H,o (3) if u,v E Q and 1T(U) = 1T(V), then there is an element a E H such that v = ua; and (4) every point x of M has a neighborhood U and a cross section a: U --+ P such that a( U) c Q. Then Q(M, H) is a reduced subbundle of P(M, G). Proof of Lemma 1. For each u E 1T-1( U), let x = 1T(U) and a E G the element determined by u = a( x) a. Define an isomorphism 1p: 1T- 1 ( U) --+ U X G by setting 1p(u) = (x, a). It is easy to see that 1p maps Q n 1T-1 ( U) 1: 1 onto U x H. Introduce a differentiable structure in Q in such a way that 'lfJ: Q n 1T-1( U) --+ U X H becomes a diffeomorphism; using Proposition 1.3 of Chapter I as in the proof of Proposition 5.3 of Chapter I, we see that Q becomes a differentiable manifold. It is now evident that Q is a principal fibre bundle over M with group H and that Q is a subbundle of P. Going back to the proof of the first assertion of Theorem 7.1, we see that, M being paracompact, the holonomy group
Let Q(M, H) be a subbundle of P( M, G) and r a connection in P. Ij,for every u E Q, the horizontal subspace of Tu(P) is tangent to Q, then r is reducible to a connection in Q. LEMMA

2.

II.

85

THEORY OF CONNECTIONS

Proof of Lemma 2. We define a connection r' in Q as follows. The horizontal subspace of Tu(Q), u E Q, with respect to r' is by definition the horizontal subspace of Tu(P) with respect to r. It is obvious that r is reducible to r'. QED. We shall call P(u) the holonomy bundle through u. It is evident that P(u) = P(v) if and only if u and v can be joined by a horizontal curve. Since the relation "-' introduced in §4 (u "-' v if u and v can be joined by a horizontal curve) is an equivalence relation, we have, for every pair of points u and v of P, either P(u) = P(v) or P(u) n P(v) = empty. In other words, P is decomposed into the disjoint union of the holonomy bundles. Since every a E G maps each horizontal curve into a horizontal curve, Ra(P(u)) = P(ua) and R a: P(u) ~ P(ua) is an isomorphism with the corresponding isomorphism ad (a-I): (u) ~ (ua) of the structure groups. It is easy to see that, given any u and v, there is an element a E G such that P(v) = P(ua). Thus the holonomy bundles P(u), u E P, are all isomorphic with each other. Using Theorem 7.1, we prove that the holonomy groups k(U), 1 < k < 00, coincide as was pointed out in Remark of § 4. This result is due to Nomizu and Ozeki [2J. THEOREM

7.2.

All the holonomy groups k(U) , 1 < k <

00,

coincide. Proof. It is sufficient to show that <1>1 (u) = co (u). We denote co (u) by (u) and the holonomy bundle through u by P(u). We know by Theorem 7.1 that P(u) is a subbundle of P with (u) as its structure group. Define a distribution Son P by setting

Su

=

Tu(P(u))

for u E P.

Since the holonomy bundles have the same dimension, say k, S is a k-dimensional distribution. We first prove LEMMA

1.

( 1) S is differentiable and involutive.

(2) For each u E P, P(u) is the maximal integral manifold through u.

of S

Proof of Lemma 1. (1) We set U E

P,

where S~ is horizontal and S~ is vertical. The distribution S' is differentiable by the very definition of a connection. To prove the

86

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

differentiability of S, it suffices to show that of S". For each u E P, let U be a neighborhood of x = 7T(U) with a cross section 0': U -+ P(u) such that a(x) = u. (Such a cross section was constructed in the proof of Theorem 7. I.) Let A l' . . . , A r be a basis of the Lie algebra g(u) of
(Ai)v = (ad (a-I)(A i)):, i = 1, ... , r, where (ad (a-I) (Ai)) * is the fundamental vector field on P corresponding to ad (a-I) (Ai) E g(v) c g, i = 1, ... , r. It is easy to see that AI, ... , Ar are differentiable and form a basis of S" on U). For each point u, P(u) is an integral manifold of S, since for every VEP(U), we have Tv(P(u)) = Tv(P(v)) =Sv. This implies that Sis involutive. (2) Let W(u) be the maximal integral manifold of S through u (cf. Proposition 1.2 of Chapter I). Then P(u) is an open submanifold of W(u). We prove that P(u) = W(u). Let v be an arbitrary point of W(u) and let u(t), 0 ~ t ~ 1, be a curve in W(u) such that u(O) = u and u( 1) = v. Let t l be the supremum of to such that 0 < t < to implies u(t) E P(u). Since P(u) is open in W(u), t l is positive. We show that U(tI) lies in P(u); since P(u) is open in W(u), this will imply that t l = 1, proving that u(l) = v lies in P(u). The point u(t I) is in P(u(t I)) and P(u(t I)) is open in W(u( t I)). There exists e > 0 such that t l - e < t < t l + e implies u(t) E P(u(t I)). Let t be any value such that t l - e < t < t i . By definition of t I, we have u(t) E P(u). On the other hand, u(t) E P(u(t I)). This implies that P(u) = P(U(tI)) so that u(t I) E P(u) as we wanted to show. We have thereby proved that P(u) is actually the maximal integral manifold of S through u. 7T- I (

Let S be an involutive, Coo -distribution on a Coo -manifold. Suppose Xt, 0 < t < 1, is a piecewise CI-curve whose tangent vectors xt belong to S. Then the entire curve X t lies in the maximal integral manifold W of S through the point Xo. LEMMA

2.

Proof of Lemma 2. We may assume that X t is a CI-curve. Take a local coordinate system Xl, . . . , xn around the point X o such that

II.

THEORY OF CONNECTIONS

87

oj ax!, ... , oj oxk, k = dim S, form a local basis for S (cf. Chevalley [1, p. 92J). For small values of t, say, 0 < t S 8, X t can be expressed by Xi = xi(t), 1 < i < n, and its tangent vectors are given by l;i (dxijdt) (oj ox t ). By assumption, we have dxijdt = 0 for k + 1 < i < n. Thus, xi(t) = xi(O) for k + 1 < i < n so that x t , 0 S t < 8, lies in the slice through X o and hence in W. The standard continuation argument concludes the proof of Lemma 2. We are now in position to complete the proof of Theorem 7.2. Let a be any element of l(U). This means that u apd ua can be joined by a piecewise Cl-horizontal curve u t , 0 < t < 1, in P. The tangent vector itt at each point obviously lies in Sue. By Lemma 2, the entire curve U t lies in the maximal integral manifold W(u) of S through u. By Lemma 1, the entire curve Ut lies in P(u). In particular, ua is a point of P(u). Since P(u) is a subbundle with structure group (u), a belongs to (u). QED. COROLLARY

7.3.

The restricted holonomy groups g(u), 1 < k <

coincide. Proof. g(u) is the connected component of the identity of k(U) for every k (cf. Theorem 4.2 and its proof). Now, Corollary 7.3 follows from Theorem 7.2. QED. 00,

Remark. In the case where P(M, G) is a real analytic principal bundle with an analytic connection, we can still define the holonomy group w(u) by using only piecewise analytic horizontal curves. The argument used in proving Theorem 7.2 and Corollary 7.3 shows that w(u) = l(U) and ~(u) = ~(u). Given a connection r in a principal fibre bundle P( M, G), we shall define the notion ofparallel displacement in the associated fibre bundle E(M, F, G, P) with standard fibre F. For each wEE, the horizontal subspace Qwand the vertical subspace F w of Tw(E) are defined as follows. The vertical subspace F w is by definition the tangent space to the fibre of E at w. To define Qw, we recall that we have the natural projection P X F -+ E = P X G F. Choose a point (u, ~) E P X F which is mapped into w. We fix this ~ E F and consider the mapping P -+ E which maps v E P into v~ E E. Then the horizontal subspace Qw is, by definition, the image of the horizontal subspace Que Tu(P) by this mapping P -+ E. We see easily that Qw is independent of the choice of

88

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(u, ~) E P X F. We leave to the reader the proof that Tw(E) F w + Qw (direct sum). A curve in E is horizontal if its tangent vector is horizontal at each point. Given a curve 7 in M, a (horizontal) lijt 7* of 7 is a horizontal curve in E such that 7TE( 7*) = 7. Given a curve 7 = Xt, 0 < t < 1, and a point W o such that 7T E( w o) = X O, there is a unique lift 7* - W t starting from WOo To prove the existence of 7*, we choose a point (u o, ~) in P X F such that uo~ = WOo Let U t be the lift of 7 = X t starting from UO' Then W t = Ut~ is a lift of 7 starting from WOo The uniqueness of 7* reduces to the uniqueness of a solution of a system of ordinary linear differential equations satisfYing a given initial condition just as in the case of a lift in a principal fibre bundle. A cross section a of E defined on an open subset U of M -.is called parallel if the image of T ro ( M) by a is horizontal for each x E U, that is, for any curve 7 = Xt, 0 < t 1, the parallel displacement of a(xo) along 7 gives a(x 1 ). 7.4. Let P(M, G) be a principal fibre bundle and E(M, GjH, G, P) the associated bundle with standard fibre GjH, where H is a closed subgroup oiG. Let a: M -+ E be a cross section and Q(M, H) the reduced subbundle qf P(M, G) corresponding to a (if. Proposition 5.6 qfChapter I). Then a connection r in P is reducible to a connection r' in Q if and only if a is parallel with respect to r. Proof. If we identify E with PjH (cf. Proposition 5.5 of Chapter I), then a( M) coincides with the image of Q by the natural projection p: P -+ E PjH; in other words, if u E Q and x = 7T(U) , then a(x) p(u) (cf. Proposition 5.6 of Chapter I). Suppose r is reducible to a connection r' in Q. We note that if ~ is the origin (i.e., the coset H) of GjH, then u~ = p(u) for every u E P and hence if u t , 0 t 1, is horizontal in P, so is p(u t ) in E. Given a curve Xt, 0 t 1, in M, choose Uo E Q with 7T(UO) = X o so that a(x o) p(uo). Let U t be the lift to P of X t starting from Uo (with respect to r), so that p( u t ) is the lift of x t to E starting from a(x o)' Since r is reducible to r', we have U t E Q and hence p(u t ) a(x t ) for all t. Conversely, assume that a is parallel (with respect to r). Given any curve x t, 0 < t ::;: 1, in M and any point U o of Q with 7T( uo) = X O, let U t be the lift of X t to P starting from Uo. Since a is parallel, p(u t ) = a(x t ) and hence U t E Q for all t. This shows that every horizontal vector at Uo E Q (with respect to r) is PROPOSITION

II.

THEORY OF CONNECTIONS

tangent to Q. By Proposition 7.2, in Q.

r

89

is reducible to a connection QED.

8. Holonomy theorem We first prove the following result of Ambrose and Singer [1] by applying Theorem 7.1.

8.1. Let P(M, G) be a principal fibre bundle, where M is connected and paracompact. Let r be a connection in P, Q the curvature form, (u) the holonomy group with reference point u E P and P(u) the holonomy bundle through u of r. Then the Lie algebra of (u) is equal to the subspace of g, Lie algebra of G, spanned by all elements of the form Qv(X, Y), where v E P(u) and X and Yare arbitrary horizontal vectors at THEOREM

v.

Proof. By virtue of Theorem 7.1, we may assume that P(u) = P, i.e., (u) = G. Let g' be the subspace of 9 spanned by all elements of the form Qv(X, Y), where v E P(u) = P and X and Y are arbitrary horizontal vectors at v. The subspace g' is actually an ideal of g, because Q is a tensorial form of type ad G (cf. §5) and hence g' is invariant by ad G. We shall prove that g' = g. At each point v E P, let Sv be the subspace of Tv(P) spanned by the horizontal subspace Qv and by the subspace g~ = {A:; A E g'}, where A * is the fundamental vector field on P corresponding to A. The distribution S has dimension n + r, where n = dim M and r = dim g'. We shall prove that S is differentiable and involutive. Let v be an arbitrary point of P and U a coordinate neighborhood ofy = 7T(V) E M such that 7T- I ( U) is isomorphic with U x G. Let Xl' ... ,Xn be differentiable vector fields on U which are linearly independent everywhere on U and Xi, ... , X~ the horizontal lifts of Xl' ... , Xn- Let AI, ... , A r be a basis for g' and Ai, ... ,Ai the corresponding fundamental vector fields. I t is clear that Xi, ... ,X~, Ai, . .. ,Ai form a local basis for S. To prove that S is involutive, it suffices to verify that the bracket of any two of these vector fields belongs to S. This is clear for [At, AI], since [Ai, A j ] E g' and [Ai, A j ]* = [Aj, Aj]. By the lemma for Theorem 5.2, [At, Xj] is horizontal; actually, [At, Xj] = 0 as X j* is invariant by R a for each a E G. Finally, set A = w([Xt, Xj]) E g, where w is the connection form of r. By Corollary 5.3, A = w([Xt, Xj]) = -2.4(Xt, X j*) E g'. Since the vertical component of [Xt, X J*] at v E P is equal to A: E S v'

90

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

[Xi, Xj] belongs to S. This proves our assertion that S is involutive. Let Po be the maximal integral manifold of S through u. By Lemma 2 in the proof of Theorem 7.2, we have Po = P. Therefore,

dim 9 = dim P - n

This implies 9

=

=

dim Po - n

=

dim g'.

QED.

g'.

Next we prove

8.2. Let P(M, G) be a principal fibre bundle, where P is connected and M is paracompact. If dim M > 2, there exists a connection in P such that all the holonomy bundles P(u), U € P, coincide with P. Proof. Let Uo be an arbitrary point of P and xl, ... , x n a local coordinate system with origin X o = 7T(U O). Let U and V be neighborhoods of X o defined by Ixtl < l.I.. and Ixil < fJ respectively, where 0 < fJ < l.I... Taking l.I.. sufficiently small, we may assume that PI U = 7T-1 ( U) is isomorphic with the trivial bundle U X G. We shall construct a connection r' in P I U such that the holonomy group of the bundle P I V coincides with the identity component of G. We shall then extend r' to a connection r in P in such a way that r coincides with r' on P I V (cf. Theorem 2.1). Let AI, ... , A r be a basis for the Lie algebra 9 ofG. Choose real numbers l.I..l' • • • , l.I.. r such that 0 < l.I.. 1 < l.I.. 2 < ... < l.I.. r < fJ and letJ:(t), i = 1, ... , r, be differentiable functions in -l.I.. - 8 < t < l.I.. + 8 such that.fi(O) = 0 for every i andJ:( l.I.. j ) = ~ij (Kronecker's symbol). On 7T- 1 ( U) = U X G, we can define a connection form W by requiring that THEOREM

r

w(x,e}(

aj ax

1

)

=

!

fj(x 2 )A j

j=l

and that W(x,e}(

aj ax i ) = 0

for i

=

2, 3, ... , n.

(Note that, by virtue of the property R:w = ad (a-I) (w), the preceding conditions determine the values of W at every point (x, a) of U X G.) Fixing t, 0 < t < fJ, and l.I..k' 1 < k < r, for the moment, consider the rectangle on the x1x2 -plane in V formed by the line segments '7"1 from (0, 0) to (0, l.I..k), '7"2 from (0, l.I.. k ) to (t, l.I.. k ), '7"3 from (t, l.I.. k ) to (t,O) and '7"4 from (t,O) to (0,0). (Here and in the

II.

91

THEORY OF CONNECTIONS

following argument, the x3 to xn-coordinates of all the points remain oand are hence omitted.) In 17-1 ( V) = V X G, we determine the horizontal lift of T = T4 . T3 . T2 . T1 starting from the point (0,0; e). The lift Ti ofT1 starting from (0,0; e) is clearly (0, s; e), o < s ~ Cl. k, since its tangent vectors oj ox 2 are horizontal. The lift Tl of T2 starting from the end point (0, Cl. k; e) of Ti is of the form (s, Cl..k; cs ), 0 < s < t, where Cs is a suitable curve with Co = e in G. Its tangent vector is of the form (oj ox 1 ) (S,Cl.k) + cs. By a similar computation to that for Proposition 3.1, we have w((ojox 1)(S,Cl.k) + cs) = ad (c - 1)w((ojox1))(S,Cl.k;e) + c- 1 . C s

= ad (c s- 1)

s

s

(.f ji(Cl..k)A i) + c 1 . Cs = ad (c;1)A k + c 1 • cs· s-

s-

)=1

Therefore we have Cs • cs- 1 = -A k, that is, Cs = exp (-sA k). The end point of T~ is hence (t,Cl..k;eXp (-tA k)). The lift T: of T3 starting from (t, Cl. k ; exp (-tA k)) is (t, Cl..k - s; exp (-tA k)), 0 < s ~ Cl. k. Finally, the lift T4 of T4 starting from the end point (t,O; exp (-tA k)) of T: is (t - s, 0; exp (-tA k)), 0 < s < t, since oj ox1 is horizontal at the points with x2 = O. This shows that the end point of the lift T* of T is (0, 0; exp (-tA k )), proving that exp (-tA k ) is an element of the holonomy group of 17-1 ( V) with reference point (0, 0; e). Since this is the case for every t, we see that A k is in the Lie algebra of the holonomy group. The result being valid for any A k , we see that the holonomy group of the connection in 17-1 ( V) coincides with the identity component of G. Let r be a connection in P which coincides with r' on 17-1 ( V) . Since the holonomy group (u o) of r obviously contains the identity component of G, the holonomy bundle P(u o) of r has the same dimension as P and hence is open in P. Since P is a disjoint union ofholonomy bundles each of which is open, the connectedness of P implies that P = P(u o). QED. COROLLARY 8.3. Any connected Lie group G can be realized as the hplonomy group oj a certain connection in a trivial bundle P = M X G, where M is an arbitrary differentiable manifold with dim M > 2.

.

Theorem 8.2 was proved for linear connections by Hano and Ozeki [lJ and then in the general case by Nomizu [5J, both by making use of Theorem 8.1. The above proof which is more direct is due to E. Ruh (unpublished).

92

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

9. Flat connections Let P = M X G be a trivial principal fibre bundle. For each a E G, the set }lVf X {a} is a submanifold of P. In particular, M X {e} is a subbundle of P, where e is the identity of G. The canonical flat connection in P is defined by taking the tangent space to M X {a} at u = (x, a) E M X G as the horizontal subspace at u. In other words, a connection in P is the canonical flat connection if and only if it is reducible to a unique connection in M X {e}. Let fJ be the canonical I-form on G (cf. §4 of Chapter I). Let f: M X G ---+ G be the natural projection and set w = f*fJ.

I t is easy to verify that w is the connection form of the canonical flat connection in P. The Maurer-Cartan equation of fJ implies that the canonical flat connection has zero curvature: dw

=

d(f*fJ)

=

=

f*(dfJ)

=

f*( -![fJ, fJ])

-![f*fJ,f*fJ] = -t[w, w].

A connection in any principal fibre bundle P(M, G) is called flat if every point x of M has a neighborhood U such that the induced connection in P U = 7T- 1 ( U) is isomorphic with the canonical flat connection in U X G. More precisely, there is an isomorphism 'ljJ: 7T-1 ( U) ---+ U X G which maps the horizontal subspace at each u E 7T- 1 ( U) upon the horizontal subspace at 'ljJ(u) of the canonical flat connection in U X G.

I

9.1. A connection in P(M, G) is flat if and only if the curvature form vanishes identically. Proof. The necessity is obvious. Assume that the curvature form vanishes identically. For each point x of M, let U be a simply connected open neighborhood of x and consider the induced connection in P I U = 7T- 1 ( U). By Theorems 4.2 and 8.1, the holonomy group of the induced connection in P I U consists of the identity only. Applying the Reduttion Theorem (Theorem 7.1), we see that the induced connection in P I U is isomorphic with the canonical flat connection in U X G. QED. THEOREM

9.2. Let r be a connection in P(M, G) such that the curvature vanishes identically. If M is paracompact and simply connected, COROLLARY

II.

93

THEORY OF CONNECTIONS

then P is isomorphic with the trivial bundle M with the canonical fiat connection in M X G.

X

G and

r

is isomorphic

We shall study the case where M is not necessarily simply connected. Let r be a fiat connection in P( M, G), where M is connected and paracompact. Let Uo E P and M* = P(u o)' the holonomy bundle through U o ; M* is a principal fibre bundle over M whose structure group is the holonomy group (u o). Since (uo) is discrete by Theorems 4.2 and 8.1 and since M* is connected, M* is a covering space of M. Set X o = 1T( uo), X o E M. Every closed curve of M starting from X o defines, by means of the parallel displacement along it, an element of (u o). Since the restricted holonomy group is trivial by Theorems 4.2 and 8.1, any two closed curves at X o representing the same element of the first homotopy group 1Tl(M, xo) give rise to the same element of (u o) . Thus we obtain a homomorphism of 1T 1(M, xo) onto (u o). Let N be a normal subgroup of (u o) and set M' = M* IN. Then M' is a principal fibre bundle over M with structure group (u o)IN. In particular, M' is a covering space of M. Let P' (M', G) be the principal fibre bundle induced from P( M, G) by the covering projection M' ---+ M. Letf: P' ---+ P be the natural homomorphism (cf. Proposition 5.8 of Chapter I). 9.3. There exists a unique connection r' in P' (M', G) which is mapped into r by the homomorphism f: P' ---+ P. The connection r' is fiat. If u~ is a point of P' such that f (u~) = uo, then the holonomy group (u~) of r' with reference point u~ is isomorphically mapped onto N byf. Proof. The first statement is contained in Proposition 6.2. By the same proposition, the curvature form of r' vanishes identically and r' is fiat. We recall that P' is the subset of M' X P defined as follows (cf. Proposition 5.8 of Chapter I) : PROPOSITION

P'

=

{(x', u) EM' X P; f-l(x')

=

1T(U)},

where f-l: M' ---+ M is the covering projection. The projection 1T': P' ---+ M' is given by 1T' (x', u) = x' and the homomorphism f: P' ---+ P is given by f (x', u) = u so that the corresponding homomorphism f: G ---+ G of the structure groups is the identity automorphism. To prove thatf maps (u~) isomorphicallyonto

94

FOUNDATIONS OF DIFFERENTIAL GEOMETRY


N, it is therefore sufficient to prove u~ =

Since fl(X~)

N. Write

(x~,

uo) E P' c M' X P. = 7T(UO)' there exists an element a E
v(uoa) ,

where v: M* = P(u o) ---+ M' = P(uo)jNis the covering projection. Let 'T = 0 < t < 1, be a horizontal curve in P' such that 7T' (u~) = 7T' (u~). For each t, we set

u;,

u;

=

(x;, ut )

E

P'

c

M'

X

P.

Then the curve u t , 0 < t < 1, is horizontal in P and hence is contained in M* = P(u o). Since fl(X;) = 7T(U t ) = fl 0 v(u t ) and x~ = v(uoa), we have x; = v(uta) for 0 < t < 1. We have

v(u1a) =

x~ =

7T'(uD

=

7T'(U~) = x~

v(u 1)

=

v(u o),

= v(uoa)

and, consequently, which means that U 1 = uob for some bEN. This shows that
u;,

where x; = v(uta). Then u~ = u~b, showing that b E
QED.

10. Local and infinitesimal holonomy groups Let r be a connection in a principal fibre bundle P( M, G), where M is connected and paracompact. For every connected open subset U of M, let rube the connection in P I U = 7T- 1 ( U) induced from r. For each u E 7T- 1 ( U), we denote by
n

II.

95

THEORY OF CONNECTIONS

all connected open neighborhoods of the point x = 7T(U). If {Uk} is a sequence of connected open neighborhoods of x such that

n Uk = {x}, then we have obviously °(u, U 00

Uk

:::>

(jk+l and

1)

:::>

k=1 2 ) :::> • • • :::>

<1>0 (u, U <1>0 (u, Uk) :::> • • •• Since, for every open neighborhood U of x, there exists an integer k such that Uk C U,

n °(u, Uk). Since each group °(u, Uk) is a 00

we have <1>* (u) =

k=1

connected Lie subgroup of G (Theorem-4.2), it follows that dim °(u, Uk) is constant for sufficiently large k and hence that <1>* (u) = °(u, Uk) for such k. The following proposition is now obvious. PROPOSITION

10.1.

The local holonomy groups have the following

properties: (1) <1>* (u) is a connected Lie subgroup of G which is contained in the restricted holonomy group °(u); (2) Every point x = 7T( u) has a connected open neighborhood U such that *(u) = <1>0 (u, V) for any connected open neighborhood V of x contained in U; (3) If U is such a neighborhood of x = 7T(U), then *(u) :::> *(v) for every v E P(u, U); (4) For every a E G, we have *(ua) = ad (a-I) (<1>* (u)) ; (5) For every integer m, the set {7T(U) EM; dim *(u) < m} is open. As to (5), we remark that dim *(u) is constant on each fibre of P by (4) and thus can be considered as an integer valued function on M. Then (5) means that this integer valued function is upper semicontinuous.

10.2. Let g(u) and g*(u) be the Lie algebras of °(u) and *(u) respectively. Then °(u) is generated by all <1>* (v), v E P(u), and g(u) is spanned by all g*(v), v E P(u). Proof. If v E P(u), then °(u) = °(v) :::> *(v) and g(u) = g(v) :::> g*(v). By Theorem 8.1, g(u) is spanned by all elements of the form 0v(X*, Y*) where v E P(u) and x* and y* are horizontal vectors at v. Since 0v(X*, Y*) is contained in the Lie algebra of <1>0 (v, V) for every connected open neighborhood V of 7T(V) , it is contained in g* (v). Consequently, g(u) is spanned by all g* (v) THEOREM

96

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

where v E P(u). The first assertion now follows from the following len1ma. LEMMA. If the Lie algebra 9 of a connected Lie group G is generated by aJamily of subspaces {m;.}, then every element of G can be written as a product exp Xl . exp X 2 • ••• • exp X k' where each Xi is contained zn some m A• Proof of Lemma. The set H of all elements of G of the above form is clearly a subgroup which is arcwise connected; indeed, every element of H can be joined to the identity by a differentiable curve which lies in H. By the theorem of Freudenthal-KuranishiYamabe (proved in Appendix 4), H is a connected Lie subgroup of G. Its Lie algebra contains all mA and thus coincides with g. Hence, H = G. QED. THEOREM 10.3. If dim <1>* (u) is constant on P, then °(u) = <1>* (u) Jor every u in P. PROOF. By (3) of Proposition 10.1, x = 7T(U) has an open neighborhood Usuch that *(u) ~ *(v) for each v in P(u, U). Since dim *(u) = dim *(v), we have *(u) = *(v). By the standard continuation argument, we see that, if v E P(u), then *(u) = <1>* (v). By Theorem 10.2, we have °(u) = *(u). QED. We now define the infinitesimal holonomy group at each point u of P by means of the curvature form and study its relationship to the local holonomy group. We first define a series of subspaces mk(u) of 9 by induction on k. Let mo(u) be the subspace of 9 spanned by all elements of the form .Qu(X, Y), where X and Yare horizontal vectors at u. We consider a g-valued functionJ on P of the form J = Vk • • • VI (.Q (X, Y)), where X, Y, Vb ... , Vk are arbitrary horizontal vector fields on P. Let mk(u) be the subspace of 9 spanned by mk-l(u) and by the values at u of all functionsJofthe form (I k ). We then set g'(u) to be the union of all mk(u), k = 0, 1; 2, .... PROPOSITION 10.4. The subspace g' (u) of 9 is a subalgebra of g* (u). The connected Lie subgroup '(u) of G generated by g'(u) is called the infinitesimal holonomy group at u.

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THEORY OF CONNECTIONS

97

Proof. We show that mk(u) c g* (uj by induction on k. The case k = 0 is obvious. Assume that m k - 1 (u) c g* (u) for every point u. It is sufficient to show that, for every horizontal vector field X and for every function j of the form (Ik - 1 ) , we have Xuj E g*(u). Let ut, ItI < c for some c > 0, be the integral curve of X with Uo = u. Since U t is horizontal, we have g* (u t) C g* (u) by (3) of Proposition 10.1. Therefore,j (u t ) E mk-l(u t) c g*(u t ) C

g*(u). On the other hand, Xuj

=

lim ~ [j (u t) - j (u)] so that

t Xujis in g*(u). Consequently, g'(u) is contained in g*(u). To prove that g'(u) is a subalgebra of g, we need the following two lemmas. t-O

1. Let j be a g-valued junction of type ad G on P. Then (1) ForanyvectorfieldXonP, wehavev(X)u .j= -[wu(X),j(u)], where v(X) denotes the vertical component of X. (2) For any horizontal vector fields X and Y on P, we have LEMMA

v([X, Y]J . j = 2[Ou(X, Y) ,j (u)]. (3)

If X

and Yare vector fields on P which are invariant by all Ra , a E G, then 0 (X, Y) and Xj are junctions of type ad G. Proof of Lemma 1. (1) Let A = wu(X) E 9 and at = exp tAo Then v(X)u .j =

A~j

==;

lim~ [j(ua t) t-O

=

t

-j(u)]

lim ~ [ad (at-I) (j (u)) - j (u) ] t-O

t

= -[A,j(u)]

=

-[wu(X),j(u)].

(2) By virtue of the structure equation (Theorem 5.2), we have 20 u(X, Y) = 2(dw)u(X, Y) = Xu (w(Y)) - Y u(w(X)) - wu([X, Y]) = - w u ([X, Y]). Replacing X by [X, Y] in (1), we obtain (2). (3) Since 0 is of type ad G (cf. §5 of Chapter II), we have

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FOUNDATIONS OF DIFFERENTIAL GEOMETRY

which shows that Q(X, Y) is of type ad G, if X RaY. We have also

=

RaX and Y =

(Xf)ua = Xuaf = (RaXu)f = Xu(f R a) = ad (a-I) (Xuf) = ad (a-I) (Xf)w 0

iff is of type ad G and X is invariant by R a • This completes the proof of Lemma 1. Let Xi = oj ox i, where xl, ... , xn is a local coordinate system in a neighborhood U of x = 7T(U). Let Xt be the horizontal lift of Xi. Consider a g-valued functionf of the form

(Ilk)

f

where i, l,il' ... LEMMA

Xj: ... Xj:(Q(Xf, Xt)),

=

,Jk are

taken freely from 1, ... , n.

2. For each k, mk(u) is spanned by mk-I(u) and by the

values at u of all functions f of the form (11k). Proof of Lemma 2. The proof is by induction on k. The case k = 0 is obvious. Every horizontal vector field in 7T- I ( U) is a linear combination of Xi, ... , X; with real valued functions as coefficients. It follows that every function f of the form (l k ) is a linear combination of functions of the form (lIs), s < k, with real valued functions as coefficients, in a neighborhood of u. It is now clear that, if the assertion holds for k - 1, it holds for k. We now prove that g' (u) is a subalgebra of 9 by establishing the relation [mk(u), ms(u)] c m k+ S +2(U) for all pairs of integers k and s. In view of Lemma 2, it is sufficient to prove that, for every functionf of the form (Is) and every function g of the form (Ilk)' the function [f,g](u) = [f(u),g(u)] is a linear combination of functions of the form (Ir ), r < k + s + 2, with real valued functions as coefficients. The proof is by induction on s. Let s = 0 and let f (u) = Qu(X, Y), where X and Yare horizontal vector fields. Since g is of type ad G, we have, by (2) of Lemma 1, 2[Q u(X, Y), g(u)J = v([X, YJ)u . g. On the other hand, we have v([X, YJ) u . g

=

[X, Y] u . g - h([X, YJ) u . g

= Xu(Yg) - Yu(Xg) - h([X, Y])u . g,

II.

THEORY OF CONNECTIONS

99

where h[X, YJ denotes the horizontal component of [X, YJ. The functions X(Yg) and Y(Xg) are of the form (I k+2 ) and the function h([X, YJ)g is of the form (I k + 1 ). This proves our assertion for s = 0 and for an arbitrary k. Suppose now that our assertion holds for s - I and every k. Every function of the form (Is) can be written as XI, where f is a function of the form (Is-I) and X is a horizontal vector field. Let g be an arbitrary function of the form (Ilk)' Then

[Xu!' g(u)J

=

Xu([I, gJ) - [f(u), XugJ.

The function [J: XgJ is a linear combination of functions of the form (Ir ) , r < k + s + 1, by the inductive assumption. The function X[I, gJ is a linear combination of functions of the form (Ir ), r < s + k + 2, also by the inductive assumption. Thus, the function [XI, gJ is a linear combination of functions of the form (Ir ), r < s + k + 2. QED.

10.5. The infinitesimal holonomy groups have ·the following properties: (1) <1>' (u) is a connected Lie subgroup of the local holonomy group *(u); (2) '(ua) = ad (a-I) ('(u)) and g'(ua) = ad (a- 1 )(g'(u)); (3) For each integer m, the set {7T(U) EM; dim '(u) > m} is open; (4) If <1>' (u) = * (u) at a point u, then there exists a connected open neighborhood U of x = 7T(U) such that '(v) = *(v) = '(u) = <1>* (u) for every v E P(u, U). Proof. (1) is evident from Proposition 10.4. (2) follows from PROPOSITION

1. For each k, we have mk(ua) = ad (a-I) (mk(u)). Proof of Lemma 1. The proof is by induction on k. The case k = 0 is a consequence of the fact that .Q is of type ad G. Suppose the assertion holds for k - 1. By (3) of Lemma 1 for Proposition 10.4, every function of the form (Ilk) is of type ad G. Our lemma now follows from Lemma 2 for Proposition 10.4. (2) means that '(u) can be considered as a function on M. (3) is a consequence of the fact that, if the values of a finite number of functions of the form (I k ) are linearly independent at a point u, then they are linearly independent at every point of a neighborhood of u. Note that (3) means that dim <1>' (u), considered as a function on M, is lower semicontinuous. To prove (4), assume LEMMA

100

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

..

'(u) = *(u) at a point u. Since dim '(u) is lower semicontinuous and dim *(u) is upper semicontinuous [cf. (5) of Proposition 10.lJ, the point x = 7T(U) has a neighborhood Usuch that dim '(v) > dim '(u) and dim <1>* (v) < dim *(u) for v E 7T-1( U).

On the other hand, *(v)

:::>

'(v) for every v E

7T-

1( U). Hence,

dim *(v) = dim <1>' (v) = dim *(u) = dim '(u) and, consequently, *(v) = '(v) for every v E 7T- 1( U). Applying Theoren1 10.3 to P U, we see that °(u, U) = <1>* (u) and °(v, U) = *(v). If v E P(u, U), then °(u, U) = °(v, U) so that <1>* (u) = <1>* (v). QED.

I

THEOREM 10.6. If dim <1>' (v) is constant in a neighborhood of u in P, then <1>' (u) = <1>* (u). Proof. We first prove the existence of an open neighborhood U of x = 7T(U) such that g' (u) = g' (v) for every v E P(u, U). Let 11, ... ,Is be a finite number of functions of the form (Ilk) such that 11(u), ... ,ls(u) form a basis of g'(u). At every point v of a small neighborhood of u,11 (v), ... ,ls(v) are linearly independent and, by the assumption, they form a basis of g'(v). Since11, ... ,Is are of type ad G, 11(va), ... ,11(va) form a basis of g'(va) = ad (a-1)(g'(v)). This means that there exists a neighborhood Uof x = 7T(U) such thatl1(v), ... ,ls(v) form a basis of g'(v) for every point v E 7T- 1( U). Now, let v be an arbitrary point of P(u, U) and let u t , 0 < t < 1, be a horizontal curve from u to v in 7T-1( U) so that u = U o and v = u 1 • We may assume that U t is differentiable; the case where U t is piecewise differentiable follows easily. Set gi(t) = h(u t ), i = 1, ... ,s, and X = Ute Since X is horizontal, we have i = 1, ... , s. Sinceg1(t), ... ,gs(t) form a basis for g'(u t ), dgddt can be expressed by (dgddt) t = ~J=l Aij(t)gj(t), where Aij(t) are continuous functions of t. By the lemma for Proposition 3.1, there exists a unique curve (aij(t))i,j=l, ... ,s in

II.

101

THEORY OF CONNECTIONS

GL(s; R) such that dai:Jldt = 2:~=1 Aikak:J and aij'(O) di:J. (Note that (Aij'(t)) E gl(s; R) corresponds to Y t E T e ( G) in the lemma for Proposition 3.1.) Let (bi:J(t)) be the inverse matrix of S (aij(t)) so that db 1,3..Idt = _2: k=1 b.1,k A k3·. Then g k) d (S b ) S (dbi:J) S b (d j j dt 2: =1 iigj = 2: =1 dt gj + 2:k=1 ik dt =

~j~l(d~;i + ~1~1 bikAki)g; =

O.

dij , we have

2:j=1 bii(t)gj(t)

=

gi(O).

This means that g'(u t ) = g'(u) and, in particular, g'(v) = g'(u). Taking U sufficiently small, we may assume that

g*(u)

:::l

g*(v)

:::l

g'(v)

:::l

mo(v)

for every v E P(u, U).

By Theorem 8.1, the Lie algebra of °(u, U) is spanned by all mo(v), v E P(u, U). A fortiori, g*(u) is spanned by all g'(v), v E P(u, U). Since g'(v) = g'(u) for every v E P(u, U) as we have just shown, we may conclude that g*(u) = g'(u) and *(u) = '(u). QED. COROLLARY 10.7. If dim '(u) is constant on P, then °(u) = *(u) = '(u). Proof. This follows from Theorems 10.3 and 10.6. QED. THEOREM 10.8. For a real analytic connection in a real analytic principal fibre bundle P, we have °(u) = *(u) = '(u) for every u E P. Proof. We may assume that P = P(u) and, in particular, P is connected. It suffices to show that dim ' (u) is locally constant; it then follows that dim ' (u) is constant on P and, by Corollary 10.8, that °(u) = *(u) = '(u) for every u E P. Let xl, ... , xn be a real analytic local coordinate system with origin x 1T( u). i Let U be a coordinate neighborhood of x given by 2: i (X )2 < a2 for some a O. We want to show that dim '(u) is constant on 1 1T- ( U). Let Xi alax i and let Xt be the horizontal lift of Xi'

102

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

For any set of numbers (a\ ... , an) with ~i (a i )2 = 1, consider the vector field X = ~i aiXi on U. Let X t be the ray given by xi(t) = ait and let U t be the horizontal lift of X t such that U = Uo. We prove that g' (u) = g' (u t ) for every t with ItI < a. Consider all the functions f of the form (Ilk)' k > 0,

f

=

X:J:Jk ... X:J:(Q(X:f: X*)) 11 '/, , l

defined on 'TT'-1( U). We set h(t) = f(u t ). Then the functions h(t) are analytic functions of t. For each to with Itol < a, there exists d > 0 such that all the functions h (t) can be expanded in a common neighborhood It - tol < d in the Taylor series:

If x* is the horizontal lift of X, then we can write h' (t) = X~tj, h"(t) = X~t(X*f) and so on. The fact that there exists such a d common to all h(t) follows from the lemma we prove below. Now, if It - tol < d, then all h(m)(t) belong to g'(utJ. The first power series shows that g'(u t) is contained in g'(u to ). Similarly, the second power series shows that g' (utJ is contained in g' (u t). This means that g'(u t ) = g'(u to ) for It - tol < d. The standard continuation argument shows that g'(u t ) = g'(u) for every t with It I < a, proving our theorem.

In a real analytic manifold, let X t be the integral curve of a real analytic vector field X such that X o = x, where Xx =1= O. For any real analytic function g and for a finite number of real analytic vector fields X l' . . . , X s' consider all the functions of the form LEMMA.

f (x)

=

(X jk

h(t) =

•••

f

X j1g)(x)

(x t ),

where j l' . . . ,jk are taken freely from 1, 2, . . . , s. Then there exists d > 0 such that the functions h(t) can be expanded into power series in a m common neighborhood It I < d as follows: h(t) = ~::=o t , !z<m)(o). m.

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THEORY OF CONNECTIONS

103

Proof. Since Xx =1= 0, we may take a local coordinate system x\ ... ,xn such that x=a/ax l and xt=(t,O, ... ,O) in a neighborhood of x. The preceding expansions of h(t) are nothing but the expansions off (x) into power series of Xl. Each Xi is of the form Xi = "'i:.jhi . a/ ax j . Since f andhj are all real analytic, they can be expanded into power series of (xl, ... , xn ) in a common neighborhood Ixil < a for some a > 0. Our lemma then follows from the fact that if fl and h are real analytic functions which can be expanded into power series of xl, ... , x n in a neighborhood Ixil < a, then the functions flh and afl/ ax i can be expanded into power series in the same neighborhood. QED. The results in this section are due to Ozeki [1 J.

11. Invariant connections Before we treat general invariant connections, we present an important special case.

Let G be a connected Lie group and H a closed subgroup oj G. Let 9 and I) be the Lie algebras oj G and H respectively. (1) If there exists a subspace m oj 9 such that 9 = I) + m (direct sum) and ad (H) m = m, then the I)-component OJ oj the canonical 1-Jorm f) oj G (cf. §4 oj Chapter I) with respect to the decomposition 9 = I) + m difmes a connection in the bundle G(G/H, H) which is invariant by the left translations oj G; (2) Conversely, any connection in G(G/H, H) invariant by the left translations oj G (if it exists) determines such a decomposition 9 = I) + m and is obtainable in the manner described in (1); (3) The curvature form Q ofthe invariant connection defined by OJ in (1) is given by Q(X, Y) = -leX, Y]1) (I)-component oj-lex, Y] € g), where X and Yare arbitrary left invariant vector fields on G belonging to THEOREM

11.1.

m; (4) Let 9(e) be the Lie algebra oj the holonomy group
reference point e (identity element) of the invariant connection defined in (1). Then 9(e) is spanned by all elements oj the form [X, Y] 1)' X, Y € m. Proo£ (1) The proof is straightforward and is similar to that of Proposition 6.4. Under the identification 9 ~ Te(G), the subspace m corresponds to the horizontal subspace ate.

104

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(2) Let W be a connection form on G( GjH, H) invariant by the left translations of G. Let m be the set of left invariant vector fields on G such that w(X) = o. It is easy to verify that 9 = 1) + m is a desired decomposition. (3) A left invariant vector field is horizontal if and only if it is an element of m. Now, (3) follows from Corollary 5.3. (4) Let 91 be the subspace of 9 spanned by the set {Qe(X, Y); X, Y Em}. Let 92 be the subspace of 9 spanned by the set {Qu(X, Y); X, Y E m and u E G}. By Theorem 8.1, we have 91 c 9(e) c 92. On the other hand, we have 91 = 92 as Qu(X, Y) = Qe(X, Y) for any X, Y E m and u E G. Now, (4) follows from (3). QED. Remark. (1) can be considered as a particular case of Proposition 6.4. Let P = (GjH) X G be the trivial bundle over GjH with group G. We imbed the bundle G( GjH, H) into P by the mapping f defined by U E G, f(u) = (7T(U), u), where 7T: G ~ GjH is the natural projection. Let qJ be the form defining the canonical flat connection (cf. §9) of P. Its 1)-component, restricted to the subbundle G(G j H, H), defines a connection (Proposition 6.4) and agrees with the form w in (1). Going back to the general case, we first prove the following proposition which is basic in many applications.

11.2. Let qJt be a I-parameter group of automorphisms of a principal fibre bundle P( M, G) and X the vector field on P induced by qJt. Let r be a connection in P invariant by qJt. For an arbitrary point U o of P, we define curves ut, X t, V t and at as follows: Ut = qJt(uo), Xt = 7T(U t ), PROPOSITION

Vt

= the horizontal lift of X t such that Vo = uo,

Then at is the I-parameter subgroup of G generated by A = where w is the connection form of r. Proof. As in the proof of Proposition 3.1, we have w(u t)

=

(ad (at-I) )w(V t)

Since Vt is horizontal, we have w(u t)

=

W Uo

(X),

+ at-lat. at-lat. On the other hand,

II.

105

THEORY OF CONNECTIONS

we have ut = CPt(Xuo ) and hence w(u t ) = w(Xuo ) = A, since the connection form W is invariant. by CPt. Thus we obtain at-1a t = A. QED. Let K be a Lie group acting on a principal fibre bundle P(M, G) as a group of automorphisms. Let Uo be an arbitrary point of P which we choose as a reference point. Every element of K induces a transformation of M in a natural manner. The set J of all elements of K which fix the point X o = 7T(U O) of M forms a closed subgroup of K, called the isotropy subgroup of K at X O• We define a homomorphism A: J -+ G as follows. For each j E J, juo is a point in the same fibre as Uo and hence is of the form juo = uoa with some a E G. We define A(j) = a. If j,j' E J, then

UoA(jj')

=

(jj')u o = j(UoA(j'))

=

(juo)A(j')

= (UoA(j))A(j')

=

Uo(A(j)A(j')).

Hence, A(Jj') = A(j) A(j'), which shows that A: J -+ G is a homomorphism. It is also easy to check that A is differentiable. The induced Lie algebra homomorphism i -+ 9 will be also denoted by the same A. Note that Adepends on the choice of U o ; the reference point U o is chosen once for all and is fixed throughout this section.

11.3. Let K be a group of automorphisms of P( M, G) and r a connection in P invariant by K. We define a linear mapping A: f -+ 9 by X E f, PROPOSITION

where X is the vector field on P induced by X. Then (1) A(X) = A(X) (2) A(ad (j)(X))

for X E L· ad (A(j))( A(X))

J and X E f, where ad (j) is the adjoint representation of J in f and ad (A (j)) is that of Gin g. =

for j

E

Note that the geometric meaning of A(X) is given by Proposition

11.2. Proof. (1) We apply Proposition 11.2 to the l-paran1eter subgroup CPt of K generated by X. If X E j, then the curve X t = 7T(cpt(UO)) reduces to a single point X o = 7T(U O). Hence we have CPt(uo) = UoA(CPt)· Comparing the tangent vectors of the orbits CPt(uo) and UoA(cpt) at UO, we obtain A(X) = A(X).

106

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(2) Let X € f and j € J. We set Y = ad (j) (X). Then Y generates the I-parameter subgroup j
OJ uo (Y) = OJ uo (j(R;'(j-l)XuJ) = OJj-1uo (R;'(j-l)XuJ = ad (A(j)) (OJuo(XuJ) = ad (A(j))(A(X)). QED. With the notation of Proposition 11.3, the curvature form 0 of r satisfies the following condition: 20 u/%, Y) = [A(X), A(Y)] - A([X, Y]) for X,Y € f. Proof. From the structure equation (Theorem 5.2) and Proposition 3.11 of Chapter I, we obtain PROPOSITION

11.4.

20(%, Y) = 2dOJ(X, Y) + [OJ(X), OJ(Y)] = X (OJ (Y)) - Y(OJ (X)) - OJ ([X, Y])

+ [OJ (X), OJ ( Y) ] .

Since OJ is invariant by K, we have by (c) of Proposition 3.2 of Chapter I (cf. also Proposition 3.5 of Chapter I)

X (OJ (Y)) - OJ ([X, Y]) = (LxOJ) (Y) = 0, Y(OJ(X)) - OJ([Y, X]) = (LyOJ)(X) = O. On the other hand, X we have

---+

X being

a Lie algebra homomorphism,

OJuo ([ X, Y]) = A ([ X, Y]). Thus we obtain

20 uo (X, Y) = [OJuo(X), wuo(Y)] - A([X, Y]) = [A(X), A(Y)] - A([X, Y]). QED. We say that K acts fibre-transitively on P if, for any two fibres of P, there is an element of K which maps one fibre into the other, that is, if the action of K on the base M is transitive. If J is the isotropy subgroup of K at X o = 7T(uo) as above, then M is the homogeneous space K/J. The following result is due to Wang [1 J.

If a connected Lie group K is a fibre-transitive automorphism group of a bundle P( M, G) and if J is the isotropy subgroup of THEOREM

11.5.

II.

107

THEORY OF CONNECTIONS

K at X o = 1T ( uo), then there is a' 1: 1 correspondence between the set of Kinvariant connections in P and the set of linear mappings A: f ---+ 9 which satisfies the two conditions in Proposition 11.3; the correspondence is given by for X € f,

where X is the vector field on P induced by X. Proof. In view of Proposition 11.3, it is sufficient to show that, for every A: f ---+ 9 satisfying (1) and (2) of Proposition 11.3, there is a K-invariant connection form w on P such that A(X) = wuo(X) for X € f. Let X* € Tu(P). Since K is fibre-transitive, we can write U o = kua = k Rau 0

k RaX* = 0

XUo + A*o' U

where k € K, a € G, X € k and A* is the fundamental vector field corresponding to A € g. We then set

+ A).

w(X*) = ad (a) (A(X)

We first prove that w(X*) is independent of the choice of X and A. Let where Y € f and B € g, Yu + B u* , Xu - Yu = B u* - Au* . From the definition of A: i

Xu o

+ Au*

0

=

0

0

so that ---+ g, 000 0 it follows that A(X - Y) = B - A. By condition (1) of Proposition 11.3, we have A(X - Y) = A(X - Y) = A(X) - A(Y). Hence, A(X) + A = A( Y) + B. We next prove that w(X*) is independent of the choice of k and a. Let U o = kua = k1ua l (k l € K and a l € G), so that k1k-1u O = uoa1 l a and k1k- 1 € J. We set j = k1k- l . Then A(j) = alIa. We have kl

0

Ra1X* = jk Ra).(j-l)X* 0

=

j

0

R).(j-l)(k RaX*) 0

=

j

0

R).(j-l)(XuO

+ A~o).

By Proposition 1. 7 of Chapter I, we have

j

0

R).(j-l)(XuO ) = j(XUO).(j-l») = Zuo'

where Z = ad (j) (X).

108

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

By Proposition 5.1 of Chapter I, we have

j

0

R),(j-l)(A~o)

=

R),(j-l)(jA~J

= R),(j-l)A}'u o =

R),(j-l)A~o),(j)

= C;;'o'

where C = ad (A(j)) (A). Hence we have

kloRa1 X* =0 Zu + C;;' , 0 ~

ad (al)(A(Z)

+ C)

+ ad (A(j))A) ad (al) [ad (A(j)) (A(X) + A)] ad (a) (A(X) + A).

= ad (a l ) (A(ad (j)(X)) =

=

This proves our assertion that w(X*) depends only on X*. We now prove that w is a connection form. Let X* E Tu(P) and U o = kua as above. Let b be an arbitrary element of G. We set

y* so that U o

=

=

R b X*

kub(b-1a)

=

E

where v = ub,

Tv (P),

kv(b-1a). We then have

k Rb-lay* = k Rb-laRbX* = k RaX* = (guo 0

0

0

+ A~o)

and hence

w(RbX*) = w(Y*) = ad (b-1a) (A(X)

+ A)

= ad (b-l)(w(X*)) ,

which shows that w satisfies condition (b') of Proposition 1.1. Now, let A be any element of 9 and let U o = kua. Then

k

0

Ra(A~)

= Ra

0

k(A~)

= Ra(AZu) =

B~o'

where B = ad (a-I) (A).

Hence we have w(A~)

= ad (a) (B) = A,

which shows that w satisfies condition (a') of Proposition 1.1. To prove that w is differentiable, let U 1 be an arbitrary point of P and let U o = k1u1a l . Consider the fibre bundle K(M, J), where M = K/J. Let 0': U ---+ K be a local cross section of this bundle defined in a neighborhood U of 1T( u l ) such that 0'(1T( u l )) = kl . For each u E 1T-1 ( U), we define k E K and a E G by

k

=

0'(1T(U))

and

Uo =

kua.

Then both k and a depend differentiably on u. We decompose the vector space f into a direct sum of subspaces: f = i + m. For ,an

II.

arbitrary x*

E

109

THEORY OF CONNECTIONS

Tu(P), we set

k R a (X*)

=

0

XUo + A*Uo'

where X

E

m.

Then both X and A are uniquely determined and depend differentiably on X*. Thus w(X*) = ad (a) (A(X) + A) depends differentiably on X*. Finally, we prove that w is invariant by K. Let X* E Tu(P) and Uo = kua. Let k1 be an arbitrary element of K. Then k1X* E Tk1u(P) and Uo = kk11(k1u)a. Hence, kk1 1 Ra(k1X*) = k Ra(X*). 0

0

From the construction of w, we see immediately that w(k1X*) QED. w(X*). In the case where K is fibre-transitive on P, the curvature form 0, which is a tensorial form of type ad G (cf. §5) and is invariant by K, is completely determined by the values 0u0 (X, Y), X, Y E f. .... .... Proposition 11.4 expresses 0uo(X, Y) in terms of A. As a consequence of Proposition 11.4 and Theorem 11.5, we obtain

11.6. The K-invariant connection in P defined by A is flat if and only if A: f -4- 9 is a Lie algebra homomorphism. Proof. A connection is flat if and only if its curvature form vanishes identically (Theorem 9.1). QED. COROLLARY

Assume in Theorem 11.5 that f admits a subspace m such that f = i + m (direct sum) and ad (J) (m) = m, where ad (J) is' the adjoint representation of J in f. Then (1) There is a 1: 1 correspondence between the set of K-invariant connections in P and the set of linear mappings Am: m -4- 9 such that THEOREM

11.7.

Am(ad (j)(X)) = ad (A(j))(Am(X))

for X Em

and j EJ;

the correspondence is given via Theorem 11.5 by A(X) = {A(X) Am(X)

°

if X E i, if X E m.

(2) The curvature form of the K-invariant connection defined by Am satisfies the following condition: 20 uo (X, Y)

=

[Am(X), Am(Y)] - Am([X, Y]m) - A([X, Y]j) for X, Y Em,

110

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

where [X, Y]m (resp. [X, Y]j) denotes the m-component (resp. j-component) of [X, Y] E f. Proof. Let A: f --+ 9 be a linear mapping satisfying (1) and (2) of Proposition 11.3. Let Am be the restriction of A to m. It is easy to see that A --+ Am gives a desired correspondence. The statement (2) is a consequence of Proposition 11.4. QED. In Theorem 11. 7, the K-invariant connection in P defined by Am = is called the canonical connection (with respect to the decomposition f = j + m). Remark. ( 1) and (3) of Theorem 11.1 follow from Theorem 11. 7 if we set P( M, G) = G( G/ H, H) and K = G; the invariant connection in Theorem 11.1 is the canonical connection just defined. Finally, we determine the Lie algebra of the holonomy group of a K-invariant connection.

°

11.8. With the same assumptions and notation as in Theorem 11.5, the Lie algebra g(u o) of the holonomy group (u o) of the K-invariant connection defined by iA: f --+ 9 is given by THEOREM

+ [A(f), mo] + [A(f),

mo

[A(f), mo]]

+ ... ,

where mo is the subspace of 9 spanned by {[A (X), A ( Y )] - A ( [X, Y]); X, Y E f}.

Proof. Since K is fibre-transitive on P, the restricted holonomy group <1>0 (u o) coincides with the infinitesimal holonomy group <1>' (u o) by virtue of Corollary 10.7. We define a series of subspaces m k , k = 0, 1, 2, ... , of 9 as follows:

+ [ACf), mo], mo + [A (f), mo] + [A (f), [A (f), m 1 = mo

m2

=

mo]]

and so on. We defined in §10 an increasing sequence of subspaces mk(u O), k = 0,1,2, ... , ofg. Since.the union of these subspaces mk(u O) is the Lie algebra g'(u o) of the infinitesimal holonomy group '(u o), it is sufficient to prove that m k = mk(uO) for k = 0, 1, 2, .... By Proposition 11.4, _the _ subspace m _ o is_spanned by {Q _ o Y); _ u (X, X, Y E f}. Since Q u o(X, Y) = Q u 0 (hX, hY), where h){ and hY

II.

111

THEORY OF CONNECTIONS

denote the horizontal components of coincides with mo( uo). We need the following lemmas.

X

and

Y

respectively, m o

If Y

is a horizontal vector field on P and X is the vector field on P induced by an element X of f, then [X, Y] is horizontal. LEMMA

1.

Proof of Lemma 1. By (c) of Proposition 3.2 of Chapter I (cf. also Proposition 3.5 of Chapter I), we have

X( w(Y))

(L XW ) (Y)

=

+ w([X,

Y]).

Since w(Y) = 0 and Lxw = 0, we have w([X, Y])

2. Let V, W, Y l'

=

o.

Y r be arbitrary horizontal vector fields on P and let X be the vector field on P induced by an element X of f. Then Xu o(Yr· .. Y 1 (Q(V, W))) E mr(uO). LEMMA

Proof of Lemma 2.

... ,

We have

Xuo(Yr ·· . Y 1 (Q(V, W))) (Yr)u o(XYr- 1

Y 1 (Q(V, W)))

• ••

mod mr(uo),

since [X, Y r] is horizontal by Lemma 1 and [X, Y r]uo(Yr- 1 Y 1 (Q(V, W))) is in mr(uO). Repeating this process, we obtain

Xu o(Yr

• • •

•••

Y1 (Q (V, W)))

= (Yr)Uo (Yr- 1

• • •

Y 1 X( Q (V, W)))

By the same argument as in the proof of Lemma 1, we have

X(Q(V, W)) Since LxQ

=

(Yr)uo(Yr- 1

=

(LxQ) (V, W)

+ Q([X,

V], W)

+ Q(V,

[X,W]).

0, we have •••

Y 1 X(Q(V, W)))

= (Yr)u o(Yr- 1

•.•

Y 1 (Q([X, V], W)))

+ (Yr)uo(Yr-l···

Y1 (Q(V,

[X, W]))).

The two terms on the right hand side belong to mr(uo) as [X, V] and [X, W] are horizontal by Lemma 1. This completes the proof of Lemma 2. Let Xi = aj ax i , where xl, ... , xn is a local coordinate system in a neighborhood of Xo = 7T(U O). Let Xi be the horizontal lift of

112

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

f =

Xj~

... Xj;(O(Xt, X[))

be a function of the form (llr) as defined in §10. If hX and vX denote the horizontal and the vertical components of X respectively, then Lemma 1 for Proposition 10.4 implies

(hX)uof = -(vX)uof + Xuof = [wuo(X),f (u o)] + Xuof. Since Xuof € mr(uo) by Lemma 2 and since wuo(X) = A(X), we have (hX) uof == [A( X), f (uo)] mod m r(uo). Assuming that m r = mr(uo) for all r < s, we shall show that m s ms(uo). Since K is fibre-transitive on P, every horizontal vector at Uo is of the form (hX)u o for some_ X € f. Hence, msCuo) is spanned by m S -l(uO) and the set of all (hX)uof, where X € f andf is a function of the form (lIs-I). On the other hand, m s is spanned by m s- 1 m s- 1 (u O) and by [A(f), m S - 1 ] [A(f), m s- 1 (u O)]. In other words, m s is spanned by m s- 1 = m s- 1 (u o) and the set of all [A(X),f(u o)], where X € f andfis a function of the form (lIs_I)' Our assertion m s = ms(uo) follows from the congruence (hX)uof [A(X),f (u o)] mod m s- 1 (u O). QED. (4) of Theorem 11.1 is a corollary to Theorem 11.8 (cf. Remark made after the proof of Theorem 11.7). Remark.

CHAPTER III

Linear and Affine Connections

1. Connections in a vector bundle Let F be either the real number field R or the complex number field C, Fm the vector space of all m-tuples of elements of F and GL(m; F) the group ofall m x m non-singular matrices with entries from F. The group GL(m; F) acts on Fm on the left in a natural manner; if a = (aJ) E GL(m; F) and ~ = (~\ ... , ~m) EFm, then a~ = (I;j aJ~', ... , I;, aj~') EFm. Let P(M, G) be a principal fibre bundle and p a representation of G into GL(m; F). Let E(M, Fm, G, P) be the associated bundle with standard fibre Fm on which G acts through p. We call E a real or complex vector bundle over M according as F - R or F = C. Each fibre 7Ti1(x), x E M, of E has the structure of a vector space such that every U E P with 7T(U) = x, considered as a mapping of Fm onto 7Ti1(x), is a linear isomorphism of Fm onto 7Ti1(x). Let S be the set of cross sections cp: M --+ E; it forms a vector space 1) with addition and scalar over F (of infinite dimensions if m multiplication defined by

(cp

+ 1jJ) (x)

=

cp(x)

+ 1jJ(x) ,

(2cp) (x)

=

2( cp(x)),

CP,"P

E

S,

X

cp E S, 2 E F,

E

M,

x EM.

We may also consider S as a module over the algebra ofF-valued functions; if 2 is an F-valued function on M, then

(2cp) (x) = 2(x) . cp (x),

cp E S,

X

E

M.

Let r be a connection in P. We recall how r defined the notion of parallel displacement of fibres of E in §7 of Chapter II. If T = xt , a t b, is a curve in M and T* = U t is a horizontal 113

114

FOUNDATIONS OF DIFFERENrnAL GEOMETRY

lift of T to P, then, for each fixed ~ € Fm, the curve T' = Ut~ is, by definition, a horizontal lift of T to E. Let q; be a section of E defined on T = Xt so that 1TE q;(x t ) = Xt for all t. Let xt be the vector tangent to T at Xt. Then, for each fixed t, the covariant derivative \/Xtq; of q; in the direction of (or with respect to) x t is defined by 0

\/x q; t

=

lim h-+O

hI

[T~+h( q;(Xt+h))

- q;(x t)],

where T~+h: 1Ti1(xt+h) ~ 1Ti1(xt) denotes the parallel displacement of the fibre 1Ti1(xt+h) along T from x t +h to Xt. Thus, \/ Xtq; € 1Ti] (x t ) for every t and defines a cross section of E along T. The cross section q; is parallel, that is, the curve q;(x t) in E is horizontal, ifand only if\/Xtq; = 0 for all t. The following formulas are evident. If q; and "P are cross sections of E defined on TXt, then

\/Xt (q;

+ "P)

=

\/ Xt q;

If It is an F-valued function defined on

\/xt"P· T,

then

\/1;t(Itq;) - It(x t ) . \/Xtq; (xtlt)· q;(x t)· The last formula follows immediately from T~+h(It(XHh) . q;(x t+h)) It(x t+h) . T:+ h(q;(X Hh )). Let X € To: (M) and q; a cross section of E defined in a neighborhood of x. Then the covariant derivative \/ xq; of q; in the directif1n of X is defined as follows. Let T = Xt, c t c, be a curve such that X = xO' Then set \/xq; \/ Xo q;. It is easy to see that \/ xq; is independent of the choice of T. A _cross section q; of E defined on an open subset U of M is parallel if and only if \/ xq; = 0 for all X € T ro ( U), X € U. 1.1. Let X, Y € T ro ( M) and let q; and"P be cross sections of E defined in a neighborhood of x. Then (1) \/ yq; - \/ xq; + \/ yq; ; (2) \/ x(q; "P) = \/ xq; \/ x"P; (3) \/),xq; = It . \/ xq;, where It € F~' (4) \/ x (ltq;) = It(x) . \/ xq; + (XIt) . q;(x), where It is an F-valued function defined in a neighborhood of x. PROPOSITION

III.

LINEAR AND AFFINE CONNECTIONS

115

Proof. We proved (2) and (4). (3) is obvious. Finally, (1) will follow immediately from the following alternative definition of covariant differentiation. Suppose that a cross section cP of E is defined 011 an open subset U of M. As in Example 5.2 of Chapter II, we associate with cp an Fm-valued functionf on 7T-1( U) as follows: f(v) = V- 1(cp(7T(V))) , VE7T- 1(U). Given X E Tre(M), let x* E Tu(P) be a horizontal lift of X. Sincef is an Fm-valued function, X*f is an element of Fm and u(x* f) is an element of the fibre 7Ti1(x). We have V xCP

u(X*f). 8, be a curve such Proof of Lemma. Let T = Xt, - 8 t that X = xO' Let T* = U t be a horizontal lift of T such that Uo = u so that X* uO' Then we have LEMMA.

=

X*f = lim hI [f(u h ) -feu)] h-O

and

cp(X)]. In order to prove the lemma, it is sufficient to prove T~(CP(Xh)) = U 0 Uh 1(CP(Xh)). Set ~ = Uh 1(cp(xh)). Then Ut~ is a horizontal curve in E. Since Uh~ = cp(xh), cp(xh) is the element of E obtained by the parallel displacement of uo~ = u 0 Uh1(CP(Xh)) along T from X o to Xh. This implies ~(CP(Xh)) = U0 Uh 1(CP(Xh)), thus completing the proof of the lemma. Now, (1) of Proposition 1.1 follows fi~om the lemma and the fact that, if X,Y E Tre(M) and X*,Y* E Tu(P) are horizontal lifts of X and Y respectively, then X* -t- y* is a horizol1tallift of X + Y.

QED. If cP is a cross section of E defined on M and X is a vector field on M, then the covariant derivative V xCP of cP in the direction of (or with respect to) X is defined by

(V xcp) (x)

=

Vx cpo :l;

116

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Then, as an immediate consequence of Proposition 1.1, we have

1.2. Let X and Y be vector fields on M, cp and 'IjJ cross sections of E on M and A an F-valued function on M. Then (1) V x+ycp·= V xcp + Vycp; (2) V x(cp + 1p) = V xcp + V x1p; (3) V;.xCP = A . V xCP; (4) V x (Acp) = A . V xCP + (XA) cpo PROPOSITION

Let X be a vector field on M and x* the horizontal lift of X to P. Then covariant differentiation V x corresponds to Lie differentiation Lx. in the following sense. In Example 5.2 of Chapter II, we saw that there is a 1: 1 correspondence between the set of cross sections cp: M ---+ E and the set of Fm-valued functions f on P such thatf(ua) = a-1(f(u)), a E G (a-1 means p(a- 1) E GL (m; F)). The correspondence is given by f(u) = U- 1(cp(7T(U))), UE P. We then have

1.3. If cp: M ---+ E is a cross section and f: P ---+ Fm is the corresponding function, then X*f is the function corresponding to the cross section V xcp. Proof. This is an immediate consequence of the lemma for QED. Proposition 1.1. PROPOSITION

A fibre metric g in a vector bundle E is an assignment, to each x E M, of an inner product gal in the fibre 7Ti1(x), which is differentiable in x in the sense that, if cp and 1p are differentiable cross sections of E, then ga;( cp(x), 1p(x)) depends differentiably on x. When E is a complex vector bundle, the inner product is understood to be hermitian:

1.4. If M is paracompact, every vector bundle E over M admits a fibre metric. Proof. This follows from Theorem 5.7 of Chapter I just as the existence of a Riemannian metric on a paracompact manifold. We shall give here another proof using a partition of unity. Let {Ui}iEI be a locally finite open covering of M such that 7Ti1( Ui) is isomorphic with U i X Fm for each i. Let {Si} be a partition of unity subordinate to {Ui} (cf. Appendix 3). Let hi be a fibre PROPOSITION

III.

I Ui = 'Tri

metric in E

g(3 1, 3 2)

~i

=

117

LINEAR AND AFFINE CONNECTIONS 1

(

Ut ). Set g = ~i sihi, that is,

st(x)h i (3 1, 3 2)

3 b 3 2 E'Tri 1 (x),

for

XEM.

Since {Ut} is locally finite and Si vanishes outside Ui' g is a well defined fibre metric. QED. Given a fibre metric g in a vector bundle E(M, Fm, G, P), we construct a reduced subbundle Q(M, H) of P(M, G) as follows. In the standard fLbre Fm of E, we consider the canonical inner product ( , ) defined by (~,

r;) =

~i ~ir;i

for

~

=

(~\

,

~m),

r; = (r;\

, r;m)

E

Rm,

(~,

r;)

~i ~if/

for

~

=

(~\

,

~m),

r; = (r;\

, r;m)

E

Cm.

=

Let Q be the set of u E P such that g(u(~), u(r;)) = (~, r;) for ~,r; E Fm. Then Q is a closed submanifold of P. It is easy to verify that Q is a reduced subbundle of P whose structure group H is given by H = {a E G; p(a) E O(m)} ifF = R,

H

=

{a

E

G; p(a)

E

U(m)}

ifF

=

C,

where p is the representation of G in GL(m; F). Given a fibre metric g in E, a connection in P is called a metric connection if the parallel displacement of fibres of E preserves the fibre metric g. More precisely, for every curve T = Xt, 0 < t < 1, of M, the parallel displacement 'Tri 1 (x o) ~ 'Tri1(X 1) along T is isometric.

1.5. Let g be a fibre metric in a vector bundle E(M, Fm, G, P) and Q(M, H) the reduced subbundle of P(M, G) defined by g. A connection r in P is reducible to a connection r' in Q if and only if r is a metric connection. Proof. Let T = Xt, 0 ~ t < 1, be a curve in M. Let ~,r; E Fm and Uo E Q with 'Tr(u o) = X o. Let T* = Ut be the horizontal lift of T to P starting from Uo so that both T' = Ut(~) and Til = ut(r;) are horizontal lifts of T to E. If r is reducible to a connection r' in Q, then U t E Q for all t. Hence, PROPOSITION

g(uo(~), uo(r;))

proving that

r

=

(~,

r;)

= g(Ut(~),

ut(r;)),

is a metric connection. Conversely, if r is a metric

118

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

connection, then Hence, U t € Q. This means that r is reducible to a connection in Q by Proposition 7.2 of Chapter II. QED. Proposition 1.5, together with Theorem 2.1 of Chapter II, implies that, given a fibre metric g in a vector bundle E over a paracompact manifold K, there is a metric connection in P. Let E(M, Fin, G, P) be a vector bundle such that G = GL(m; F). Let E{ € gI(m; F), Lie algebra of GL(m; F), be the m X m matrix such that the entry at the J-th column and the Z,j-th row is 1 and other entries are all zero. Then {Ei; i,j = 1, ... , m} form a basis of the Lie algebra gI(m; F). Let OJ and 0 be the connection form and the curvature form of a connection r in P. Set OJ = ~.~ ,J'OJ~E~ 0 = ~'~,J. Q~E~ J ~' ') ~ • It is easy to verify that the structure equation of the connection r (cf. §5 of Chapter II) can be expressed by

Let g be a fibre metric in E and Q the reduced subbundle of P defined by g. If r is a metric connection, then the restriction of OJ to Q defines a connection in Q by Proposition 6.1 of Chapter II and Proposition 1.5. In particular, both OJ and 0, restricted to Q, take their values in the Lie algebra oem) or u(m) according as F = R or F = C. In other words, both (OJj) and (OJ), restricted to Q, are skew-symmetric or skew-hermitian according as F = R or

F

=

C.

2. Linear connections Throughout this section, we shall denote the bundle of linear frames L(M) by P and the general linear group GL(n; R), n = dim M, by G. The canonicalform e of P is the Rn-valued I-form on P defined by for X € Tu(P),

III.

119

LINEAR AND AFFINE CONNECTIONS

where u is considered as a linear mapping of Rn onto T (u)(M) (cf. Example 5.2 of Chapter I). 1T

2.1. The canonicalform 0 of P is a tensorial I-jorm of type (GL(n; R), Rn). It corresponds to the identity transformation of the tangent space Tx(M) at each x E M in the sense of Example 5.2 of Chapter II. Proof. If X is a vertical vector at u E P, then 1T(X) = 0 and hence O(X) = O. If X is any vector at U E P and a is any element of G = GL(n; R), then RaX is a vector at ua E P. Hence, (R:O) (X) = O(RaX) = (ua) -1 (1T(R aX) ) = a- 1 u- 1 (1T(X)) =- a- 1 (O(X)), thus proving our first assertion. The second assertion is clear. QED. A connection in the bundle of linear frames P over M is called a linear connection of M. Given a linear connection r of M, we associate with each ~ ERn a horizontal vector field B(~) on P as follows. For each u E P, (B(~))u is the unique horizontal vector at u such that 1T((B(~))u) = u(~). We call B(~) the standard horizontal vector field corresponding to ~. Unlike the fundamental vector fields, the standard horizontal vector fields depend on the choice of connections. PROPOSITION

2.2. The standard horizontal vector fields have the following properties: (1) If 0 is the canonical form of P, then O(B(~)) = ~ for ~ E Rn; (2) Ra(B(~)) = B(a-l~) for a E G and ~ ERn; (3) If ~ i=- 0, then B(~) never vanishes. Proof. (1) is obvious. (2) follows from the fact that if X is a horizontal vector at u, then Ra(X) is a horizontal vector at ua and 1T(R a(X) ) = 1T(X). To prove (3), assume that (B(~)) u = 0 at some point u E P. Then u(~) = 1T( (B(~)) u) = O. Since u: Rn --* T (u)(M) is a linear isomorphism, ~ = 0. QED. PROPOSITION

1T

Remark. The conditions O(B(~)) = ~ and OJ(B(~)) = 0 (where OJ is the connection form) completely determine B(~) for each ~ ERn. 2.3. If A * is the fundamental vector field corresponding 9 and if B (~) is the standard horizontal vector field corresponding

PROPOSITION

to A

E

120 to

~

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

€

Rn, then

[A*,

B(~)]

=

B(A~),

where A~ denotes the image of ~ by A € 9 = gI(n; R) (Lie algebra of all n X n matrices) which acts on Rn. Proof. Let at be the I-parameter subgroup of G generated by A, at = exp tAo By Proposition 1.9 of Chapter I and (2) of Proposition 2.2,

[A*,

B(~)]

=

lim ~ t--O t

[B(~)

-

Rat(B(~))]

=

lim~ [B(~) -B(at-l~)]. t--O t

Since ~ ---+ (B(~)) u is a linear isomorphism ofRn onto the horizontal subspace Qu (cf. (3) of Proposition 2.2), we have lim ~ t~O t

[B(~)

-

B(at-l~)]

= B(lim ~ t--O t

(~

-

at-l~))

=

B(A~). QED.

We define the torsion form 0 of a linear connection

o=

De

r

by

(exterior covariant differential of e).

By Proposition 5.1 of Chapter II and Proposition 2.1, 0 is a tensorial 2-form on P of type (GL(n; R), Rn).

2.4 (Structure equations). Let w, 0, and Q be the connection form, the torsion form and the curvature form of a linear connection r of M. Then THEOREM

1st structure equation: de (X, Y) = -!(w(X) . e(Y) - w(Y) . e(x))

+ 0(X, Y),

2nd structure equation: dw(X, Y) = -![ w(X), w( Y)]

+ Q(X, Y),

where X, Y € Tu(P) and u € P. Proof. The second structure equation was proved in Theorem 5.2 of Chapter II (see also §l). The proof of the first structure equation is similar to that of Theorem 5.2 of Chapter II. There are three cases which have to be verified and the only nontrivial case is the one where X is vertical and Y is horizontal. Choose A € 9 and ~ € Rn such that X = A: and Y = B(~) u.

III.

121

LINEAR AND AFFINE CONNECTIONS

8(X, Y) = 0, w(Y) . O(X) = 0 and w(X) . O(Y) = w(A*) . O(B(~)) = A~, since w(A *) = A and O(B(~)) = ~. On the other hand, 2dO(X, Y) = A*(O(B(~))) - B(~)(O(A*)) - O([A*, B(~)]) = -O([A*, B(~)]) = -O(B(A~)) = -A~ by Proposition Then

2.3. This proves the first structure equation. With respect to the natural basis e1 ,

o=

~i

Oiei ,

8

=

... ,

~i

QED.

en of Rn, we write

8 iei ·

As in §I, with respect to the basis E{ of gI(n; R), we write ~ 2,1 . .w1~Et J t'

w=

0 =

~ 2,1 . .0~Et J t·

Then the structure equations can be written as (1)

dO i =

(2)

dW j

i

+ 8 i, i k i -~k wk A W + OJ' -~ 1

=

wj A 01

i = 1, ... , n,

j

i,j = 1, ... , n.

Considering 0 as a vector valued form and w as a matrix valued form, we also write the structure equations in the following simplified form:

+8 w A w + O.

(1')

dO = -w

(2')

dw

=

-

A

0

In the next section, we shall give an interpretation of the torsion form and the first structure equation from the viewpoint of affine connections. THEOREM

2.5 (Bianchi's identities). For a linear connection, we have

1st identity: D8

=

0

A

0, that is,

3D8(X, Y, Z) = O(X, Y) O(Z)

+ O(Y, Z)

O(X)

+ O(Z, X)

where X,Y,Z 2nd identity: DO

=

E

O(Y),

Tu(P).

o.

Proof. The second identity was proved in Theorem 5.4 of Chapter II. The proof of the first identity is similar to that of Theorem 5.4. If we apply the exterior differentiation d to the first structure equation dO = -w A 0 + 8, then we obtain

o=

-dw

A

0

+ w A dO + d8.

122

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Denote by hX the horizontal component of X. Then w(hX) = 0, ()(hX) = ()(X) and dw(hX, hY) = O(X, Y). Hence,

D8(X, Y, Z)

=

d8(hX, hY, hZ)

=

(dw

A ()) (hX,

hY, hZ)

=

(0 A

()) (X,

Y, Z).

QED. Let B 1, ••• , B n be the standard horizontal vector fields corresponding to the natural basis e1 , • • • , en of Rn and {Et*} the fundamental vector fields corresponding to the basis {En of gI(n; R). It is easy to verify that {B i , Et*} and {()i, wJ} are dual to each other in the sense that ()k(B i ) = bT, ()k(Et*) = 0,

wr(B i ) = 0,

wr(E{*) = bTb{.

2.6. The n2 + n vector fields {B k, E{*; i, j, k = 1, ... , n} define an absolute parallelism in P, that is, the n2 + n vectors {(Bk)u, (Ei*)u}form 'a basis of Tu(P) for every u E P. Proof. Since the dimension of P is n2 + n, it is sufficient to prove that the above n 2 + n vectors are linearly independent. Since A ---+ A~ is a linear isomorphism of 9 onto the vertical subspace of Tu(P) (cf. §50fChapter I), {El*} are linearly independent at every point of P. By (3) of Proposition 2.2, {B k } are linearly independent at every point of P. Since {B k } are horizontal and {Ej*} are vertical, {B k, Ei*} are linearly independent at every point of P. PROPOSITION

QED.

Let T~(M) be the tensor bundle over M of type (r, s) (cf. Example 5.4 of Chapter I). It is a vector bundle with standard fiber T~ (tensor space over R n of type (r, s)) associated with the bundle P of linear frames. A tensor field K of type (r, s) is a cross section of the tensor bundle T~(M). In §1, we defined covariant derivatives of a cross section in a vector bundle in general. As in §1, we can define covariant derivatives of K in the following three cases: (1) V' XtK, when K is defined along a curve T = X t of M; (2) V' xK, when X E Tx(M) and K is defined in a neighborhood of X; (3) V' xK, when X is a vector field on M and K is a tensor field on M.

III.

123

LINEAR AND AFFINE CONNECTIONS

For the sake of simplicity, we state the following proposition in case (3) only, although it is valid in cases (1) and (2) with obvious changes. PROPOSITION 2.7. Let:r(M) be the algebra of tensor fields on M. Let X and Y be vector fields on M. Then the covariant dijferentiation has the following properties: (1) V x: :r (M) ~:r (M) is a type preserving derivation; (2) V x commutes with every contraction; (3) V xf = Xf for every function f on M; (4) V x + y = V x + V y ; (5) VfxK = f· V xKfor every functionf on M and K E :r(M). Proof. Let 7' = X t , 0 < t < 1, be a curve in M. Let T(x t ) be the tensor algebra over TXt (M) , T (x t) = ~T~ (x t) (cf. §3 of Chapter I). The parallel displacement along 7' gives an isomorphism of the algebra T(x o) onto the algebra T(x 1 ) which preserves type and commutes with every contraction. From the definition of covariant differentiation given in §l, we obtain (1) and (2) by an argument similar to the proof of Proposition 3.2 of Chapter I. (3), (4) and (5) were proved in Proposition 1.2. OED.

---

By the lemma for Proposition 3.3 of Chapter I, the operation of V x on :r(M) is completely determined by its operation on the algebra of functions ~(M) and the module of vector fields X(M). Since V x f = Xf for every f E ~ (M), the operation of V x on :r( M) is determined by its operation on X( M). As an immediate corollary to Proposition 1.2, we have PROPOSITION 2.8. If X, Y and Z are vector fields on M, then (1) Vx(Y+Z) =VxY+VxZ; (2) V x+yZ = V xZ + VyZ; (3) VfXY = f· V x Y for every f E ~(M); (4) V x (fY) =f' V x Y + (Xf) Y for every f E ~(M). We shall prove later in §7 that any mapping X(M) X X(M) ~ X(M), denoted by (X, Y) -> VxY, satisfying the four conditions above is actually the covariant derivative with respect to a certain linear connection. The proof of the following proposition, due to Kostant [1], is similar to that of Proposition 3.3 of Chapter I and hence is left to the reader.

124

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

2.9.

Let M be a manifold with a linear connection. Every derivation D (preserving type and commuting with contractions) of the algebra ~(M) of tensor fields into the tensor algebra T(x) at x E M can be uniquely decomposed as follows: PROPOSITION

D

=

Vx

+ S,

where X E Tx(M) and S is a linear endomorphism of Tx(M). Observe that, in contrast to Lie differentiation Lx with respect to a vector field, covariant differentiation V x makes sense when X is a vector at a point of M. Given a tensor field K of type (r, s), the covariant dijferential VK of K is a tensor field of type (r, s + 1) defined as follows. As in Proposition 2.11 of Chapter I, we consider a tensor of type (r, s) at a point x E M as a multilinear mapping of Tw(M) X •.. X Tx(M) (s times product) into T~(x) (space of contravariant tensors of degree r at x). We set

PROPOSITION

2.10.

If K

is a tensor field of type (r, s), then

where X,Xi E X(M). Proof. This follows from the fact that V x is a derivation commuting with every contraction. The proof is similar to that of Proposition 3.5 of Chapter I and is left to the reader. QED. A tensor field K on M, considered as a cross section of a tensor bundle, is parallel if and only if V xK = 0 for all X E Tx(M) and x E M (cf. §1). Hence we have PROPOSITION

VK =

o.

2.11.

A tensor field K on M is parallel if and only if

The second covariant differential V2K of a tensor field K of type (r, s) is defined to be V(V K), which is a tensor field of type (r, s + 2). We set (V2K) (;X; Y)

=

(Vy(VK))(;X),

where X,Y E Tx(M),

III.

125

LINEAR AND AFFINE CONNECTIONS

that is, if we regard K as a multilinear mapping of Tx(M) X Tx(M) (s times product) into T~(x), then

X .••

(V2K) (Xl' ... , X s ; X; Y) = (Vy(VK)) (Xl' ... , X s ; X). Similarly to Proposition 2.10, we have 2.12. X and Y, we have PROPOSITION

For any tensor field K and for any vector fields

(V2K) (;X; Y) = Vy(V xK) - VvyxK. In general, the m-th covariant differential vm K is defined inductively to be v(vm-lK). We use the notation (vmK) (;X l ; . .. ; X m- l ; X m) for (V x m (vm-l K)) (;Xl ; ... ; X m- l ).

3. Affine connections A linear connection of a manifold M defines, for each curve T = Xt, < t < 1, of M, the parallel displacement of the tangent space T xo (M) onto the tangent space T X1 (M); these tangent spaces are regarded as vector spaces and the parallel displacement is a linear isomorphism between them. We shall now consider each tangent space Tx(M) as an affine space, called the tangent affine space at x. From the viewpoint of fibre bundles, this means that we enlarge the bundle 0'£ linear frames to the bundle of affine frames, as we shall now explain. Let Rn be the vector space of n-tuples of real numbers as before. When we regard Rn as an affine space, we denote it by An. Similarly, the tangent space of M at x, regarded as an affine space, will be denoted by Ax (M) and will be called the tangent affine space. The group A(n; R) of affine transformations of An is represented by the group of all matrices of the form

°

a=

(~

:),

where a = (aj) EGL(n;R) and ~ = (~i), ~ERn, is a column vector. The element if maps a point 'YJ of An into a'YJ + ~. We have the following sequence: )

ex f3 0---+ Rn ---+ A(n; R) ---+ GL(n; R) ---+ 1,

126

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

n

where r:t. is an isomorphism of the vector group Rn into A(n; R) which maps

~ € Rn

into

(~n

n

€

A(n; R)

(In =

identity of

GL(n; R)) and f3 is a homomorphism of A(n; R) onto GL(n; R) which maps

(~

€

A(n; R) into a € GL(n; R). The sequence is

exact in the sense that the kernel of each homomorphism is equal to the image of the preceding one. It is a splitting exact sequence in the sense that there isahomomorphismy: GL(n; R) ---+ A(n; R) such that f3 y is the identity automorphism of GL(n; R); indeed, 0

we define y byy(a) =

(~ ~)

€

A(n; R), a € GL(n; R). The group

A(n; R) is a semidirect product of Rn and GL(n; R), that is, for every if E A(n; R), there is a unique pair (a, ~) E GL(n; R) X Rn such that ii = r:t..(~) • y(a). An affineframe ofa manifold M at x consists ofa pointp E Ax(M) and a linear frame (Xl' ... ' X n ) at x; it will be denoted by (p; Xl' ... , X n). Let 0 be the origin of Rn and (e l , . .. ,en) the natural basis for R n. We shall call (0; e1, • • • , en) the canonical frame of An. Every affine frame (p; Xl' ... ,Xn) at x can be identified with an affine transformation ii: An ---+ Ax(M) which maps (0; el , ... ,en) into (p; X l' • • • ,Xn), because (p; Xl' ... ,

X n ) ~ ii gives a 1: 1 correspondence between the set of affine frames at x and the set of affine transformations of Anon to Ax (M) . We denote by A(M) the set of all affine frames of M and define the projection iT: A(M) ---+ M by setting iT(ii) = x for every affine frame ii at x. We shall show that A(M) is a principal fibre bundle over M with group A(n; R) and shall call A(M) the bundle of affine frames over M. We define an action of A(n; R) on A(M) by (ii, if) ---+ iiif, ii E A(M) and ii E A(n; R), where iiif is the composite of the affine transformations if: An ---+ An and ii: An ---+ Ax(M). It can be proved easily (cf. Example 5.2 of Chapter I) that A(n; R) acts freely on A(M) on the right and that A(M) is a principal fibre bundle over M with group A(n; R). Let L (M) be the bundle of linear frames over M. Corresponding to the homomorphisms f3: A(n; R) ---+ GL(n; R) and y: GL(n; R) ---+A (n; R), we have homomorphisms f3: A(M) ---+ L(M) and y: L(M) ---+ A(M). Namely, f3: A(M) ---+ L(M) maps (p; Xl' ... ' X n) into (Xl' ... ' X n) and y: L(M) ---+A(M) maps

III.

127

LINEAR AND AFFINE CONNECTIONS

(Xl' ... , X n) into (ox; Xl' ... ,Xn), where Ox E Ax(M) is the point corresponding to the origin of Tx(M). In particular, L(M) can be considered as a subbundle of A(M). Evidently, (3 y is the identity transformation of L(M). A generalized affine connection of M is a connection in the bundle A(M) of affine frames over M. We shall study the relationship 0

between generalized affine connections and linear connections. We denote by Rn the Lie algebra of the vector group Rn. Corresponding to the splitting exact sequence 0 ---+ Rn ---+ A (n; R) ---+ GL(n; R) ---+ 1 of groups, we have the following splitting exact sequence of Lie algebras:

o ---+ Rn ---+ a(n; R)

---+

gI(n; R)

---+

O.

Therefore,

a(n; R)

=

gI(n; R)

+ Rn

(semidirect sum).

Let wbe the connection form of a generalized affine connection of M. Then y*w is an a(n; R)-valued I-form on L(M). Let

y*w

=

OJ

+ cp

be the decomposition corresponding to a(n; R) = gI(n; R) + Rn, so that OJ is a gI(n; R)-valued I-form on L(M) and cp is an Rn_ valued I-form on L(M). By Proposition 6.4 of Chapter II, OJ defines a connection in L (M). On the other hand, we see easily that cp is a tensorial I-form on L(M) of type (GL(n; R), Rn) (cf. §5 of Chapter II) and hence corresponds to a tensor field of type (1, 1) of M as explained in Example 5.2 of Chapter II. PROPOSITION

affine connection

Let w be the connection form of a generalized of M and let

3.1.

r

y*w =

OJ

+ cp,

where OJ is gI(n; R)-valued and cp is Rn-valued. Let r be the linear connection of M defined by OJ and let K be the tensor field of type (1, 1) of M defined by cpo Then (1) The correspondence between the set of generalized affine connections of M and the set of pairs consisting of a linear connection of M and a tensor field of type (1, 1) of M given by r ---+ (r, K) is 1: 1. (2) The homomorphism (3: A(M) ---+ L(NI) maps into r (if. §6 of Chapter II).

r

128

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. (1) It is sufficient to prove that, given a pair (r, K), there is which gives rise to (r, K). Let OJ be the connection form of rand cp the tensorial I-form on L(M) of type (GL(n; R), Rn) corresponding to K. Given a vector X E Ta(A(M)), choose X E Tu(L(M)) and a E A(n; R) such that ii = ua and X - Ra(X) is vertical. There is an element A E a(n; R) such that

r

X = Ra(X)

+ A;,

where A* is the fundamental vector corresponding to A. We define w by w(X) = ad (a-I) (OJ (X) + cp(X)) + A. It is straightforward to verify that w defines the desired connection

r.

(2) Let g E Ta(A(M)). We set u P(il) and X = P(X) so that X E Tu(L(M)). Since p: A(M) -+ L(M) is the homomorphism associated with the homomorphism p: A(n; R) -+ GL(n; R) = A(n; R)jRn, L(M) can be identified with A(M)jRn and p: A(M) -+ L(M) can be considered as the natural projection A(M) -+ A(M)jRn. Since X = P(X) = P(X), there exist a E Rn c A(n; R) and A E Rn c a(n; R) such that ii ua and X = R'U,(X) + A;. Assume that X is horizontal with respect to so that 0 = w(X) = w(Ra(X)) + w(A;) = ad (a- 1 )(w(X)) + A. Hence, w(X) = ad (a) (A) and OJ(X) + cp(X) = ad (a) (A). Since both cp(X) and ad (a) (A) are in Rn and OJ(X) is in gl(n; R), we have OJ(X) = O. This proves that if X is horizontal with respect to then P(X) is horizontal with respect to r. QED.

r

r,

3.2. In Proposition 3.1, let nand Q be the curvature forms of rand r respectively. Then PROPOSITION

y*n -

Q

+ Dcp,

where D is the exterior covariant differentiation with respect to r. Proof. Let X,Y E Tu(L(M)). To prove that (y*n)(~ Y) = Q(X, Y) + Dcp(X, Y), it is sufficient to consider the following two cases: (1) at least one of X and Y is vertical, (2) both X and Y are horizontal with respect to r. In the case (1), both sides vanish. In the case (2), OJ(X) = OJ(Y) = 0 and hence w(X) = p(X) and w(Y) p(Y). From the structure equation of r, we

III.

129

LINEAR AND AFFINE CONNECTIONS

have

dev(X, Y)

-t[m(X),m(Y)] -t[cp(X), cp(Y)]

+ Q(X, Y) + Q(X, Y).

(Here, considering L(M) as a subbundle of A(M), we identified y(X) with X.) On the other hand, y* dm = dw + dcp and hence dm(X, Y) = dw(X, Y) + dcp(X, Y). Since Rn is abelian, [cp(X), cp(Y)] = O. Hence, dw(X, Y) + dcp(X, Y) = Q(X, Y). Since both X and Yare horizontal, Dw(X, Y) + Dcp(X, Y) = Q(X, Y).

QED. Consider again the structure equation of a generalized affine connection dill = -t[m,m] + Q.

:r:

By restricting both sides of the equation to L(M) and by comparing the gI(n; R)-components and the Rn-components we obtain

dcp(X, Y) = -t([w(X), cp(Y)] - [w(Y), cp(X)]) dw(X, Y) = -t[w(X), w(Y)]

+ O(X, Y),

+ Dcp(X, Y),

X,Y

E

Tu(L(M)).

Just as in §2, we write

+ Dcp -w A w + O.

dcp = -w dw

=

A

cp

:r

A generalized affine connection is called an affine connection if, with the notation of Proposition 3.1, the Rn-valued I-form cp is the canonical form f) defined in §2. In other words, is an affine connection if the tensor field K corresponding to cp is the field of identity transformations of tangent spaces of M. As an immediate consequence of Proposition 3.1, we have

:r

The homomorphism f3: A(M) --+ L(M) maps every qffine connection of M into a linear connection r of M. Moreover, --+ r gives a 1: 1 correspondence between the set of affine connections :r of M and the set of linear connections r of M. THEOREM

:r

3.3.

:r

Traditionally, the words "linear connection" and "affine connection" have been used interchangeably. This is justified by Theorem 3.3. Although we shall not break with this tradition, we

130

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

shall make a logical distinction between a linear connection and an affine connection whenever necessary; a linear connection of M is a connection in L(M) and an affine connection is a connection in A(M). From Proposition 3.2, we obtain

°

3.4. Let 0 and be the torsionform and the curvature form of a linear connection r of M. Let 0 be the curvature form of the corresponding affine connection. Then PROPOSITION

y*O = 0 where y: L(M)

---+

+ 0,

A (M) is the natural injection.

Replacing cp by the canonical form () in the formulas:

dcp =

-OJ

A cp

+ Dcp,

dOJ =

-OJ

A

OJ

+ 0,

we rediscover the structure equations of a linear connection proved in Theorem 2.4. Let
r

3.5. The homomorphism (3: A(n; R) maps
---+

GL(n; R)

4. Developments We shall study in this section the parallel displacement arising from an affine connection of a manifold M. Let T = X t , 0 < t < 1, be a curve in M. The affine parallel displacement along T is an affine transformation of the affine tangent space at Xo onto the affine tangent space at Xl defined by the given connection in A (M). I t is a special case of the parallelism in an associated bundle which is, in our case, the affine tangent bundle whose fibres are Ax (M), X € M. We shall denote this affine parallelism by f.

III.

LINEAR AND AFFINE CONNECTIONS

131

The total space (i.e., the bundle space) of the affine .tangent bundle over M is naturally homeomorphic with that of the tangent (vector) bundle over M; the distinction between the two is that the affine tangent bundle is associated with A(M) whereas the tangent (vector) bundle is associated with L(M). A cross section of the affine tangent bundle is called a point field. There is a natural 1: 1 correspondence between the set of point fields and the set of vector fields. Let f; be the affine parallel displacement along the curve T from X t to X S ' In particular, f~ is the parallel displacement Ax/M) ---+ Axo(M) along T (in the reversed direction) from Xt to X o' Let p be a point field defined along T so that PXt is an element of Aa:t(M) for each t. Then f~(Px) describes a curve in Axo(M). We identify the curve T = X t with the trivial point field along T, that is, the point field corresponding to the zero vector field along T. Then the development of the curve T in M into the affine tangent space Axo(M) is the curve f~(Xt) in Axo(M). The following proposition allows us to obtain the development of a curve by means of the linear parallel displacement, that is, the parallel displacement defined by the corresponding linear connection.

4.1. Given a curve T = Xt, a < t ::;;: 1, in M, set Y t = T~(Xt), where T~ is the linear parallel displacement along T from Xt to Xo and xt is the vector tangent to T at Xt. Let Ct, 0 < t < 1, be the curve in Axo(M) starting from the origin (that is, Co = xo) such that Ct is parallel (in the affine space Axo(M) in the usual sense) to Ytfor every t. Then Ct is the development of T into A xo (M). Proof. Let Uo be any point in L(M) such that 7T(U O) = X o and U t the horizontal lift of X t in L(M) with respect to the linear connection. Let fit be the horizontal lift of Xt in A(M) with respect to the affine connection such that fio = UO• Since the homomorphism f3: A(M) ---+ L(M) = A(M)jRn (cf. §3) maps fit into ut, there is a curve at in Rn c A (n; R) such that fit = utat and that ao is the identity. As in the proof of Proposition 3.1 of Chapter II, we shall find a necessary and sufficient condition for at in order that fit be horizontal with respect to the affine connection. From . . fit = uta t ~ utat PROPOSITION

132

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

which follows from Leibniz's formula as in the proof of Proposition 3.1 of Chapter II, we obtain

w(u t)

=

=

+ ilt-lat ad (ilt- l ) (CO (u t ) + O(u t)) + ilt-lat = ad (at l )(w(u t ))

ad (ilt- l )(O(u t))

+ ilt-lat,

where wand co are the connection forms of the affine and the linear connections respectively. Thus ut is horizontal if and only ifO(u t) = -atilt- l . Hence,

Y t = T~(.it) = uo(ut-l(i t)) = uo(O(u t))

= -uo(il tilt- l ) = -uo(dilt! dt). On the other hand, we have Ct = T~(Xt) = uo(ut-l(x t)) = uo(ilt-l(Ut-l(Xt))) = uo(ilt-l(O)).

Hence, QED.

4.2. The development of a curve T = X t , 0 < t < 1, into Axo(M) is a line segment if and only if the vectorfields it along T = Xt is parallel. Proof. In Proposition 4.1, C t is a line segment if and only if Y t is independent of t. On the other hand, Y t is independent of t if and only if it is a parallel vector field along T. QED. COROLLARY

5. Curvature and torsion tensors We have already defined the torsion form 8 and the curvature form n of a linear connection. We now define the torsion tensor field (or simply, torsion) T and the curvature tensor field (or simply, curvature) R. We set

T (X, Y)

=

u(28(X*, Y*))

for X,Y

E

Ta,(M),

where u is any point of L(M) with 7T(U) = x and x* and y* are vectors of L(M) at u with 7T(X*) = X and 7T( Y*) = Y. We already know that T( X, Y) is independent of the choice of u, X*, and y* (cf. Example 5.2 of Chapter II); this fact can be easily verified directly also. Thus, at every point x of M, T defines a skew symmetric bilinear mapping Tx(M) x Tx(M) ~ Tx(M). In

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133

LINEAR AND AFFINE CONNECTIONS

other words, Tis a tensor field of type (1, 2) such that T(X, Y) = - T(Y, X). We shall call T(X, Y) the torsion translation in Tx(M) determined by X and Y. Similarly, we set

R(X, Y)Z = u((20(X*, Y*)) (u-1Z)

for X,Y,Z

€

Tx(M),

where u, x* and y* are chosen as above. Then R(X, Y)Z depends only on X, Yand Z, not on u, x* and Y*. In the above definition, (20(X*, Y*)) (u-1Z) denotes the image of u-1Z € Rn by the linear endomorphism 20(X*, Y*) € glen; R) of Rn. Thus, R(X, Y) is an endomorphism of Tx(M) and is called the curvature transformation of T x(M) determined by X and Y. I t follows that R is a tensor field of type (1, 3) such that R (X, Y) =

-R(Y, X). 5.1. In terms of covariant differentiation, the torsion T and the curvature R can be expressed as follows: THEOREM

T(X, Y)

=

VxY - VyX- [X, Y]

and where X, Y and Z are vector fields on M. Proof. Let X*, y* and z* be the horizontal lifts of X, Y and Z, respectively. We first prove (V xY)x = u(X:(O(Y*))), where 7T(U) = x. Proof of Lemma. In the lemma for Proposition 1.1, we proved that (V xY)x = u(X:f), wherefis an Rn-valued function defined by feu) = u-1(Yx). Hence, feu) = O(Y:) for u € L(M). This completes the proof of the lemma. LEMMA.

We have therefore

T(Xx, Y x)

=

u(20(X:, Y:))

=

u(X:(O(Y*)) - Y:(O(X*)) - O([X*, Y*]J

=

(V xY)x - (VyX)x -[X, Y]x,

since 7T([X*, Y*]) = [X, Y]. To prove the second equality, we setf = O(Z*) so thatfis an Rn-valued function on L(M) of type (GL(n; R), Rn). We have

134

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

then

([V x, V y ]Z - V[X,Y]Z) ro

= u(X;(Y*f) - y;(x*f) - (h[X*, Y*])uf)

=

u((v[x*, Y*])uf),

where h (resp. v) denotes the horizontal (resp. vertical) component. Let A be an element of gI(n; R) such that A; = (v[X*, Y*])u, where A* is the fundamental vector field corresponding to A. Then by Corollary 5.3 of Chapter II, we have

20(X;, Y;) = -w([X*, Y*]J = -A. On the other hand, if at is the I-parameter subgroup of GL(n; R) generated by A, then

A;f = lim t-+O

= lim t-+O

=

~ t

[f(ua t ) -f(u)]

~ [at- 1 (f(u)) t

-f(u)]

-A(f(u)),

where A(f(u)) denotes the result of the linear transformation A: Rn ~ Rn applied tof (u) € Rn. Therefore, we have

([V x, Vy]Z - V[X,y]Z)ro = u((v[X*, Y*])uj) = u(-A(j(u)))

= u(20(X;, Y;)(f(u))) = u(20(X;, Y;)(u-1Z)) = R(X, Y)Z. QED. 5.2. Let X,Y,Z, W € Tro(M) and u € L(M) with 1T(U) = x. Let X*, Y*, z* and W* be the standard horizontal vector fields on L (M) corresponding to u-1 X, u-1 Y, u-1 Z and u-1 W respectively, so that 1T( X;) = X, 1T( Y;) = Y, 1T( Z;) = Z and 1T( W;) = W. Then (VxT)(Y, Z) = u(X;(28(Y*, Z*))) and ((VxR)(Y, Z))W = u((X;(20(Y*, Z*)))(u-1W)). PROPOSITION

Proof. We shall prove only the first formula. The proof of the second formula is similar to that of the first. We consider the torsion T as a cross section of the tensor bundle T~ (M) whose standard fibre is the tensor space T~ of type (1, 2) over Rn. Let f be the T~-valued function on L(M) corresponding to the

III.

135

LINEAR AND AFFINE CONNECTIONS

torsion T as in Example 5.2 of Chapter II so that, if we set r; = u-1 Y and ~ = u-1 Z, then

fu(r;, ~) By Proposition 1.3,

u-1(T(X, Z)) =%>(Y~, Z~).

=

X~

f

corresponds to V x T. Hence, (X:f)(r;,~)

u-1((VxT)(Y, Z)) = =

X~(f (r;, ~))

X~(20(Y*,

=

Z*))

QED.

thus proving our assertion.

Using Proposition 5.2, we shall express the Bianchi's identities (Theorem 2.5) in terms of T, R and their covariant derivatives.

5.3. Let T and R be the torsion and the curvature of a linear connection of M. Then, for X,Y,Z E Tx(M), we have Bianchi's 1st identity: THEOREM

6{R(X, Y)Z} ~ianchi's

6{T(T(X, Y), Z)

=

+ (VxT)(Y,

Z)};

2nd identity : 6{(V xR) (Y, Z)

+ R( T (X,

Y), Z)} = 0,

where 6 denotes the cyclic sum with respect to X, Y and Z. In particular, if T = 0, then Bianchi's 1st identity: 6{R(X, Y)Z} = 0; Bianchi's 2nd identity: 6{(V xR) (Y, Z)} = o. Proof. Let u be any point of L(M) such that 7T(U)

x. We lift X to a horizontal vector at u and then extend it to a standard horizontal vector field x* on L(M) as in Proposition 5.2. Similarly, we define y* and Z*. We shall derive the first identity from D0 = 0 A 0 (Theorem 2.5). We have

6(0

A

0) (X~, Y~, Z:)

=

=

6{20(X~, Y~) O(Z:)}

=

6{u- 1 (R(X, Y) Z)}.

On the other hand, by Proposition 3.11 of Chapter I, we have 6D0(X~, Y~,

Z:) = =

6d0(X~, Y~,

Z:)

6{X~(20(Y*,

Z*)) - 20([X*, Y*]u, Z:)}.

136

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

By Proposition 5.2, X:(28(Y*, Z*)) = u-I((VxT)(Y, Z)). It is therefore sufficient to prove that

-28([X*, Y*]w Z:) = u-I(T(T(X, Y), Z)). We observe first that

7T([X*, Y*]u) = u(O[X*, Y*]J) = -u(2dO(X:, Y:)) = -u(28(X:, Y:))

=

-

T (X, Y).

Hence we have

-28([X*, Y*]u, Z:)

-u- 1(T(7T[X*, Y*]u, Z))

=

= u- l ( T( T(X, Y), Z)). We shall derive the second identity from

DO.

=

0

(Theorem 2.5).

We have

o=

3Do.(X:, Y:, Z:)

= 6{X:(o.(y*, Z*)) - o.([X*, Y*]u' Z:)}. On the other hand, by Proposition 5.2, we have

X:(O.(Y*, Z*))

=

tu-I((VxR)(Y, Z)).

As in the proof of the first identity, we have

-o.([X*, Y*]u, Z:) = iU-1(R (T(X, Y), Z)). The second identity follows from these three formulas.

QED.

Remark.

Theorem 5.3 can be proved from the formulas in Theorem 5.1 (see, for instance, Nomizu [7, p. 611]).

5.4. Let Band B' be arbitrary standard horizontal vector fields on L (M ). Then we have . (1) If T = 0, then [B, B'] is vertical; (2) If R = 0, then [B, B'] is horizontal. Proof. (1) O([B, B']) = -2dO(B, B') = -28(B, B') = O. Hence, [B, B'] is vertical. (2) w([B, B']) = -2dw(B, B') = -2o.(B, B') = O. Hence, [B, B'] is horizontal. QED. PROPOSITION

Let P(u o) be the holonomy subbundle of L(M) through a point U o E L(M) and 'Y(u o) the linear holonomy group with reference point U o• Let AI, ... , A r be a basis of the Lie algebra of

III.

LINEAR AND AFFINE CONNECTIONS

137

o/(uo) and At, ... ,Ai the corresponding fundamental vector fields. Let B 1, ••• ,Bn be the standard horizontal vector fields corresponding to the basis e1, ••• , en of Rn. These vector fields At, ... ,A~, B b . . . , B n (originally defined on L(M)), restricted to P(uo), define vector fields on P(uo). Just as in Proposition 2.6, they define an absolute parallelism on P(uo). We know that [Ai, At] is the fundamental vector field corresponding to [Ai, A;] and hence is a linear combination of At, ... ,Ai with constant coefficients. By Proposition 2.3, [A{, B;] is the standard horizontal vector field corresponding to Aie; € R n. The following proposition gives some information about [B i , B;]. 5.5. Let P(uo) be the holonomy subbundle of L( M) through Uo. Let Band B' be arbitrary standard horizontal vector fields. Then we have (1) If V T = 0, then the horizontal component of [B, B'] coincides with a standard horizontal vector field on P(uo). (2) If VR = 0, then the vertical component of [B, B'] coincides with the fundamental vector field A * on P(uo), which corresponds to an element A of the Lie algebra of the linear holonomy group 0/ (u o). Proof. (1) Let X* be any horizontal vector at u € L(M). Set X = 7T(X*), Y = 7T(B u ) and Z = 7T(B~). By Proposition 5.2, we have X*(20(B, B')) = u-1 ((V xT)(Y, Z)) = 0. PROPOSITION

This means that 0(B, B') is a constant function (with values in Rn) on P(uo). Since e([B, B']) = -20(B, B'), the horizontal component of [B, B'] coincides on P(u o) with the standard horizontal vector field corresponding to the element -20(B, B') ofRn. (2) Again, by Proposition 5.2, VR = implies

°

X*(O(B, B')) = 0. This means that O(B, B') is a constant function on P(u o) (with values in the Lie algebra of o/(uo)). Since w([B, B']) = -20(B, B'), the vertical component of [B, B'] coincides on P (u o) with the fundamental vector field corresponding to the element -20(B, B') of the Lie algebra of o/(uo). QED. I t follows that, if V T = and VR - 0, then the restriction of [Bi' B j] to P (u o) coincides with a linear combination of At, . . . ,

°

138

Ai, B 1 ,

FOUNDATIONS OF DIFFERENTIAL GEOMETRY ••• ,

B n with constant coefficients on P(u o). Hence we have

5.6. Let 9 be the set of all vector fields X on the holonomy bundle P(u o) such that O(X) and w(X) are constant functions on P (u o) (with values in R n and in the Lie algebra ofT (u o), respectively). If V T = 0 and VR = 0, then 9 forms a Lie algebra and dim 9 = dim P(u o). COROLLARY

The vector fields Ai, ... , Ai, B 1, a basis for g.

... ,

B n defined above form

6. Geodesics A curve T = X t , a < t < b, where - 00 :s;: a < b :s;: 00, of class C in a manifold M with a linear connection is called a geodesic if the vector field X = xt defined along T is parallel along T, that is, if VxX exists and equals 0 for all t, where xt denotes the vector tangent to T at X t • In this definition of geodesics, the parametrization of the curve in question is important. 1

6.1. Let T be a curve of class C1 in M. A parametrization which makes T into a geodesic, if any, is determined up to an affine transformation t s = if.t + (3, where if. i=- 0 and (3 are constants. Proof. Let X t and y s be two parametrizations of a curve T which make T into a geodesic. Then s is a function of t, s = s(t), PROPOSITION

-)0-

and Y8(') =

X t·

The vector j8 is equal to

displacement along

T

&

~: Xt.

Since the parallel

is independent of parametrization (cf. §3

-

of Chapter II), ds must be a constant different from zero. Hence,

s = if.t

+ (3, where

if.

i=- O.

QED.

If T is a geodesic, any parameter t which makes T into a geodesic is called an affine parameter. In particular, let x be a point of a geodesic T and X E Tx(M) a vector in the direction of T. Then there is a unique affine parameter t for T, T = X t , such that X o = x and Xo = X. The parameter t is caned the affine parameter for T determined by (x, X).

6.2. A curve T of class C1 through x E M is a geodesic its development into Tx(M) is (an open interval of) a

PROPOSITION

if and

only if straight line.

III.

Proof.

LINEAR AND AFFINE CONNECTIONS

139

This is an immediate consequence of Corollary 4.2. QED.

Another useful interpretation of geodesics is given in terms of the bundle of linear frames L(M). 6.3. The projection onto M of any integral curve of a standard horizontal vector field of L (M) is a geodesic and, converse(y, every geodesic is obtained in this way. Proof. Let B be the standard horizontal vector field on L (M) which corresponds to an element ~ ERn. Let bt be an integral curve of B. We set X t = 7T(b t). Then x t = 7T(b t ) = 7T(Bb) = bt~, where bt~ denotes the image of ~ by the linear mapping bt : Rn ~ TXt(M). Since bt is a horizontal lift of X t and ~ is independent of t, bt~ is parallel along the curve X t (see §7, Chapter II, in particular, before Proposition 7.4). Conversely, let X t be a geodesic in M defined in some open interval containing O. Let Uo be any point of L(M) such that 7T(U O) = X O• We set ~ = U01x o ERn. Let Ut be the horizontal lift of X t through UO• Since X t is a geodesic, we have x t = Ut~. Since Ut is horizontal and since O(u t) = Ut- 1(7T(U t)) = ut-1X t = ~, Ut is an integral curve of the standard horizontal vector field B corresponding to ~. QED. PROPOSITION

As an application of Proposition 6.3, we obtain the following 6.4. For any point x E M andfor any vector X E Tx(M), there is a unique geodesic with the initial condition (x, X), that is, a unique geodesic x t such that X o = x and Xo = X. THEOREM

Another consequence of Proposition 6.3 is that a geodesic, which is a curve of class Cl, is automatically of class Coo (provided that the linear connection is of class COO). In fact, every standard horizontal vector field is of class Coo and hence its integral curves are all of class Coo. The projection onto M of a curve of class Coo in L(M) is a curve of class Coo in M. A linear connection of M is said to be complete if every geodesic can be extended to a geodesic T = X t defined for - 00 < t < 00, where t is an affine parameter. In other words, for any x E M and X E Tx(M), the geodesic T = X t in Theorem 6.4 with the initial condition (x, X) is defined for all values of t, - 00 < t < 00.

140

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Immediate from Proposition 6.3 is the following 6.5. A linear connection is complete if and only every standard horizontal vector field on L(M) is complete. PROPOSITION

if

We recall that a vector field on a manifold is said to be complete ifit generates a global I-parameter group of transformations of the manifold. When the linear connection is complete, we can define the exponential map at each point x € M as follows. For each X € Tx(M), let T = X t be the geodesic with the initial condition (x, X) as in Theorem 6.4. We set exp X = Xl' Thus we have a mapping of Tx(M) into M for each x. We shall later (in §8) define the exponential map in the case where the linear connection is not necessarily complete and discuss its differentiability and other properties.

7. Expressions in local coordinate systems In this section, we shall express a linear connection and related concepts in terms of local coordinate systems. Let M be a manifold and U a coordinate neighborhood in M with a local coordinate system xl, ... ,xn • We denote by Xi the vector field oj ox i , i = 1, ... , n, defined in U. Every linear frame at a point x of U can be uniquely expressed by (~i Xl (Xi) x,

••. ,

~i X~(Xi)X)'

where det (XD =J= O. We take (Xi, XD as a local coordinate system in 'TT-l(U) c L(M). (cf. Example 5.2 of Chapter I). Let (Y{) be the inverse matrix of (Xi) so that ~i X~YJ = ~iY1XJ = ~~. We shall express first the canonical form e in terms of the local coordinate system (Xi, Xi). Let el , • . . , en be the natural basis for Rn and set e = ~i eie i· 7.1. In terms of the local coordinate system (Xi, X~), the canonical form e = ~i eie i can be expressed as follows: PROPOSITION

ei

=

~j

Y} dx i .

III.

141

LINEAR AND AFFINE CONNECTIONS

Proof. Let u be a point of L(M) with coordinates (Xi, Xi) so that u maps ei into ~j X{(XjL" where x = 1T(U). If X* € Tu(L(M)) and if

x* = so that 1T(X*)

~j

=

L; A{J~;t +

Lj,k

A{(a~J

Aj(Xj)x, then

O(X*) = U-l(~j Aj(Xj)x) = ~i,j(Yj Aj) ei •

QED. Let w be the connection form of a linear connection With respect to the basis {En of gI(n; R), we write W --

r

of M.

'" k.Ji,j WjiEji.

Let a be the cross section of L(M) over U which assigns to each x € U the linear frame ((XI)X, ... , (Xn)x). We set W

u = a*w.

Then W u is a gI(n; R)-valued I-form defined on U. We define n3 functions rlk' i, j, k = 1, . . . , n, on U by

Wu

=

~ IJ, .. k(r~k dxj)E~. 3 I

These functions r;k are called the components (or Christoffel's symbols) of the linear connection r with respect to the local coordinate system xl, ... , x n • It should be noted that they are not the components of a tensor field. In fact, these components are subject to the following transformation rule.

7.2. Let r be a linear connection of M. Let rJk and rJk be the components of r with respect to local coordinate systems xl, ... , xn and xI,· ... , xn, respectively. In the intersection of the two coordinate neighborhoods, we have PROPOSITION

. ox j ox k ox '~krjk oxf3 oir ox i 1,3,

lX

rpy

=

02 Xi oxlX

+ ~i oxf3 oxYoxi .

Proof. We derive the above formula from Proposition 1.4 of Chapter II. Let V be the coordinate neighborhood where the coordinate system Xl, ... , xn is valid. Let ii be the cross section of L(M) over V which assigns to each x € V the linear frame

142

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

((iJjiJi1)x, ... , (iJjiJin)x). We define a mapping VJuv: Un V-)GL(n; R) by a(x) = a(x) . VJuv(x) for x € U n V. Let cp be the (left invariant gI(n; R)-valued) canonical I-form on GL (n; R) defined in §4 of Chapter I; this form was denoted by e in §4 of Chapter I and in §1 of Chapter II. If (s1) is the natural coordinate system in GL(n; R) and if (tj) denotes the inverse matrix of (s}), then . • k m ~. . k t~J dskJ E· T = ~,J,

~,

the proof being similar to that of Proposition 7.1. It is easy to verify that and hence

'Pfrvrp =

~.'P(~i ~; d(;;:) )E~ = ~.A~i'Y ~~: a::~~Y dXY)~.

With our notation, the formula in Proposition 1.4 of Chapter II can be expressed as follows:

Wv

=

(ad (VJuf) )w u

+ VJtvCP·

By a simple calculation, we see that this formula is equivalent to the transformation rule of our proposition. QED. From the components r}k we can reconstruct the connection form w. 7.3. Assume that, for each local coordinate system xl, . . . , x , there is given a set of functions r}k' i, j, k = 1, ... , n, in such a way that they satisfy the transformation rule of Proposition 7.2. Then there 'is a unique linear connection r whose components with respect to xl, ... , xn are precisely the given functions r}k. Moreover, the connection form w = ~i, j wJ E{ is given in terms of the local coordinate system (Xi, Xi) by PROPOSITION n

wj =

~k Yi(dXJ

+ ~l,m r~lXJ dx m ) ,

i,j = 1, ... , n.

Proof. It is easy to verify that the form w defined by the above formula defines a connection in L(M), that is, w satisfies the conditions (a /) and (b /) of Proposition 1.1 of Chapter II. The fact that w is independent of the local coordinate system used

III.

LINEAR AND AFFINE CONNECTIONS

143

follows from the transformation rule of rjk; this can be proved by reversing the process in the proof of Proposition 7.2. The cross section a: U ---+ L (M) used above to define wu is given in terms of the local coordinate systems (Xi) and (xi, X{) by (Xi) ---+ (xi, £51). Hence, a*wj = ~mr~Lj dx m • This shows that the components of the connection r defined by ware exactly the functions rjk' QED. The components of a linear connection can be expressed also in terms of covariant derivatives.

7.4. Let xl, ... , xn be a local coordinate system in M with a linear connection r. Set Xi = 0/ ox i, i = 1, ... ,n. Then the components rJk of r with respect to Xl, . • . , xn are given by PROPOSITION

V x 1 X.'t .L

=

~k r~,Xk' J't

Proof. Let Xj be the horizontal lift of Xj' From Proposition 7.3, it follows that, in terms of the coordinate system (Xi, X{), X* is given by j Xj = (%x ) - ~i,k,~ rjkXf(%XD· To apply Proposition 1.3, let f be the Rn-valued function on 7T- I ( U) c L(M) which corresponds to Xi' Then

f

=

~k

Y7 e

k'

A simple calculation shows that

Xjf = ~k,~rjiYfek' By Proposition 1.3, Xjf is the function corresponding to V x,Xi and hence V x , X.'t = ~k r~. J't X k • QED.

7.5. Assume that a mapping X( M) X 1:( M) ---+ X(M), denoted by (X, Y) ---+ V xY, is given so as to satisfy the conditions (1), (2), (3) and (4) of Proposition 2.8. Then there is a unique linear connection r of M such that V x Y is the covariant derivative of Y in the direction of X with respect to r. Proof. Leaving the detail to the reader, we shall give here an outline of the proof. Let x € M. If X, X', Yand Y' are vector fields on M and if X = X' and Y = Y' in a neighborhood of x, then (V xY)oo = (V x,Y')w' This implies that the given mapping PROPOSITION

144

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

X(M) X X(M) ~ X(M) induces a mapping X( U) X X( U) -+ X( U) satisfying the same conditions of Proposition 2.8 (where U is any open set of M). In particular, if U is a coordinate neighborhood with a local coordinate system xl, ... ,xn , we define n3 functions rjk on U by the formula given in Proposition 7.4. Then these functions satisfy the transformation rule of Proposition 7.2. By Proposition 7.3, they define a linear connection, say, r. It is clear that V xY is the covariant derivative of Y in the direction of X with respect to r. QED. Let r/ be the components of a vector field Y with respect to a local coordinate system xl, ... ,xn , Y = ~i r/( 0 j ox i ). Let 'YJ~J be the components of the covariant differential V Y so that V x Y = ~i 'YJfjXi , where Xi = ojox i • From Propositions 7.4 and 2.8, we obtain the following formula: 3

'YJ~j = o'YJ i j ox

j

+ ~ k rJk'YJk.

If X is a vector field with components ~i, then the components of V xY are given by ~j 'YJ~j~j. More generally, if K is a tensor field of type (r, s) with components Kj~:: :1:, then the components of VK are given by K~1 ... ~r. = 11·. ·Ja,k

OK~1." ~rjoxk

J1" ·Ja

+ ~ra;-1 (~ _

l

riIXK~1'" Z." kZ h··· Ja _ ~s _ (~

/3 -1

. ir)

rm:kJ pK~1'" mi ••• r

m

J1

•••

.) Ja '

where l takes the place of ia; and m takes the place ofJ/3. The proof of this formula is the same as the one for a vector field, except that Proposition 2.7 has to be used in place of Proposition 2.8. If X is a vector field with components ~i, then the components of V xK are given by The covariant derivatives of higher order can be defined similarly. For a t~nso! field K with components KJ~:::j:, VmK has components KJ~1" 'J~r. k . ·k. 1· .. s' 1'···' m The components TA of the torsion T and the components R}kl of the curvature R are defined by T(Xj, X k) = ~i TYkXz, R(Xk, ~)Xj = ~i R;kZXi· Then they can be expressed in terms of the components rJk of the linear connection r as follows.

III. PROPOSITION

We have

7.6.

i -ri T jle - jle -

Ifjlel

=

145

LINEAR AND AFFINE CONNECTIONS

(or;)OX k

-

rio lej'

+ 1: (rli r 1m -

arkjjox l )

m

riJr~m)'

Proof. These formulas follow immediately from Theorem 5.1 and Proposition 7.4. QED. The proof of the following proposition is a straightforward calculation. PROPOSITION

7.7.

(1)

Iff

is afunction defined on M, then

!le;j - !j;le = 1: i T1j!i' (2)

If X

is a vector field on M with components ~~l;le - ~~le;l = 1: j Rjlel~j

+ 1:

j

~i,

then

Tile~~j'

wt,

Since n2 + n I-forms ()i, i,j, k = 1, ... , n, define an absolute parallelism (Proposition 2.6), every differential form on L(M) can be expressed in terms of these I-forms and functions. Since the torsion form 0 and the curvature form n are tensorial forms, they can be expressed in terms of n I-forms ()i and functions. We define a set of functions Tjle and R}kl on L (M) by

0 i = 1:.J,k 1.2 T~le()j J

A ()k' J T~le

= - T lei l'.

These functions are related to the components of the torsion T and the curvature R as follows. Let a: U ---+ L(M) be the cross section over U defined at the beginning of this section. Then a * T-iJOle = T JiOle ,

i i a *R-jkl = R jlel'

These formulas follow immediately from Proposition 7.6 and from a* d()i = -1:.J a*w~J A a*()j + a*0 i , a* dw~J

=

a*()i = dx i PROPOSITION

7.8.

Let

-1: k a*w ile

and Xi

A a*w~J

+ a*n~

a*w} = 1: k

'1'

r1j dx k •

= Xi (t) be the equations of a curve

146 T

=

FOUNDATIONS OF DIFFERENTIAL GEOMETRY Xt

of class C2. d2x i dt 2

is a geodesic if and only if . dx i dx k ~j,krjk dt dt = 0, i = 1, ... ,- n.

Then

+

T

Proof. The components of the vector field xt along Tare given by dxijdt. From the formula for the components of V xY given above, we see that, if we set X = Xt, then V xX = 0 is equivalent to the above equations. QED. We shall compare two or more linear connections by their components. PROPOSITION 7.9. Let r be a linear connection of M with components r}k. For each fixed t, 0 < t < 1, the set of functions r*jk = trjk + (1 - t) r1j defines a linear connection r* which has the same geodesics as r. In particular, r*jk = t(r]k + r1j) define a linear connection with vanishing torsion. Proof. Our proposition follows immediately from Propositions 7.3, 7.6 and 7.8. QED. In general, given two linear connections r with components r]k and r' with components r'}k' the set of functions trjk + (1 - t) r'}k define a linear connection for each t, 0 < t < 1. Proposition 7.8 implies that rand r' have the same geodesics if rjk + r1j = r';k + r'ij· The following proposition follows from Proposition 7.2. PROPOSITION 7.10. If r]k and r'}k are the components of linear connections rand r' respectively, then SJk = r'jk - rjk are the components of a tenor field of type (1, 2). Conversely, if rjk are the components of a linear connection rand Sjk are the components of a tensor field S of type (1, 2), then r'}k = rjk + S]k define a linear connection r'. In terms of covariant derivatives, they are related to each other as follows: V~Y = V x Y + S(X, Y) for any vector fields X and Y on M,

where V and V' are the covariant differentiations with respect to rand respectively.

r'

8. Normal coordinates In this section we shall prove the existence of normal coordinate systems and convex coordinate neighborhoods as well as the differentiability of the exponential map.

III.

LINEAR AND AFFINE CONNECTIONS

147

Let M be a manifold with a linear connection r. Given X E Tx(M), let T = Xl be the geodesic with the initial condition (x, X) (cf. Theorem 6.3). We set exp tX = x t • As we have seen already in §6, exp tX is defined in some open interval -£1 < t < £2' where £1 and £2 are positive. If the connection is complete, the exponential map exp is defined on the whole of T x (M) for each x E M. In general, exp is defined only on a subset of Tx(M) for each x E M. 8.1. Identifying each x E M with the zero vector at x, we consider M as a submanifold of T(M) = U Tx(M). Then there PROPOSITION

x€M

is a neighborhood N of M in T( M) such that the exponential map is defined on N. The exponential map N ~ M is dijJerentiable of class Coo, provided that the connection is of class Coo. Proof. Let X o be any point of M and Uo a point of L(M) such that 7T(U O) xo. For each ~ ERn, we denote by B(~) the corresponding standard horizontal vector field on L(M) (cf. §2). By Proposition 1.5 of Chapter I, there exist a neighborhood u* of U o and a positive number 0 such that the local I-parameter group of local transformations exp tB(~): u* ~ L(M) is defined for It I < o. Given a compact set K of Rn, we can choose U* and 0 for all ~ E K simultaneously, because B(~) depends differentiably on ~. Therefore, there exist a neighborhood u* of U o and a neighborhood V of 0 in Rn such that exp tB(~): u* ~ L(M) is defined for ~ E V and It I 1. Let U be a neighborhood of X o in M and Ga cross section of L(M) over U such that G(xo) = U o and G( U) c U*. Given x E U, let N x be the set of X E Tx(M) such that G(X)-IX E V and set N(x o) = U N x • Given X E N x , set x€U

~ =

G(X) -IX. Then 7T( (exp tB(~)) . G(x)) is the geodesic with the initial condition (x, X) and hence exp X = 7T«exp B(~)) . G(x)). It is now clear that exp: N(x o) ~ M is differentiable of class Coo. Finally, we set N = U N(x o). QED. xo€M

8.2. For every point x E M, there is a neighborhood N x of x (more precisely, the zero vector at x) in Tx(M) which is mapped difJeomorphically onto a neighborhood Ux of x in M by the exponential map. PROPOSITION

148

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. From the definition of the exponential map, it is evident that the differential of the exponential map at x is nonsingular. By the implicit function theorem, there is a neighborhood N ro of x in Tro(M) which has the property stated above. QED. Given a linear frame u = (Xl' ... ,Xn) at x, the linear isomorphism u: Rn -)- Tro(M) defines a coordinate system in Tro(M) in a natural manner. Therefore, the diffeomorphism exp: N ro -)Uro defines a local coordinate system in Uro in a natural manner. We call it the normal coordinate system determined by the frame u. PROPOSITION 8.3. Let xl, ... ,xn be the normal coordinate system

determined by a linear frame u = (Xl, ... ,Xn) at x € M. Then the geodesic T = Xl with the initial condition (x, X), where X = ~i aiXi, is expressed by n Xi = ait ,; - 1 ,

c,-

,

•••

,

•

Conversely, a local coordinate system xl, ... , xn with the above property is necessarily the normal coordinate system determined by u = (Xl, ... , Xn). Proof. The first assertion is an immediate consequence of the definition of a normal coordinate system. The second assertion follows from the fact that a geodesic is uniquely determined by the initial condition (x, X). QED. Remark. In the above definition of a normal coordinate system, we did not specify the neighborhood in which the coordinate system is valid. This is because if xl, ... , xn is the normal coordinate system valid in a neighborhood U of x andy!, ... ,yn is the normal coordinate system valid in a neighborhooEl--Y of x and if the both are determined by the frame u = (Xl' ... , X n), then they coincide in a neighborhood of x. 8.4. Given a linear connection r on M, let rjk be its components with respect to a normal coordinate system with origin XO• Then PROPOSITION

r}k

+ rtj

=

0

at

X O•

Consequently, if the torsion of r vanishes, then r;k = 0 at XO• Proof. Let xl, . .. ,xn be a normal coordinate system with origin XO• For any (a\ ... , an) € Rn, the curve defined by Xi = ait, i = 1, ... ,n, is a geodesic and, hence, by Proposition 7.8,

III.

~j,k

149

LINEAR AND AFFINE CONNECTIONS

r}k(alt, ... , ant)aia k

=

~j,k

O. In particular,

r}k

(x o)aia k = O.

Since this holds for every (a\ ... , an), r;k + r1j = 0 at Xo. If the torsion vanishes, then r}k = 0 at X o by Proposition 7.6. QED.

8.5. Let K be a tensor field on M with components n wJth K;~ :::}: with respect to a normal coordinate system Xl, ••• origin Xo. If the torsion vanishes, then .the ~ovariant derivative Kj~:: :j:; k coincides with the partial derivative oK;~::: j:/ ox k at Xo. Proof. This is immediate from Proposition 8.4 and the formula for the covariant differential of K in terms of rjk given in §7. QED. COROLLARY

8.6. torsion vanishes, then COROLLARY

,x.

Let w be any differential form on M.

If

the

1-1,r

dw

''vA (V w),

where V w is the covariant differential of wand A is the alternation defined in Example 3.2 of Chapter 1. Proof. Let X o be an arbitrary point of M and xl, ... ,xn a normal coordinate system with origin X o• By Corollary 8.5, dw = A(Vw) at X o• QED. THEOREM 8.7. Let Xl, ••• , xn be a normal coordinate system with origin Xo. Let U(x o; p) be the neighborhood of Xo defined by ~i (Xi) 2 < p2. Then there is a positive number a such that if 0 < P < a, then (1) U(x o; p) is convex in the sense that any two points of U(x o; p) can be joined by a geodesic which lies in U( Xo; p). (2) Each point of U(x o; p) has a normal coordinate neighborhood containing U(x o; p). Proof. By Proposition 7.9, we may assume that the linear connection has no torsion. LEMMA 1. Let S(x o; p) denote the sphere defined by ~i (X i )2 = p2. Then there exists a positive number c such that, if 0 < P < c, then any geodesic which is tangent to S(xo; p) at a point, sayy, of S(x o; p) lies outside S(x o; p) in a neighborhood ofy. Proof of Lemma 1. Since the torsion vanishes by our assumption, the components r}k of the linear connection vanish at Xo by Proposition 8.4. Let Xi = xi(t) be the equations of a geodesic

150

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

which is tangent to S(xo; p) at a point y = (x1(0), ... , xn(O)) (p will be restricted later). Set

F(t) Then

~i

(X i (t))2.

F(O) = p2,

2~i xi(O) (dX ) i

= ( dF) dt t=O

dt

0,

=

t=O

Because of the equations of a geodesic given in Proposition 7.8, we have k 2 i d F) ( . XZ. dx dX ) ( dt 2 t=O = ~j,k (d;k - ~i rjk ) -dt -dt t=O . Since r;k vanish at xo, there exists a positive number c such that the quadratic form with coefficients (d jk - ~i rJkxi) is positive definite in U(x o ; c). If 0 < P < c, then (d 2Fjdt 2)t=o 0 and hence F(t) > p2 when t =1= 0 is in a neighborhood of O. This completes the proof of the lemma.

2. Choose a positive number c as in Lemma 1. Then there exists a positive number a < c such that (1) Any two points of U( Xo; a) can be joined by a geodesic which lies in U(x o; c) ; (2) Each point of U(X o ; a) has a normal coordinate neighborhood containing U(Xo; a). Proof of Lemma 2. We consider M as a submanifold of T( M) in a natural manner. Set LEMMA

lp(X)

(x, exp X)

for X

€

TflJM).

If the connection is complete, lp is a mapping of T(M) into M X M. In general, lp is defined only in a neighborhood of M in T(M). Since the differential of lp at X o is nonsingular, there exist a neighborhood V of X o in T(M) and a positive number a < c such that lp: V -* U(x o ; a) X U(x o; a) is a diffeomorphism. Taking V and a small, we may assume that exp tX € U(x o; c) for all X € V and ItI 1. To verify condition (1), let x andy be points of U(x o; a). Let X lp-l(X, y), X € V. Then the geodesic with the initial

III.

151

LINEAR AND AFFINE CONNECTIONS

condition (x, X) joins x and y in lJ(x o; c). To verify (2), let Vx; = V n T x U'vl). Since exp: Vx ---+ U(x o; a) is a diffeomorphism, condition (2) is satisfied. To complete the proof of Theorem 8.7, let 0 < P < a. Let x andy be any points of U(x o; p). Let Xi = xi(t), 0 < t < 1, be the equations of a geodesic from x to y in U(x o; c) (see Lemma 2). We shall show that this geodesic lies in U(x o; p). Set

F(t) =

~~

(X i (t))2

for 0 < t < 1.

Assume that F(t) > p2 for some t (that is, xi(t) lies outside U(x o; p) for some t). Let to, 0 < to < 1, be the value for which F( t) attains the maximum. Then

o=

dF) . (dx~) ( -dt t=t = 2~i x~(to) -dt t=t . o o

This means that the geodesic xi(t) is tangent to the sphere S(x o; Po), where P5 = F(to), at the point xt(to). By the choice of to, the geodesic xi(t) lies inside the sphere S(x o; Po), contradicting Lemma 1. This proves (1). (2) follows from (2) of Lemma 2. QED. The existence of convex neighborhoods is due to Whitehead [1].

J.

H. C.

9. Linear infinitesimal holonomy groups Let r be a linear connection on a manifold M. For each point u of L(M), the holonomy group 'Y(u), the local holonomy group 'Y* (u) and the infinitesimal holonomy group 'Y' (u) are defined as in §10 of Chapter II. These groups can be realized as groups of linear transformations of Tx(M), x = 7T(U), denoted by 'f(x), 'f* (x) and 'f' (x) respectively (cf. §4 of Chapter II).

The Lie algebra g(x) of the holonomy group 'Y(x) is equal to the subspace of linear endomorphisms of Tx(M) spanned by all elements of the form (TR) (X, Y) = T- 1 R( T X, T Y) T, where X, Y € Tx(M) and T is the parallel displacement along an arbitrary piecewise differentiable curve T starting from x. THEOREM

9.1.

0

0

152

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. This follows immediately from Theorem 8.1 of Chapter II and from the relationship between the curvature form Q on L(M) and the curvature tensor field R (cf. §5 of Chapter III).

QED. It is easy to reformulate Proposition 10.1, Theorems 10.2 and 10.3 of Chapter II in terms ofo/(x) and \}1'*(x). We shall therefore proceed to the determination of the Lie algebra of 0/' (x).

9.2. The Lie algebra g'(x) of the infinitesimal holonomy group o/'(x) is spanned by all linear endomorphisms of Tx(M) of the form (VkR) (X, Y; VI; ... ; Vk), where X, Y, VI' ... , Vk E Tx(M) and 0 < k < OC). THEOREM

Proof.

The proof is achieved by the following two lemmas.

1. By a tensor field of type A k (resp. B k), we mean a tensor field of type (1, 1) of the form V Yk · .. Vv1(R(X, Y)) (resp. (VkR) (X, Y; VI; ... ; Vk)), where X, Y, VI' ... ' Vk are arbitrary vector fields on M. Then every tensor field of type A k (resp. B k) is a linear combination (with differentiable functions as coefficients) of afinite number of tensor fields of type B j (resp. Ai), 0 s;.j < k. Proof of Lemma 1. The proof is by induction on k. The case k = 0 is trivial. Assume that V Yk _1 · .. Vy1(R(X, Y)) is a sum of terms like LEMMA

f(VjR)(U, V; WI; ... ; Wi)'

0

<j < k - 1,

wherefis a function. Then we have Vyk(f(VjR )(U, V; WI'··. ; Wi)) =

(Vkf) . (ViR) (U, V; WI; ... ; Wj)

+ (Vj+lR) (U, V; WI; ; Wi; V + (ViR) (VykU, V; WI; ; Wi) + (ViR)(U, VVkV; WI' ; Wi) + ~{=1 (VjR)(U, V; WI;··· ; VykW k)

i ;··· ;

Wi)·

This shows that every tensor field of type A k is a linear combination of tensor fields of type B i' 0 <j < k.

III.

153

LINEAR AND AFFINE CONNECTIONS

Assume now that every tensor field of type B k-I is a linear combination of tensor fields of type A j , 0 <j < k - 1. We have (VkR)(X, Y; VI; ... ; V k )

=

VVk((Vk-IR) (X, Y; VI;··· ; Vk- I ))

- (Vk-IR) (VV kX , Y; VI; - (Vfc-IR) (X, Vv kY ; VI;

-

~r:-t

(Vk-IR) (X, Y; VI;

; Vk - I ) ; Vk - I ) ; VY. k Vi; ... ; V k -

I ).

The first term on the right hand side is a linear combination of tensor fields of type A j, 0 <j ::;;: k. The remaining terms on the right hand side are linear combinations of tensor fields of type A j , o <j < k - 1. This completes the proof of Lemma 1. By definition, g'(u) is spanned by the values at u of all glen; R)valued functions of the form (lk) , k = 0, 1, 2, ... (cf. §10 of Chapter II). Theorem 9.2 will follow from Lemma 1 and the following lemma.

LEMMA 2. If X, Y, VI'...' Vk are vector fields on M and if X*, Y*, Vi, ... , VZ are their horizontal lifts to L (M), then we have (VV k · .. Vv 1 (R(X, Y))L:Z =

u

0

(VZ ... Vi(20(X*, Y*)))u u- I (Z) 0

for Z

E

Too(M).

Proof of Lemma 2. This follows immediately from Proposition 1.3 of Chapter III; we take R(X, Y) and 2Q(X*, Y*) as cp and f in Proposition 1.3 of Chapter III. QED. By Theorem 10.8 of Chapter II and Theorem 9.2, the restricted holonomy group ,¥O(x) of a real analytic linear connection is completely determined by the values of all successive covariant differentials VkR, k = 0, 1, 2, ... , at the point x. The results in this section were obtained by Nijenhuis [2].

CHAPTER IV

Riemannian Connections

1. Riemannian metrics Let M be an n-dimensional paracompact manifold. We know (cf. Examples 5.5, 5.7 of Chapter I and Proposition 1.4 of Chapter III) that M admits a Rienlannian metric and that there is a 1: 1 correspondence between the set of Riemannian metrics on M and the set of reductions of the bundle L(M) of linear frames to a bundle 0 (M) of orthonormal frames. Every Riemannian metric g defines a positive definite inner product in each tangent space Tx(M); we write gx(X, Y) or, simply, g(X, Y) for X,Y E Tx(M) (cf. Example 3.1 of Chapter I). Example 1.1. The Euclidean metric g on Rn with the natural coordinate system xl, ... , xn is defined by g( oj ox i , oj oX1 ) = ~1-j (Kronecker's symbol). Example 1.2. Letf: N ----+ M be an immersion of a manifold N into a Riemannian manifold M with metric g. The induced Riemannian metric h on N is defined by heX, Y) = g(!*X, f*Y), X,Y E Tre(N). Example 1.3. A homogeneous space GjH, where G is a Lie group and H is a compact subgroup, admits an invariant metric. Let H be the linear isotropy group at ~the origin 0 (i.e., the point represented by the coset H) of GjH; H is a group of linear transformations of the tangent space To (GjH) , each induced by an element of H which leaves the point 0 fixed. Since H is compact, so is H and there is a positive definite inner product, say go, in To (GjH) which is invariant by H. For each x E GjH, we take an element a E G such that a(o) = x and define an inner product gx in Tx(GjH) by gx(X, Y) = go(a-1X, a-1Y), X,Y E Tx(GjH). It ~

154

IV.

RIEMANNIAN CONNECTIONS

155

is easy to verify that gx is independent of the choice of a E G such that a(o) = x and that the Riemannian metric g thus obtained is invariant by G. The homogeneous space GjH provided with an invariant Riemannian metric is called a Riemannian homogeneous space. Example 1.4. Every compact Lie group G admits a Riemannian metric which is invariant by both right and left translations. In fact, the group G X G acts transitively on G by (a, b) . x = axb- 1 , for (a, b) E G x G and x E G. The isotropy subgroup of G X G at the identity e of G is the diagonal D = {(a, a); a E G}, so that G = (G x G)jD. By Example 1.3, G admits a Riemannian metric invariant by G x G, thus proving our assertion. If G is compact and semisimple, then G admits the following canonical invariant Riemannian metric. In the Lie algebra 9, identified with the tangent space Te(G), we have the Killing-Cartan form cp(X, Y) = trace (ad X ad Y), where X,Y E 9 = Te(G). The form cp is bilinear, symmetric and invariant by ad G. When G is compact and semisimple, cp is negative definite. We define a positive definite inner product ge in T e(G) by ge( X, Y) = -cp(X, Y). Since cp is invariant by ad G, ge is invariant by the diagonal D. By Example 1.3, we obtain a Riemannian metric on G invariant by G x G. We discuss this metric in detail in Volume II. By a Riemannian metric, we shall always mean a positive definite symmetric covariant tensor field of degree 2. By an indefinite Riemannian metric, we shall mean a symmetric covariant tensor field g of degree 2 which is nondegenerate at each x E M, that is, g(X, Y) = 0 for all Y E Tx(M) implies X = o. Example 1.5. An indefinite Riemannian metric on Rn with the coordinate system xl, ... , xn can be given by 0

""p koIi=l

(dX i) 2

_

""n koIj=p+l (dX i) 2 ,

where 0 < p < n - 1. Another example of an indefinite Riemannian metric is the canonical metric on a noncompact, semisimple Lie group G defined as follows. It is known that for such a group the Killing-Cartan form cp is indefinite and nondegenerate. The construction in Example 1.4 gives an indefinite Riemannian metric on G invariant by both right and left translations.

156

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Let M be a manifold with a Riemannian metric or an indefinite Riemannian metric g. For each x, the inner product goo defines a linear isomorphism "P of Too(M) onto its dual T;(M) (space of covectors at x) as follows: To each X E T oo (M), we assign the covector a E T:(M) defined by

The inner product goo in Too(M) defines an inner product, denoted also by goo, in the dual space T;(M) by means of the isomorphism

"P: for a, f3

E

Ti(M).

Let xl, ... , xn be a local coordinate system in M. The components gij of g with respect to xl, ... , xn are given by

i, j = 1, ... , n. The contravariant components gi1 of g are defined by

gi1 = g(dx i, dx j),

i,j = 1, ... ,n.

We have then In fact, define "Pi j by "P ( 0/ ox i ) = ~ j "P i j dx j • Then we have

gij = g(%x i, %x i ) = <%xi , "P(%x i) = "Pij· On the other hand, we have ~f

= <0/ ox i, dx k) = g(dxk, "P( 0/ ox i )) = g( dxk, ~ j "Pij dx j ) = ~ i "Piig ik ,

thus proving our assertion. If ~i are the components of a vector or a vector field X with respect to xl, ... ,xn , that is, X = ~i ~i( ojox i ), then the components ~i of the corresponding covector or the corresponding I-form a = "P(X) are related to ~i by.

ti

c;

=

gi1t . ,

~, .. c ; ,

t

C;i =

~i

gijC;· ti

The inner product g in Too(M) and in T; (M) can be extended to an inner product, denoted also by g, in the tensor space T~(x) at x for each type (r, s). If K and L are tensors at x of type (r, s) with components K{I··· ~r and L~I ... ~r (with respect to Xl ' ••• , x n ) , 21"·)8 )1···)8

IV.

157

RIEMANNIAN CONNECTIONS

then The isomorphism 'ljJ: Tx(M) --+ Ti(M) can be extended to tensors. Given a tensor K E T;(x) with components Kj~:: j;, we obtain a tensor K' E T;+Hx) with components K'i 1... ji r- 1 -

j1·.·

or K"

E

S+!-

~

k r- 1 k g j1 kK j2i 1 jiS+1'

T;~Hx) with components

'Example 1.6.

Let A and B be skew-symmetric endomorphisms of the tangent space T x (M), that is, tensors at x of type (1, 1) such that

g(AX, Y) = -g(AY, X)

and

g(BX, Y) = -g(BY, X) for X, Y

E

Tx(M).

Then the inner product g(A, B) is equal to -trace (AB). In fact, take a local coordinate system Xl, . . . , x n such that gij = bi; at x and let aj and bj be the components of A and B respectively. Then

g(A, B) = ~ gikgHajbf = ~ ajbJ = -~ ajb1 = -trace (AB), since B is skew-symmetric, i.e., bj = -bi. On a Riemannian manifold M, the arc length of a differentiable curve T = x t , a < t < b, of class CI is defined by L

= fg(x"

x,)! dt.

In terms of a local coordinate system

r L = Ja

b

(

Xl, ••• ,

x n , L is given by

i dx dXj)t ~iJjgij dt dt dt.

This definition can be generalized to a piecewise differentiable curve of class CI in an obvious manner. Given a Riemannian metric g on a connected manifold M, we define the distance function d(x,y) on M as follows. The distance d(x,y) between two points x andy is, by definition, the infinimum of the lengths of all piecewise differentiable curves of class CI

158

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

joining x andy. Then we have

d(x,y) > 0,

d(x,y)

=

dey, x),

d(x,y)

+ dey, z)

°

> d(x, z).

We shall see later (in §3) that d(x,y) = only when x = y and that the topology defined by the distance function (metric) d is the same as the manifold topology of M.

2. Riemannian connections Although the results in this section are valid for manifolds with indefinite Riemannian metrics, we shall consider (positive definite) Riemannian metrics only for the sake of simplicity. Let M be an n-dimensional Riemannian manifold with metric g and OeM) the bundle of orthonormal frames over M. Every connection in OeM) determines a connection in the bundle L(M) of linear frames, that is, a linear connection of M by virtue of Proposition 6.1 of Chapter II. A linear connection of M is called a metric connection ifit is thus determined by a connection in OeM).

A linear connection r of a Riemannian manifold M with metric g is a metric connection if and only if g is parallel with respect to r. Proof. Since g is a fibre metric (cf. §l of Chapter III) in the tangent bundle T(M), our proposition follows immediately from Proposition 1.5 of Chapter III. QED. PROPOSITION

2.1.

Among all possible metric connections, the most important is the Riemannian connection (sometimes called the Levi-Civita connection) which is given by the following theorem. THEOREM

2.2. Every Riemannian manifold admits a unique metric

connection with vanishing torsion. We shall present here two proofs, one using the bundle OeM) and the other using the forn1alisn1 of covariant differentiation. Proof (A). Uniqueness. Let () be the canonical form of L(M) restricted to OeM). Let w be the connection form on OeM) definining a metric connection of M. With respect to the basis e1 , • • • ,en ofRn and the basis E!, i <j, i,j = 1, ... ,n, of the Lie algebra o(n), we represent () and w by n forms ()i, i = 1, ... , n, and a skew-symmetric matrix of differential forms wi respectively.

IV.

RIEMANNIAN CONNECTIONS

159

The proof of the following lemma is similar to that of Proposition 2.6 of Chapter III and hence is left to the reader.

The n forms ()i, i = 1, ... , n, and the -In(n - 1) forms w~, 1 <j < k < n, define an absolute parallelism on 0 (M) . LEMMA.

Let ep be the connection form defining another metric connection of M. Then ep - w can be expressed in terms of ()i and w~ by the lemma. Since ep - w annihilates the vertical vectors, we have m~ -

w~ = ~k F~k()k ,J J J' where the FJk'S are functions on O(M). Assume that the connections defined by wand ep have no torsion. Then, from the first structure equation of Theorem 2.4 of Chapter III, we obtain

o=

~.J,J (m~

-

Wi) J A ()j

=

~.J, k F~k J ()k A ()j.

This implies that Fjk = Flj' On the other hand, F;k = -F{k since (w;) and (epj) are skew-symmetric. It follows that Fjk = 0, proving the uniqueness. Existence. Let ep be an arbitrary metric connection form on OeM) and (0 its torsion form on O(M). We write (0i

=

l.~. A ()k , 2 J.k T~k()j J

TJ~k

and set

= - TkiJ.

and We shall show that w = (wj) defines the desired connection. Since both (TJk + T{i) and TJi are skew-symmetric in i and j, so is Tj. Hence w is o(n)-valued. Since () annihilates the vertical vectors, so does T = (Tj). It is easy to show that R;T = ad (a-1) (T) for every a E O(n). Hence, w is a connection form. Finally, we verify that the metric connection defined by w has zero torsion. Since (TJi + l'li) is symmetric inj and k, we have ~ J. T~J A ()j

and hence d()i

=

-~j

proving our assertion.

epj

A

()j

= -

+ (0i

(0i ,

=

-~j

w) A ()j,

160

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof (B). Existence. Given vector fields X and Y on M, we define V xY by the following equation:

2g(V xY, Z)

=

X· g(Y, Z)

+ y. g(X, Z)

- Z . g(X, Y)

+g([X, Y], Z) +g([Z,X], Y) +g(X, [Z, Y]), which should hold for every vector field Z on M. It is a straightforward verification that the mapping (X, Y) ~ V xY, satisfies the four conditions of Proposition 2.8 of Chapter III and hence determines a linear connection r of M by Proposition 7.5 of Chapter III. The fact that r has no torsion follows from the above definition of V xY and the formula T(X, Y) = V xY VyX - [X, Y] given in Theorem 5.1 of Chapter III. To show that r is a metric connection, that is, Vg = 0 (cf. Proposition 2.1), it is sufficient to prove

X· g(Y, Z)

=

g(V xY, Z)

+ g(Y,

V xZ)

for all vector fields X, Y and Z, by virtue of Proposition 2.10 of Chapter III. But this follows immediately from the definition of V x Y. Uniqueness. It is a straightforward verification that if VxY satisfies V xg = 0 and V xY - VyX - [X, Y] = 0, then it satisfies the equation which defined V xY. QED. In the course of the proof, we obtained the following PROPOSITION

2g(V x Y, Z)

=

2.3.

With respect to the Riemannian connection, we have

+ Y·g(X, Z) - Z·g(X, Y) + g([X, Y], Z) + g([Z, X], Y) + g(X,

X·g(Y, Z)

[Z, Y])

for all vector fields X, Y and Z of M.

2.4. In terms of a local coordinate system xl, ... , xn , the components rjk of the Riemannian connection are given by COROLLARY

~

r~. = ~ (a gkij

ZgZk J~

2 ax

+

ag jk _ agji ) ax~ ax k ·

j Proof. Let X = aI ax , Y = aI ax i and Z = aI ax k in ProposiQED. tion 2.3 and use Proposition 7.4 of Chapter III.

IV.

RIEMANNIAN CONNECTIONS

161

Let M and M' be Riemannian manifolds with Riemannian metrics g and g' respectively. A mapping f: M -+ M' is called isometric at a point x of M if g(X, Y) = g' (f*X,f* Y) for all X,Y E Tx(M). In this case, f* is injective at x, because f*X = 0 implies that g(X, Y) = 0 for all Yand hence X = O. A mapping f which is isometric at every point of M is thus an immersion, which we call an isometric immersion. If, moreover, f is 1: l, then it is called an isometric imbedding of Minto M'. Iff maps M 1 : 1 onto M', then f is called an isometry of M onto M'. 2.5. Iff is an isometry of a Riemannian manifold M onto another Riemannian manifold M', then the differential off commutes with the parallel displacement. More precisely, if T is a curve from x to y in M, then the following diagram is commutative: PROPOSITION

Tx(M) f* ~

T

~

Ty(M)

, f* ~

Tx,(M') ~ Ty,(M'),

where x' = f(x) , y' = fey) and T' = f( T). Proof. This is a consequence of the uniqueness of the Riemannian connection in Theorem 2.2. Being a diffeomorphism between M and M', f defines a 1: 1 correspondence between the set of vector fields on M and the set of vector fields on M'. From the Riemannian connection r' on M', we obtain a linear connection ron M by V xY = f- 1 (Vjx (fY )) , where X and Y are vector fields on M. It is easy to verify that r has no torsion and is metric with respect to g. Thus, r is the Riemannian connection of M. This means thatf(V xY) = Vjx(fY) with respect to the Riemannian connections of M and M'. This implies immediately our proposition. QED.

2.6. If f is an isometric immersion of a Riemannian manifold M into another Riemannian manifold M' and iff(M) is open in M', then the differential of f commutes with the parallel displacement. Proof. Since f( M) is open in M', dim M = dim M'. Since f is an immersion, every point x of M has an open neighborhood U such that f( U) is open in M' and f: U -+ f( U) is a diffeomorphism. Thus,f is an isometry of U onto f( U). By Proposition 2.5, the differential off commutes with the parallel displacement PROPOSITION

162

..

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

along any curve in U. Given an arbitrary curve T from x to y in M, we can find a finite number of open neighborhoods in M with the above property which cover T. It follows that the differential off commutes with the parallel displacement along T. QED.

Remark. It follows immediately that, under the assumption of Proposition 2.6, every geodesic of M is mapped by f into a geodesic of M'. Example 2.1. Let M be a Riemannian manifold with metric g. Let M* be a covering manifold of M with projection.l. We can introduce a Riemannian metric g* on M* in such a way that p: M* M is an isometric immersion. Every geodesic of M* projects on a geodesic of M. Conversely, given a geodesic T from x to y in M and a point x* of M* with p(x*) = x, there is a unique curve T* in M* starting from x* such that p( T*) = T. Since p is a local isometry, T* is a geodesic of M*. A similar argument, together with Proposition 2.6, shows that if p(x*) = x, then the restricted linear holonomy group of M* with reference point x* is isomorphic by p to the restricted linear holonomy group of M with reference point x. Proposition 2.5 and 2.6 were stated with respect to Riemannian connections which are special linear connections. Similar statements hold with respect to the corresponding affine connections. The statement concerning linear holonomy groups in Example 2.1 holds also for affine holonomy groups. -)0

3. Normal coordinates and convex neighborhoods Let M be a Riemannian manifold with metric g. The length of a vector X, i.e., g(X, X)t, will be denoted by IIXII. Let T = X t be a geodesic in M. Since the tangent vectors xtare parallel along T and since the parallel displacement is isometric, the length of xt is constant along T. If Ilxtll = 1, then t is called the canonical parameter of the geodesic T. By a normal coordinate system at x of a Riemannian manifold M, we always mean a normal coordinate system xl, ... ,xn at x such that 0; ox!, ... , 0; oxn form an orthonormal frame at x. However, %x!, ... , o;oxn may not be orthonormal at other points. Let U be a normal coordinate neighborhood of x with a normal coordinate system xl, ... , xn at x. We define a cross section (J of

IV.

RIEMANNIAN CONNECTIONS

o(M)

163

over U as follows: Let u be the orthonormal frame at x given by (0/ OXl)~, ... , (0/ n ) x. By the parallel displacement of u along the geodesics through x, we attach an orthonormal frame to every point of U. For the study of Riemannian manifolds, the cross section 0': U ---+ OeM) thus defined is more useful than the cross section U ---+ L(M) given by %x l , • • • , %x n • Let 0 = (Oi) and w = (wD be the canonical form and the Riemannian connection form on OeM) respectively. We set

ox

o=

0'*0 = (O~)

w = O'*w = (wD,

and

where Oi and wi are I-forms on U. To compute these forms explicitly, we introduce the polar coordinate system (pI, ... ,pn; t) by

i=l, ... ,n;

Then, Oi and wi are linear combinations of dpl, ... , dpn and dt with functions of pI, ... ,pn, t as coefficients. PROPOSITION

3.1.

(1) Oi = pi dt

+ qi,

where qi, i = 1, ... , n,

do not involve dt; (2) wi do not involve dt; (3) cpi = 0 and wi = 0 at t = 0 (i.e., at the origin x) ; (4)

+ ... , -~k.l Rjklpkcpl A dt + ... ,

dcpi = - (dpi dwj

=

+~

j

wj pi)

A

dt

where the dots ... indicate terms not involving dt and RJkl are the components of the curvature tensor field with respect to the frame field 0'. Proof. (1) For a fixed direction (pI, ,pn), let T = X t be the geodesic defined by x~ = p~t, i = 1, ,n. Set U t = O'(x t ). To prove that Oi - pi dt do not involve dt, it is sufficient to prove that Oi(X t) = pi. From the definition of the canonical form 0, we have O(u t) = O(x t) = ut-l(x t). Since both U t and x t are parallel along T, O( Xt) is independent of t. On the other hand, we have Oi(X O) = pi and hence Oi(X t) = pi for all t. (2) Since U t is horizontal by the construction of 0', we have

wi(x t)

=

wi(u t)

This means that wi do not involve dt.

=

O.

164

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(3) Given any unit vector X at x (i.e., the point where t = 0), let T = X t be the geodesic with the initial corldition (x, X) so that X = x o' By (1) and (2), we have qi(xo) = 0 and iiJt(xo) = o. (4) From the structure equations, we obtain

d(pi dt

+ cpi)

cpi)

-i dOJ· J

where n~'J

-2: j iiJ} A (pi dt

2: k,l tRjklOk A Ol = 2: k,l tRlkl(Pk dt

+ cpk) A (pl dt + cpl) (cf. §7 of Chapter III) QED.

and hence (4).

In terms of dt and cpi, we can express the metric tensor g as follows (cf. the classical expression ds 2 = 2: g ii dx i dx j for g as explained in Example 3.1 of Chapter I). PROPOSITION

3.2.

The metric tensor g can be expressed by ds 2

(dt)2

+ 2:

i

(cpi)2.

Proo£ Since O(X) = (a(y))-l(X) for every X € Ty(M), Y € U, and since a(y) is an isometric mapping ofRn onto Ty(M), we have for X,Y € Ty(M)

and y

€

U.

In other words,

ds 2

2: i (Oi) 2.

By Proposition 3.1, we have ds 2 = (dt) 2 2: i (cpi) 2

+ 2 2: i picpi dt.

Since cpi = 0 at t = 0 by Proposition 3.1, we shall prove that 2: i picpi 0 by showing that 2: i picpi is independent of t. Since 2: i picpi does not involve dt by Proposition 3.1, it is sufficient to show that d(2: i picpi) does not involve dt. We have, by Proposition 3.1, . . ., d(2: i picpi) = -2: i pi(dpi + 2: j iiJ}pi) A dt where the dots' .. indicate terms not involving dt. From 2: i (p i) 2 = 1, we obtain o = d(2: i (pi) 2) = 2 2: i pi dpi.

IV.

165

RIEMANNIAN CONNECTIONS

On the other hand, because (w~) is skew-symmetric. This proves that not involve dt.

d(~i piqi)

does QED.

From Proposition 3.2, we obtain 3.3. Let Xl, . . . , x n be a normal coordinate system at x. Then every geodesic T = Xt, Xi = ait (i = 1, ... , n), through x is perpendicular to the sphere Sex; r) defined by ~t (X i )2 = r2• PROPOSITION

For each small positive number r, we set

N(x; r) = the neighborhood of 0 in Tx(M) defined by IIXII < r, U(x; r) = the neighborhood of x in M defined by ~i (X i)2 < r2• By the very definition of a normal coordinate system, the exponential map is a diffeomorphism of N(x; r) onto U(x; r). PROPOSITION

3.4.

Let r be a positive number such that exp: N (x; r)

-+

U (x; r)

is a diffeomorphism. Then we have (1) Every pointy in U(x; r) can be joined to x (origin of the coordinate system) by a geodesic lying in U(x; r) and such a geodesic is unique; (2) The length of the geodesic in (1) is equal to the distance d(x,y) ; (3) U(x; r) is the set of points y E M such that d(x,y) < r. Proof. Every line in N(x; r) through the origin 0 is mapped into a geodesic in U(x; r) through x by the exponential map and vice versa. Now, (1) follows from the fact that exp: N (x; r) ---+ U(x; r) is a diffeomorphism. To prove (2), let (a\ ... , an; b) be the coordinates of y with respect to the polar coordinate system (pI, ... ,pn; t) introduced at the beginning of the section. Let T = x s , ex < s < (3, be any piecewise differential curve from x to y. We shall show that the length of T is greater than or equal to b. Let pI = pI(S), ... ,pn = pn(s), t = t(s), ex < s < (3, be the equation of the curve T. If we denote by L (T) the length of T, then Proposition 3.2 implies the following inequalities: L ( T) >

i

f3

(X

-dt ds >

ds

J.bdt = 0

b.

166

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

We shall now prove (3). Ify is in U(x; r), then, clearly, d(x, y) < r. Conversely, let d(x,y) < r and let T be a curve from x to y such that L( T) < r. Suppose T does not lie in U(x; r). Let y' be the first point on T which belongs to the closure of U(x; r) but not to U(x; r). Then, d(x,y') = r by (1) and (2). The length of T from x to y' is at least r. Hence, L( T) > r, which is a contradiction. Thus T lies entirely in U(x; r) and hence y is in U(x; r). QED. 3.5. d(x,y) is a distance function (i.e., metric) on M and defines the same topology as the manifold topology of M. Proof. As we remarked earlier (cf. the end of §1), we have PROPOSITION

d(x,y) > 0,

d(x,y)

=

d(y, x),

d(x,y)

+ d(y, z)

> d(x, z).

From Proposition 3.4, it follows that if x -=I- y, then d(x,y) > O. Thus d is a metric. The second assertion follows from (3) of Proposition 3.4. QED. A geodesic joining two points x andy of a Riemannian manifold M is called minimizing if its length is equal to the distance d(x,y). We now proceed to prove the existence of a convex neighborhood around each point of a Riemannian manifold in the following form. 3.6. Let xl, ... , x n be a normal coordinate system at x of a Riemannian manifold M. There exists a positive number a such that, if 0 < p < a, then (1) Any two points of U( x; p) can be joined by a unique minimizing geodesic,. and it is the unique geodesic joining the two points and lying in U(x; p),. (2) In U(x; p), the square of the distance d(y, z) is a differentiable function ofy and z. Proof. (1) Let a be the positive number given in Theorem 8.7 of Chapter III and let 0 < p < a. Ify and z are points of U(x; p), they can be joined by a geodesic T lying in U(x; p) by the same theorem. Since U(x; p) is contained in a normal coordinate neighborhood ofy (cf. Theorem 8.7 of Chapter III), we see from Proposition 3.4 that T is a unique geodesic joining y and z and lying in U(x; p) and that the length of T is equal to the distance, that is, T is minimizing. It is clear that T is the unique minimizing geodesic joiningy and z in M. THEOREM

IV.

RIEMANNIAN CONNECTIONS

167

(2) Identifying every pointy of M with the zero vector aty, we consider y as a point of T(M). For eachy in U(x; p), let Ny be the neighborhood of y in Ty(M) such that exp: Ny ~ U(x; p) is a diffeomorphism (cf. (2) of Theorem 8.7 of Chapter III). Set V = U Ny. Then the mapping V ~ U(x; p) X U(x; p) y€U(x;p)

which sends Y E Ny into (y, exp Y) is a diffeomorphism (cf. Proposition 8.1 of Chapter III). If z = exp Y, then d(y, z) = I YII· In other words, I YII is the function ~n V which corresponds to the distance function d(y, z) under the diffeomorphism V ~ U(x; p) X U(x; p). Since I YI1 2 is a differentiable function on V, d(y, Z)2 is a differentiable function on U(x; p) X U(x; p). QED. As an application of Theorem 3.6, we obtain the following THEOREM 3.7. Let M be a paracompact differentiable manifold. Then every open covering {Uo:} of M has an open refinement {Vi} such that (1) each Vi has compact closure ,(2) {Vi} is locallyfinite in the sense that every point of M has a neighbor. hood which meets only a finite number of V/ s ,(3) any nonempty finite intersection of V/s is diffeomorphic with an open cell ofRn. Proof. By taking an open refinement if necessary, we may assume that {Uo:} is locally finite and that each Uo: has compact closure. Let {U~} be an open refinement of {Ua} (with the same index set) such that O~ c Ua for all 0( (cf. Appendix 3). Take any Riemannian metric on M. For each x E M, let W x be a convex neighborhood of x (in the sense ofTheorem 3.6) which is contained in some U~. For each 0(, let Wo: = {Wx ; W x n a~ is non-empty}. Since a~ is compact, there is a finite subfamily ma of W a which covers a~. Then the family m = U ma is a desired open refine0:

ment of {Ua }. In fact, it is clear from the construction that m satisfies (1) and (2). If VI' ... , Vk are members of mand if x and yare points of the intersection VI n ... n Vk , then there is a unique minimizing geodesic joining x and y in M. Since the geodesic lies in each Vt , i = 1, ... , k, it lies in the intersection VI n ... n Vk • It follows that the intersection is diffeomorphic with an open cell of R n. QED.

168

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Remark. A covering {Vi} satisfying (1), (2) and (3) is called a simple covering. Its usefulness lies in the fact that the Cech cohomology of M can be computed by means of a simple covering of M (cf. Weil [1]). In any metric space M, a segment is defined to be a continuous image xCt) of a closed interval a < t < b such that v-

d(x (t1), x (t2))

+ d(x (t2), x (t3))

=

d(x (t 1), x (t3))

for a < t l < t2 < t3 < b, where d is the distance function. As an application of Theorem 3.6, we have

3.8. Let M be a Riemannian manifold with metric g and d the distance function defined by g. Then every segment is a geodesic (as a point set) . The parametrization of a segment may not be affine. Proof. Let x(t), a < t < b, be a segment in M. We first show that x(t) is a geodesic for a < t < a + c for some positive c. Let U be a convex neighborhood of x(a) in the sense of Theorem 3.6. There exists c > 0 such that x(t) E U for a < t < a + Co Let T be the minimizing geodesic from x(a) to x(a + c). We shall show that T and x(t), a < t < a + c, coincide as a point set. Suppose there is a number c, a < c < a + c, such that x(c) is not on T. Then d(x(a), x(a + c)) < d(x(a), x(c)) + d(x(c) , x(a + c)), PROPOSITION

contradicting the fact that x(t), a < t < a + c, is a segment. This shows that x(t) is a geodesic for a < t < a + Co By continuing this argumen t, we see that x (t) is a geodesic for a < t :s;; b. QED.

Remark. If X t is a continuous curve such that d(x t1 , x t2 ) = It 1 - t2 for all t l and t2 , then X t is a geodesic with arc length t as parameter. 1

3.9. Let T = Xt, a < t < b, be a piecewise dijferentiable curve of class CI from x to y such that its length L (T) is equal to d(x,y). Then T is a geodesic as a point set. If, moreover, II xtll is constant along T, then T is a geodesic including the parametrization. Proof. It suffices to show that T is a segment. Let a < t l < t2 < t3 < b. Denoting the points Xt. by Xi' i = 1, 2, 3, and the COROLLARY

~

IV.

169

RIEMANNIAN CONNECTIONS

arcs into which 7 is divided by these points by respectively, we have

d(x,x 1 ) < L(7 1 )'

d(X 1 ,X 2 ) < L(7 2 ),

7 1 , 72' 7 3

and

74

d(X 2 ,X3 ) < L(73 ),

d(x 3 ,y) < L(74)' If we did not have the equality everywhere, we would have

d(x, Xl)

+ d(x

1,

x2 )

+ d(x x + d(x ,y) + L( + L( + L( 2,

< L( 7 1 )

3)

3

72)

7 3)

74)

= L( 7) = d(x,y),

which is a contradiction. Thus we have

d(x 1 , x2 ) = L( 72),

d(x 2 , x3 ) = L( 7 3 ),

Similarly, we see that

d(x 1 , x3 )

=

L( 72

+

7 3 ),

Finally, we obtain

d(x 1 , x 2 )

+ d(x

2,

x3 )

=

d(x 1 , x3 )·

QED. Using Proposition 3.8, we shall show that the distance function determines the Riemannian metric.

3.10. Let M and M' be Riemannian manifolds with Riemannian metrics g and g', respectively. Let d and d' be the distance functions of M and M' respectively. If f is a mapping (which is not assumed to be continuous or differentiable) of M onto M' such that d(x,y) = d' (f(x) ,f(y)) for all x,y E M, then f is a diffeomorphism of }vl onto M' which maps the tensor field g into the tensor field g'. In particular, every mapping f of M onto itself which preserves d is an isometry, that is, preserves g. Proof. Clearly, f is a homeomorphism. Let X be an arbitrary point of M and set x' = f(x). For a normal coordinate neighborhood U' of x' let U be a normal coordinate neighborhood of x such thatf( U) c U'. For any unit tangent vector X at x, let 7 be a geodesic in U with the initial condition (x, X). Since 7 is a segment with respect to d,f( 7) is a segment with respect to d' and hence is a geodesic in U' with origin x'. Since 7 = x s is parametrized by the arc length s and since d'( f(xsJ, f(x s)) = d(x s1 , x s2 ) = IS 2 - sll, f( 7) = f(x s ) is parametrized by the arc THEOREM

170

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

length s also. Let F(X) be the unit vector tangent to f( T) at x'. Thus, F is a mapping of the set of unit tangent vectors at x into the set of unit tangent vectors at x'. It can be extended to a mapping, denoted by the same F, of Too(M) into Too,(M') by proportion. Sincefhas an inverse which also preserves the distance functions, it is clear that F is a 1: 1 mapping of Too(M) onto Too,(M'). It is also clear that

f

0

exp

x

= exp

x'

0

F

and

IIF(X) I = IIXII

for X

€

Too(M),

where expa: (resp. expx') is the exponential map of a neighborhood of 0 in Too(M) (resp. Too,(M')) onto U (resp. U'). Both expoo and expx, are diffeomorphisms. To prove thatf is a diffeomorphism of M onto M' which maps g into g', it is therefore sufficient to show that F is a linear isometric mapping of Too(M) onto Too,(M'). We first prove that g(X, Y) = g'(F (X), F(Y)) for all X, Y € Too(M). Since F(cX) = cF(X) for any X € Too(M) and any constant c, we may assume that both X and Yare unit vectors. Then both F( X) and F( Y) are unit vectors at x'. Set cos (X

=

g(X, Y)

and

cos

(x'

=

g' (F (X), F( Y)).

Let x sandys be the geodesics with the initial conditions (x, X) and (x, Y) respectively, both parametrized by their arc length from x. Set and y; = f(ys).

x;

Then and y; are the geodesics with the initial conditions (x', F(X)) and (x', F(Y)), respectively. . SIn

LEMMA.

1 2"

1 d( Xs,Ys ) an d·SIn 2(X l' - 1·1m -2 1 d(' ') (X -- 1·1m -2 X 8,Y8· 8---+-0

S

8---+-0

S

We shall give the proof of the lemma shortly. Assuming the lemma for the moment, we shall complete the proofofour theorem. Since f preserves distance, the lemma implies that sin!(X = sin !(X' and hence g(X, Y)

=

cos (X

1 - 2 sin2 !(X = 1 - 2 sin2 !(X' = cos (X' = g'(F(X),F(Y)). =

We shall now prove that F is linear. We already observed that F(cX) = cF(X) for any X € Too(M) and for any constant c.

IV.

171

RIEMANNIAN CONNECTIONS

Let Xl' ... ' X n be an orthonormal basis for TrAM). Then X~ = F(Xi ), i = 1, ... , n, form an orthonormal basis for Tx,(M') as we have just proved. Given X and Y in Too (M), we have g'(F(X

+ Y), X~)

= g(X

Y, Xi) = g(X, Xi)

g(Y, Xi)

+ g'(F(Y), X~) = g'(F(X)

= g'(F(X) , X~)

F(Y), X~)

for every i, and hence

+ Y)

F(X

=

F(X)

F(Y).

Proof of Lemma. It is sufficient to prove the first formula. Let U be a coordinate neighborhood with a normal coordinate system Xl, ••• , x n at x. Let h be the Riemannian metric in U given by 2: i (dx i ) 2 and let o(y, z) be the distance betweeny and z with respect to h. Supposing that

-1· 1 d( Xs,Ys ) > SIn . 1m -2 8 ........0

I

2 lX ,

S

we shall obtain a contradiction. (The case where the inequality is reversed can be treated in a similar manner.) Choose c > 1 such that -1· 1 d( Xs,Ys ) > c SIn . 2I lX • 1m -2 8........ 0

S

Taking U small, we may assume that ~ h < g < ch on U in the sense that c, 1

\

- h(Z, Z) < g(Z, Z)

c h(Z, Z) for Z

E

Tz(M) and

C

From the definition of the distances d and 0, we obtain 1 - o(y, z) c

c o(y, z).

d(y, z)

Hence we have

;s

<5(x"y.) >

;s

d(x"y.) > c sin

ilX

for small s.

On the other hand, h is a Euclidean metric and hence

;s

<5 (x "y.) =

sin

ilX.

Z E

U.

172

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

This is a contradiction. Hence,

-1 lim -2 d(xs,ys) = sin -la. s

8---+0

Similarly, we obtain 1 lim 2 d(xs,ys) s

8---+0

=

sin -la.

QED. Theorem 3.10 is due to Myers and Steenrod [1]; the proof is adopted from Palais [2].

4. Conlpleteness A Riemannian manifold M or a Riemannian metric g on M is said to be complete if the Riemannian connection is complete, that is, if every geodesic of M can be extended for arbitrarily large values of its canonical parameter (cf. §6 of Chapter III). We shall prove the following two important theorems.

4.1. For a connected Riemannian manifold M, the following conditions are mutually equivalent: (1) M is a complete Riemannian manifold; (2) M is a complete metric space with respect to the distancefunction d; (3) Every bounded subset of M (with respect to d) is relatively compact; (4) For an arbitrary point x of M and for an arbitrary curve C in the tangent space Tx(M) (or more precisely, the affine tangent space Ax(M)) starting from the origin, there is a curve T in M starting from x which is developed upon the given curve C. THEOREM

4.2.

If M

is a connected complete Riemannian manifold, then any two points x andy of M can be joined by a minimizing geodesic. Proof. We divide the proofs of these theorems into several steps. (i) The implication (2) --+ (1). Let x s' 0 < s < L, be a geodesic, where s is the canonical parameter. We show that this geodesic can be extended beyond L. Let {sn} be an infinite sequence such that Sn t L. Then d( X ,X ) < ISm _ Sn,I = THEOREM

sm

sn

so that {xsJ is a Cauchy sequence in M with respect to d and hence converges to a point, say x. The limit point x is independent

IV.

RIEMANNIAN CONNECTIONS

173

of the choice of a sequence {sn} converging to L. We set XL = x. By using a normal coordinate system at x, we can extend the geodesic for the values of s such that L < s < L + s for some

s>

o.

Let x be any point of M. For each

(ii) Proof of Theorem 4.2. r > 0, we set

S(r) = {y

E

M; d(x,y) < r}

and

E(r)

=

{y E S(r);y can be joined to x by a minimizing geodesic}.

We are going to prove that E (r) is compact and coincides with S(r) for every r > O. To prove the compactness of E(r), let Yi' i = 1, 2, ... , be a sequence of points of E(r) and, for each i, let T z be a minimizing geodesic from x to Yi. Let Xi be the unit vector tangent to Ti at x. By taking a subsequence if necessary, we may assume that {Xi} converges to a unit vector X o in Tx(M). Since d(X'Yi) < r for all i, we may assume, again by taking a subsequence if necessary, that d(X'Yi) converges to a non-negative number roo Since T i is minimizing, we have

Since M is a complete Riemannian manifold, exp roXo is defined. We set Yo = exp roXo. It follows that {Yz} converges to Yo and hence that d(x,yo) = roo This implies that the geodesic exp sXo, 0 < s < ro, is minimizing and that Yo is in E(r). This proves the compactness of E(r). Now we shall prove that E(r) = S(r) for all r > O. By the existence of a normal coordinate system and a convex neighborhood around x (cf. Theorem 3.6), we know that E(r) = S(r) for o < r < s for some s > O. Let r* be the supremum of ro > 0 such that E(r) = S(r) for r < roo To show that r* = 00, assume that r* < 00. We first prove that E(r*) = S(r*). Let Y be a point of S(r*) and let {Yi} be a sequence of points with d(X'Yi) < r* which converges to y. (The existence of such a sequence {Yi} follows from the fact that x andy can be joined by a curve whose length is as close to d(x,y) as we wish.) Then each Yi belongs to some E(r), where r < r*, and hence each Yi belongs to E(r*).

174

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Since E(r*) is compact, y belongs to E(r*). Hence S(r*) = E(r*). Next we shall show that S(r) = E(r) for r < r* ~ for some ~ > 0, which contradicts the definition of r*. We need the following

On a Riemannian manifold M, there exists a positive continuousfunction r(z), z € M, such that any two points of Sz(r(z)) = {y € M; d(z,y) < r(z)} can be joined by a minimizing geodesic. Proof of Lemma. For each z € M, let r(z) be the supremum of r > 0 such that any two points y and y' with d(z,y) < rand d(z,y') r can be joined by a minimizing geodesic. The existence of a convex neighborhood (cf. Theorem 3.6) implies that r(z) > O. If r(z) = 00 for some point z, then r(y) = 00 for every point y of M and any positive continuous function on M satisfies the 00 for every z € M. condition of the lemma. Assume that r(z) We shall prove the continuity ofr(z) by showing that Ir(z) - r(y) 1 < d(z,y). Without any loss of generality, we may assume that r(z) > r(y). If d(z,y) > r(z), then obviously Ir(z) r(y) 1 < d(z,y). If d(z,y) r(z), then Sy(r') = {y'; d(y,y') r'} is conr(z)tained in Sz(r(z)), where r' = r(z) - d(z,y). Hence r(y) d(z,y), that is, Ir(z) r(y) 1 < d(z,y), completing the proofof the lemma. Going back to the proof of Theorem 4.2, let r(z) be the continuous function given in the lemma and let ~ be the minimum of r(z) on the compact set E(r*). To complete the proof of Theorem 4.2, we shall show that S(r* ~) = E(r* + ~). Lety € S(r* + ~) but 1: S(r*). We show first that there exists a point y' in S(r*) such that d(x,y') = r* and that d(x,y) = d(x,y') + d(y',y). To this end, for every positive integer k, choose a curve T k from x to LEMMA.

y such that L( T k )

<

d(x,y)

1'

where L( Tk) is the length of

Let Yk be the last point on Tk which belongs to E(r*) Then d(X'Yk) = r* and d(X'Yk)

+ d(Yk'.y)

=

< L( Tk) < d(x,y)

TkO

S(r*). 1

+k·

Since E(r*) is compact, we may assume, by taking a subsequence if necessary, that {Yk} converges to a point, say y', of E(r*). We have d(x,y') = r* and d(x,y') + d(y',y) = d(x,y). Let T' be a ~ < r(y'), there minimizing geodesic from x toy'. Since d(y',y) is a minimizing geodesic T" fromy' toy. Let T be the join of T' and

IV.

175

RIEMANNIAN CONNECTIONS

Til. Then L( T) = L( T') + L( Til) = d(x,y') + d(y' ,y) = d(x,y). By Corollary 3.9, T is a geodesic, in fact, a minimizing geodesic from x to y. Hencey E E(r* + ~), completing the proofofTheorem 4.2.

Remark. To prove that E(r) = S(r) is compact for every r, it is sufficient to assume that every geodesic issuing from the particular point x can be extended infinitely. (iii) The implication (1) -+ (3) in Theorem 4.1. In (ii.) we proved that (1) implies that E(r) = S(r) is compact for every r. Every bounded subset of M is contained in S(r) for some r, regardless of the point x we choose in the proof of (ii). (iv) The implication (3) -+ (2) is evident. (v) The implication (4) -+ (1). Since a geodesic is a curve in M which is developed upon a straight line (or a segment) in the tangent space, it is obvious that every geodesic can be extended infinitely. (vi) The implication (1) -+ (4). Let Ct , 0 < t < a, be an arbitrary curve in Ta;(M) starting from the origin. We know that there is e > 0 such that Ct , 0 < t < e, is the development of a curve Xt, 0 ::;;: t < e, in M. Let b be the supremum of such e > O. We want to show that b = a. Assume that b < a. First we show that lim X t exists in M. Let tn t b. Since the development pret->-b

serves the arc length, the length of Xt, tn < t < tm , is equal to the length of Ct , tn < t < tm • On the other hand, the distance d(x tn , xtJ is less than or equal to the length of Xt, tn < t < tm • This implies that {xtJ is a Cauchy sequence in M. Since we know the implication (1) -+ (3) by (iii) and (iv) , we see that {xtJ converges to a point, say y. It is easy to see that lim X t = y. Let t->-b

C; be the curve in Ty(M) (or more precisely, in Ay(M)) obtained by the affine (not linear!) parallel displacement of the curve Ct along the curve Xt, 0 < t < b. Then C~ is the origin of Ty(M). There exist ~ > 0 and a curve Xt, b < t < b +~, which is developed upon C;, b < t < b + ~. Then the curve Xt, 0 < t < b + ~, is developed upon Ct , 0 < t < b + ~. This contradicts the definition of b. QED.

4.3. If all geodesics starting from any particular point x of a connected Riemannian manifold M are infinitely extendable, then M is complete. COROLLARY

176

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. As we remarked at the end of (ii) in the proof of Theorem 4.2, E(r) = S(r) is compact for every r. Every bounded subset of M is contained in S(r) for some r and hence is relatively compact. QED. 4.4. Every compact Riemannian manifold is complete. This follows from the implication (3) -+ (1) in Theorem QED.

COROLLARY

Proof. 4.1.

A Riemannian manifold M is said to be homogeneous if the group of isometries, i.e., transformations preserving the metric tensor g, of M is transitive on M. (Cf. Example 1.3 and Theorem 3.4, Chapter VI.) THEOREM 4.5. Every homogeneous Riemannian manifold is complete. Proof. Let x be a point of a homogeneous Riemannian manifold M. There exists r > 0 such that, for every unit vector X at x, the geodesic exp sX is defined for lsi < r (cf. Proposition 8.1 of Chapter III). Let T = x s , 0 < S < a, be any geodesic with canonical parameter s in M. We shall show that T = X s can be extended to a geodesic defined for 0 :::;: s < a + r. Let cp be an isometry of M which maps x into xa • Then cp-l maps the unit vector xa at X a into a unit vector X at x: X = cp-l(Xa ). Since exp sX is a geodesic through x, cp( exp sX) is a geodesic through xa • We set xa+ s = cp(exp sX) for 0 < S < r. Then T = x s , 0 < S < a + r, is a geodesic. QED. Theorem 4.5 follows also from the general fact that every locally compact homogeneous metric space is complete. THEOREM 4.6. Let M and M* be connected Riemannian manifolds of the same dimension. Let p: M* -+ M be an isometric immersion. (1) If M* is complete, then M* is a covering space of M with projection p and M is also complete. (2) Conversely, if p: M* -+ M is a covering projection and if M is complete, then M* is complete. Proof. The proof is divided into several steps. (i) If M* is complete so is M. Let x* E M* and set x = p(x*). Let Xbe any unit vector of M at x and choose a unit vector X* at x* such that P(X*) = X. Then exp sX = p(exp sX*) is the geodesic in M with the initial condition (x, X). Since exp sX* is defined

IV.

177

RIEMANNIAN CONNECTIONS

for all s, - 00 < s 00, so is exp sX. By Corollary 4.3, M is complete. (ii) If M* is complete, p maps M* onto M. Let x* € M* and x = p(x*). Given a point y of M, let exp sX, 0 : : ; : s ::::;;: a, be a geodesic from x to y, where X is a unit vector at x. Such a geodesic exists by Theorem 4.2 since M is complete by (i). Let X* be the unit vector of M* at x* such that p(X*) = X. Sety* = exp aX*. Thenp(y*) exp aX =y. (iii) If M* is complete, then p: M* ~ M is a covering projection. For a given x € M and for each positive number r, we set U(x; r)

=

{y

€

M; d(x,y)

Similarly, we set, for x*

€

< r},

N(~; r)

=

{X

€

Tx(M);

IIXII < r}.

M*,

U(x*; r)

=

{y*

€

N(x*; r)

=

{X*

€

M*; d(x* ,y*) T'C.(M*);

< r},

IIXII < r}.

Choose r > 0 such that exp: N(x; 2r) ~ U(x; 2r) is a diffeomorphism. Let {xt, x~, ... } be the set p-l(X). For each xt, we have the following commutative diagram: exp

N(xt; 2r) ~ U(x:; 2r)

t

p

N(x; 2r)

t

p

exp

~

U(x; 2r).

I t is sufficient to prove the following three statements: (a) p: U(xt; r) ~ U(x; r) is a diffeomorphism for every i; (b) p-l( U(x; r)) = U U(xt; r) ; i

(c) U(xt; r) n U(x!; r) is empty if xi x!. Now, (a) follows from the fact that both p: N(xt; 2r) ~ N(x; 2r) and exp: N(x; 2r) ~ U(x; 2r) are diffeomorphisms in the above diagram. To prove (b), let y* € p-l( U(x; r)) and set y = p(y*). Let exp s Y, 0 s a, be a minimizing geodesic from y to x, where Y is a unit vector aty. Let y* be the unit vector aty* such that p( Y*) = Y. Then exp sY*, 0 s < a, is a geodesic in M* starting from y* such that p(exp sy*) = exp sY. In particular, p(exp aY*) x and hence exp aY* = xt for some xi- Evidently,y* € U(xi; r), proving thatp-l(U(x; r)) c U U(xt; r). On i

178

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

the other hand, it is obvious that p( U(xi; r)) c U(x; r) for every i and hence p-l(U(x;r))::> U U(xi;r). To prove (c), suppose i

y* € U(x£; r) n U(xi; r). Then xi € U(xi; 2r). Using the above diagram, we have shown that p: U(xi; 2r) --+ U(x; 2r) is a diffeomorphism. Since P(xi) = p(xi), we must have xi = xi. (iv) Proof of (2). Assume that p: M* --+ M is a covering projection and that M is complete. Observe first that, given a curve x t , 0 < t < a, in M and given a point x6 in M* such that P(x6) = X o, there is a unique curve x7, 0 < t < a, in M* such that P(xi) = X t for 0 < t < a. Let x* € M* and let x* be any unit vector at X*. Set X = P(X*). Since M is complete, the geodesic exp sX is defined for - 00 < s < 00. From the above observation, we see that there is a unique curve xi, - 00 < s < 00, in M* such that x6 = x* and that P(xi) = exp sX. Evidently, x: = exp sX*. This shows that M* is complete. QED. 4.7. Let M and M* be connected manifolds of the same dimension and let p: M* --+ M be an immersion. If M* is compact, so is M, and p is a covering projection. Proof. Take any Riemannian metric g on M. It is easy to see that there is a unique Riemannian metric g* on M* such that p is an isometric immersion. Since M* is complete by Corollary 4.4, p is a covering projection by Theorem 4.6 and hence M is compact. QED. COROLLARY

Example 4.1. A Riemannian manifold is said to be non-prolongeable if it cannot be isometrically imbedded into another Riemannian manifold as a proper open submanifold. Theorem 5.6 shows that every complete Riemannian manifold is nonprolongeable. The converse is not true. For example, let M be the Euclidean plane with origin removed and M* the universal covering space of M. As an open submanifold of the Euclidean plane, M has a natural Riemannian metric which is obviously not complete. With respect to the natural Riemannian metric on M* (cf. Example 2.1), M* is not complete by Theoren1 4.6. It can be shown that M* is non-prolongeab1e. 4.8. Let G be a group of isometries of a connected Riemannian manifold M. If the orbit G(x) of a point x of M contains an COROLLARY

IV.

179

RIEMANNIAN CONNECTIONS

open set of M, then the orbit G(x) coincides with M, that is, M is homogeneous. Proof. I t is easy to see that G(x) is open in M. Let M* be a connected component of G(x). For any two points x* and y* of M*, there is an elementf of G such thatf (x*) =)J*. Sincef maps every connected component of G(x) onto a connected component of G(x),f(M*) = M*. Hence M* is a homogeneous Riemannian manifold isometrically imbedded into M as an open submanifold. Hence, M* = M. QED.

4.9. Let M be a Riemannian manijold and M* a submanijold of M which is locally closed in the sense that every point x of M has a neighborhood U such that every connected component of U n M* (with respect to the topology of M*) is closed in U. If M is complete, so is M* with respect to the induced metric. Proof. Let d be the distance function defined by the Riemannian metric of M and d* the distance function defined by the induced Riemannian metric of M*. Let x s be a geodesic in M* and let a be the supremum of s such that x s is defined. To show that a = 00, assume a < 00. Let Sn t a. Since d(x sn ' X s m ) < d*(x srt , xsJ ~ ISn - sml, {xsJ is a Cauchy sequence in M and hence converges to a point, say x, of M. Then x = lim X S ' Let U PROPOSITION

8--->-a

be a neighborhood of x in M with the property stated in Proposition. Then x s , b < s < a, lies in U for some b. Since the connected component of M* n U containing x s' b < s < a, is closed in U, the point x belongs to M*. Set X a = x. Then x 8' 0 < S < a, is a geodesic in M*. Using a normal coordinate system at xa , we see that this geodesic can be extended to a geodesic x s , 0 < S < a + ~, for some ~ > O. QED.

5. Holonomy groups Throughout this section, let M be a connected Riemannian manifold with metric g and 'Y(x) the linear or homogeneous holonomy group of the Riemannian connection with reference point x € M (c£ §4 of Chapter II and §3 of Chapter III). Then M is said to be reducible or irreducible according as 'Y(x) is reducible or irreducible as a linear group acting on Ta; (M). In this section, we shall study 'Y(x) and local structures ofa reducible Riemannian manifold.

180

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Assuming that M is reducible, let T; be a non-trivial subspace of Tw(M) which is invariant by o/(x). Given a pointy EM, let 'T be a curve from x to y and T~ the image of T~ by the (linear) parallel displacement along 'T. The subspace T~ of Ty(M) is independent of the choice of'T. In fact, if fl is any other curve from x to y, then fl- l • 'T is a closed curve at x and the subspace T; is invariant by the parallel displacement along fl- l . 'T, that is, fl- l . 'T( T;) = T;, and hence 'T( T;) = fl( T;). We thus obtain a distribution T' which assigns to each point y of M the subspace T~ of T y ( M) . A submanifold N ofa Riemannian manifold (or more generally, a manifold with a linear connection) M is said to be totally geodesic at a point x of N if, for every X E Tw(N), the geodesic 'T = X t of M determined by (x, X) lies in N for small values of the parameter t. If N is totally geodesic at every point of N, it is called a totally geodesic submanifold of M. PROPOSITION

5.1.

(1) The distribution T' is differentiable and

involutive; (2) Let M' be the maximal integral manifold of T' through a point of M. Then M' is a totally geodesic submanifold of M. If M is complete, so is M' with respect to the induced metric. Proof. (1) To prove that T' is differentiable, let y be any point of M and xl, ... ,xn a normal coordinate system at y, valid in a neighborhood U ofy. Let Xl' ... , X k be a basis for T~. For each i, 1
Z E

U,

where 'T is the geodesic from y to z given by x; = ait, j = 1, ... , n, (aI, ... , an) being the coordinates of z. Since the parallel displacement 'T depends differentiably on (aI, ... , an), we obtain a differentiable vector field Xi in U. It is clear that Xi, ... , Xt form a basis of T; for every point z of U. To prove that T' is involutive, it is sufficient to prove that if X and Yare vector fields belonging to T', so are V xY and VyX, because the Riemannian connection has no torsion and [X, Y VxY VyX (cf. Theorem 5.1 of Chapter III). Let X t be the integral curve of X starting from an arbitrary point y. Let 'Tb be the parallel displacement along this curve from the point X t to

IV.

the point y =

RIEMANNIAN CONNECTIONS

181

o' Since Y y and Y Xt belong to T' for every t, (V xY) = lim ~ (T~YXt - Y y ) belongs to T~. t--O t (2) Let M' be a maximal integral manifold of T'. Let T = X t be a geodesic of M with the initial condition (y, X), where y E M and X E Ty(M') = T~. Since the tangent vectors x t are parallel along T, we see that xt belongs to T~t for every t and hence X

lies in M' (cf. Lemma 2 for Theorem 7.2 of Chapter II). This proves that M' is a totally geodesic submanifold of M. From the following lemma, we may conclude that, if M is complete, so is M'. T

Let N be a totally geodesic submanifold of a Riemannian manifold M. Every geodesic of N with respect to the induced Riemannian metric of N is a geodesic in M. Proof of Lemma. Let x E N and X E Tx(N). Let T = x t , o < t < a, be the geodesic of M with the initial condition (x, X). Since N is totally geodesic, T lies in N. It now suffices to show that T is a geodesic of N with respect to the induced Riemannian metric of N. Let d and d' be the distance functions of M and N respectively. Considering only small values of t, we may assume that T is a minimizing geodesic from x = X o to X a so that d(x, xa ) = L(T), where L(T) is the arc length of T. The arc length of T measured by the metric of M is the same as the one measured with respect to the induced metric of N. From the definition of the distance functions d and d', we obtain LEMMA.

Hence, d' (x, xa ) = L( T). By Corollary 3.9, respect to the induced metric of N.

T

is a geodesic with QED.

Remark. The lemma is a consequence of the following two facts which will be proved in Volume II. (1) If M is a manifold with a linear connection whose torsion vanishes and if N is a totally geodesic submanifold of M, then N has a naturally induced linear connection such that every geodesic of N is a geodesic of M; (2) If N is a totally geodesic submanifold of a Riemannian manifold M, then the naturally induced linear connection of N is the Riemannian connection with respect to

182

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

the induced metric of N. Note that Proposition 5.1 holds under the weaker assumption that M is a manifold with a linear connection whose torsion vanishes. Let T' be a distribution defined as before. We now use the fact that the- homogeneous holonomy group consists of orthogonal transformations of Tx(M). Let T; be the orthogonal complement of T; in Tx(M). Then Tx(M) is the direct sum of two subspaces T; and T; which are invariant by 'P'(x). From the subspace T;' we obtain a distribution Til just as we obtained T' from T~. The distributions T' and Til are complementary and orthogonal to each other at every point of M. 5.2. Let y be any point of M. Let M' and Mil be the maximal integral manifolds of the distributions T' and Til defined above. Then y has an open neighborhood V such that V = V' X V", where V' (resp. V") is an open neighborhood ofy in M' (resp. Mil), and that the Riemannian metric in V is the direct product of the Riemannian metrics in V' and V". Proof. We first prove the following PROPOSITION

If T'

and Til are two involutive distributions on a manifold M which are complementary at every point of M, then, for each point y of M, there exists a local coordinate system xl, , x n with origin at y such that (ojox!, ... , ojox k) and (ojoxk+t, , ojox n) form local bases for T' and Til respectively. In other words, for any set of constants (c!, ... , c\ ck+t, ... , cn), the equations Xi = ci , 1 ::;;: i ::;;: k, (resp. x j = cj , k + 1 <J < n) define an integral manifold of Til (resp. T'). Proof of Lemma. Since T' is involutive, there exists a local · · h ongin .. y suc h t h at coor d Inate system y 1, ... ,yk,xk+l , ... , x nwIt (oj oy\ ... , oj oyk) form a local basis for T'. In other words, the equations x j = cj , k + 1 <J < n, define an integral manifold of T'. Similarly, there exists a local coordinate system xl, ... , xk, Zk+\ ... , zn with origin y such that (oj OZk+l, ... , oj ozn) form a local basis for Til. In other words, the equations Xi = ci , 1 ::;;: i < k, define an integral manifold of Til. It is easy to see that k xk+l , ... , X n·IS a Ioca I coor d·Inate system WIt . h the x1, ... , x, desired property. Making use of the local coordinate system xl, ... ,xn thus obtained, we shall prove Proposition 5.2. Let V be the neighborhood ofy defined by Ixil < c, 1 < i < n, where c is a sufficiently LEMMA.

IV.

183

RIEMANNIAN CONNECTIONS

small positive number so that the coordinate system xl, ... , x n gives a homeomorphism of V onto the cube Ixil < c in Rn. Let V' (resp. V") be the set of points in V defined by Ixil < c, 1 < i < k, and x' = 0, k + 1 <j < n (resp. Xi = 0, 1 < i < k, and Ix'i < c, k + 1 <j < n). It is clear that V' (resp. V") is an integral manifold of T' (resp. T") through y and is a neighborhood of y in M' (resp. M") and that V = V' X V". We set Xi = ajax i, 1 < i < n. To prove that the Riemannian metric of V is the direct product of those in V' and V", we show that gij = g(Xi, Xi) are independent of x k +\ ... , x n for 1 < i, j < k, that gij = g(Xi, Xi) are independent of xl, ... , xk for k + 1 < i,j < n and that gij = g(Xi, Xi) = for 1 ~ i < k and k + 1 < j < n. The last assertion is obvious since Xi' 1 < i ~ k, belong to T' and Xi' k + 1 <j < n, belong to T" and since T' and T" are orthogonal to each other at every point. We now prove the first assertion, and the proof of the second assertion is similar. Let 1 < i < k and k + 1 < m < n. As in the proof of (1) of Proposition 5.1, we see that V XmXi belongs to T' and that Vx X m belongs to T". Since the torsion is zero and since [Xi' X m] = 0, we have

°

~

V x X m - V X m Xi ~

=

V X Xm ~

V X m Xi -

[Xi' X m]

=

0.

Hence, V x X m = V X m Xi = 0. Since g is parallel, we have ~

Xm(gii) = V Xm(g(Xi, Xi))

= g(V XmXi' Xi) thus proving our assertion.

+ g(Xi, VXmXi)

=

0,

1 < i, j < k, QED.

5.3. Let T' and T" be the distributions on M used in Proposition 5.2. If M is simply connected, then the homogeneous holonomy group 'rex) is decomposed into the direct product of two normal subgroups 'f'(x) and'r"(x) such that 'r'(x) is trivial on T; and that 'r"(x) is trivial on T;. Proof. Given an element a E 'r(x) , let al (resp. a2 ) be the restriction of a to T; (resp. T;). Let a' (resp. a") be the orthogonal transformation of Ta;(M) which coincides with a l on T; (resp. with a2 on T;) and which is trivial on T; (resp. T;). If we take an orthonormal basis for Ta;(M) such that the first k vectors lie in T~ and the remaining n - k vectors lie in T;, then these linear PROPOSITION

184

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

transformations can be expressed by matrices as follows:

We shall show that both a' and a" are elements of'Y(x). Let T be a closed curve at x such that the parallel displacement along T is the given element a E 'F(x). First we consider the special case where T is a small lasso in the following sense. A closed curve T at x is called a small lasso if it can be decomposed into three curves as follows: T = fl- 1 • a . fl, where fl is a curve from x to a point y (so that fl- 1 is a curve from y to x going backward) and (J is a closed curve at y which is small enough to be contained in a neighborhood V = V' X V" ofy given in Proposition 5.2. In this special case, we denote by a' (resp. a") the image of (J by the natural projection V ~ V' (resp. V ~ V"). We set , T = fl - 1 . a' . fl, T "= -fl1." a . fl· The parallel displacement along T' (resp. T") is trivial on T; (resp. T~). The parallel displacement along a is the product of those along a' and a". Hence the parallel displacement along T is the product of those along T' and T". On the other hand, T' (resp. T") is trivial on T; (resp. T~). It follows that a' (resp. a") is the parallel displacement along T' (resp. T"), thus proving our assertion in the case where T is a small lasso. In the general case, we decompose T into a product of small lassos as follows.

If

M is simply connected, then the parallel displacement along T is the product of the parallel displacements along a finite number of small lassos at x. Proof of Lemma. This follows from the factorization lemma (cf. Appendix 7). It is now clear that both a' and a" belong to 'Y(x) in the general case. We set LEMMA.

'Y' (x) = {a'; a E 'Y (x)}, Then 'Y(x) = 'Y' (x) x 'Y" (x).

.

'Y" (x) = {a"; a E 'Y (x)}. QED.

We now proceed to define a most natural decomposition of Tx(M) and derive its consequences. Let T~O) be the set of

IV.

RIEMANNIAN CONNECTIONS

185

elements in Tx(M) which are left fixed by 'Y(x). It is the maRimal linear subspace of Tx(M) on which 'Y(x) acts trivially. Let T~ be the orthogonal complement of T~O) in Tx(M). It is invariant by 'Y (x) and can be decomposed into a direct sum T~ = ~r = I T~i) of mutually orthogonal, invariant and irreducible subspaces. We shall call T x(M) = ~1 =0 T~i) a canonical decomposition (or de Rham decomposition) of T x(M) . 5.4. Let M be a Riemannian manifold, Tx(M) ~r=o T~i) a canonical decomposition of Tx(M) and T(Z) the involutive distribution on M obtained by parallel displacement of T~i) for each i = 0, 1, ... , k. Lety be apoint of M and let, for each i = 0, 1, ... , k, M i be the maximal integral manifold of T(i) through y. Then (1) The point y 'has an open neighborhood V such that V = Vo X VI X . . • X Vk where each Vi is an open neighborhood ofy in M i and that the Riemannian metric in V is the direct product of the Riemannian metrics in the V/ s; (2) The maximal integral manifold M o is locally Euclidean in the sense that every point of M o has a neighborhood which is isometric with an open set of an no-dimensional Euclidean space, where no = dim M o:; (3) If M is simply connected, then the homogeneous holonomy group 'Y(x) is the direct product 'Yo(x) X 'Y I (x) X • • . X 'Y k ( x) of normal subgroups, where 'Yi(x) is trivial on Tfj) if i j and is irreducible on T~i) for each i = 1, ... , k, and 'Yo (x) consists of the identity only; (4) If M is simply connected, then a canonical decomposition Tx(M) ~r=0 T~i) is unique up to an order. Proof. (1) This is a generalization of Proposition 5.2. (2) Since y is an arbitrary point of M, it is sufficient to prove that Vo is isometric to an open subset of an no-dimensional Euclidean space. Since the homogeneous holonomy group of Vo consists of the identity only, T~O) is the direct sum of no I-dimensional subspaces. From the proofof Proposition 5.2, it follows that Vo is a direct product of I-dimensional submanifolds and that the Riemannian metric on Vo is the direct product of the Riemannian metrics on these I-dimensional submanifolds. On the other hand, on any I-dimensional manifold with a local coordinate system xl, every Riemannian metric is of the form g 11 dx 1 dx l • If Xl is a normal coordinate system, then the metric is of the form dx l dx l • Hence Vo is isometric to an open set of a Euclidean space. THEOREM

186

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(3) This is clear from the definition of a canonical decomposition of Tx(M) and from the proof of Proposition 5.3. (4) First we prove

Let Sx be any subspace of Tx(M) invariant by 'Y(x). Then, for each i = 1, ... ,k, either Sx is orthogonal to T~) or Sx contains T~). Proof of Lemma. (i) Assume that all vectors of Sx are left fixed by 'Yi(x). Then Sx is orthogonal to T~). In fact, let X = ~J=o Xi be any element of Sx, where Xi E T~). For an arbitrary element ai of '¥i(X), we have ai(X) = X o + Xl + ... + ai(Xi ) + ... + X k LEMMA.

since ai acts trivially on T~) for j ::j:. i. If ai(X) = X, then ai(Xi ) = Xi· Since this holds for every ai E 'Yi(x) and since 'Yi(x) is irreducible in T~i), we must have Xi = O. This shows that X is orthogonal to T~). (ii) Assume that ai(X) ::j:. X for some ai E 'Yi(x) and for some X E Sx. Let X = ~J =0 Xi' where X j E T~). Since each Xj' j ::j:. i, is left fixed by every element of 'Yi(x), X - ai(X) = Xi ai (Xi) ::j:. 0 is a vector in T~i) as well as in Sx. The subset {bi(X- aJX)); biE'Yi(x)} is in T~i) nSx and spans T~i), since 'Yi(x) is irreducible in T~i). This implies that T~i) is contained in Sx, thus proving the lemma. Going back to the proof of (4), let T x (M) = ~] =0 S~) be any other canonical decomposition. First it is clear that T~O) = S~O). I t is therefore sufficient to prove that each S~), 1 ::;;: j ::;;: I, coincides with some T~). Consider, for example, S~l). By the lemn1a, either it is orthogonal to T~) for every i:2:; 1 or it contains T~i) for some i > 1. In the first case, it must be contained in the orthogonal complement T~O) of ~r=l T~) in Tx(M). This is obviously a contradiction. In the second case, the irreducibility of S~l) implies that S~l) actually coincides with T~i). QED. The following result is due to Borel and Lichnerowicz [1]. 5.5. The restricted homogeneous holonomy group of a Riemannian manifold M is a closed subgroup of SO(n), where n = dim M. Proof. Since the homogeneous holonomy group of the universal covering space of M is isomorphic with the restricted homogeneous holonomy group of M (cf. Example 2.1), we may assume, THEOREM

IV.

RIEMANNIAN CONNECTIONS

187

without loss of generality, that M is simply connected. In view of (3) of Theorem 5.4, our assertion follows from the following result in the theory of Lie groups: Let G be a connected Lie subgroup of SO(n) which acts irreducibly on the n-dimensional vector space R n. Then G is closed in SO (n) . The proof of this result is given in Appendix 5.

6. The decolnposition theorem of de Rham Let M be a connected, simply connected and complete Riemannian manifold. Assuming that M is reducible, let T x (M) = T~ + T~ be a decomposition into subspaces invariant by the linear holonomy group 'Y(x) and let T' and T" be the parallel distributions defined by T~ and T;, respectively, as in the beginning of §5. We fix a point 0 E M and let M' and M" be the maximal integral manifolds of T' and T" through 0, respectively. By Proposition 5.1, both M' and M" are complete, totally geodesic submanifolds of M. The purpose of this section is to prove 6.1. M is isometric to the direct product M' X M". Proof. For any curve Zt, 0 < t < 1, in M with Zo = 0, we shall define its projection on M' to be the curve X t , 0 < t < 1, with X o = 0 which is obtained as follows. Let Ct be the development of Zt in the affine tangent space To(M). (For the sake of simplicity we identify the affine tangent space with the tangent (vector) space.) Since To(M) is the direct product of the two Euclidean spaces T~ and T~, Ct may be represented by a pair (At, B t), where At and B t are curves in T~ and T~ respectively. By applying (4) of Theorem 4.1 to M', we see that there exists a unique curve X t in M' which is developed upon the curve At. In view of Proposition 4.1 of Chapter III we may define the curve X t as follows. For each t, let X t be the result of the parallel displacement of the T' -component of it from Zt to 0 = Zo (along the curve Zt). The curve x t is a curve in M' with X o = 0 such that the result of the parallel displacement of xt along itself to 0 is equal to X t for each t. Before proceeding further, we shall indicate the main idea of the proof. We show that the end point Xl of the projection X t depends only on the end point Zl of the curve Zt if M is simply connected. THEOREM

188

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Thus we obtain a projection p': M ---+ M' and, similarly, a projection p": M ---+ M". The mapping p = (p', p") of Minto M' X M" will be shown to be isometric at every point. Theorem 4.6 then implies that p is a covering projection of M onto M' X M". If his a homotopy in M from a curve of M' to another curve of M', then p' (h) is a homotopy between the two curves in M'. Thus, M' is simply connected. Similarly, M" is simply connected. Thus p is an isometry of M onto M' X M". The detail now follows. 1. Let 7 = Zt, 0 ~ t < 1, be a curve in .J.\1 with Zo = 0 and let a be any number with 0 < a < 1. Let 71 be the curve Zt, o < t < a, and let 7 2 be the curve Zt, a < t < 1. Let 7; be the projection of 7 2 in the maximal integral manifold M' (za) of T' through Za. Then the projection of 7 = 7 2 . 71 in M' coincides with the projection of LEMMA

,

7

=

,

72 ·71·

Proof of Lemma 1. This is obvious from the second definition of the projection by means of the (linear) parallel displacement of tangent vectors.

2.

Let Z E M and let V = V' X V" be an open neighborhood of Z in M, where V' and V" are open neighborhoods of Z in M' (z) and M" (z) respectively. For any curve Zt with Zo = Z in V, the projection of Zt in M' (z) is given by the natural projection of V onto V'. Proof of Lemma 2. For the existence of a neighborhood V = V' X V", see Proposition 5.2. Let Zt be given by the pair (xt,Yt) where X t (resp. Yt) is a curve in V' (resp. V") with X o = Z (resp. Yo = z). Since V = V' X V", the parallel displacement of the T'-component of it from Zt to Zo = Z along the curve Zt is the same as the parallel displacement of xt from X t to Xo = Z along the curve X t. Thus X t is the projection of the curve Zt in M'(z). We introduce the following terminologies. A (piecewise differentiable) curve Zt is called a T' -curve (resp. T"-curve) if it belongs to T~t (resp. T;J for every t. Given a (piecewise differentiable) homotopy z: [0, 1] X [0, so] ---+ M which is denoted by z(t, s) = zt, we shall denote by z~S) (resp. z(t») the curve with parameter t for the fixed value of s (resp. the curve with parameter s for the fixed value of t). Their tangent vectors will be denoted by i~S) and itt), respectively. For any point Z E M, let d' (resp. d") denote the distance function on the maximal LEMMA

IV.

RIEMANNIAN CONNECTIONS

189

integral manifold M' (z) of T' (resp. M" (z) of T") through z. Let U'(z; r) (resp. U"(z; r)) denote the set of points WE M'(z) (resp. WE M"(z)) such that d'(z, w) < r (resp. d"(z, w) < r). 3. Let T' = x t, 0 < t ~ 1, be a T'-curve. Then there exist a number r > 0 and a family of isometries it, 0 < t < 1, of U" (x o; r) onto U" (x t; r) with the following properties: (1) The differential of ft at X o coincides with the parallel displacement along the curve T' from X o to xt; (2) For any curve T" = yS, 0 < S ~ so, in U" (x o; r) with yO = xo, set z; = ft(yS). Then (a) For any 0 < t l ~ 1 and 0 < SI < so, the parallel displacement along the "parallelogram" formed by the curve x t, 0 < t ~ t l , the curve z(t 1 ), 0 < S ~ SI' the inverse of the curve Z~Sl), 0 < t ~ t l , and the inverse of the curve yS, 0 < S < SI' is trivial; (b) For any sand t, i~8) is parallel to xt along the curve z(t); (c) For any sand t, i 0 sufficiently small so that X t E V' and U"(x t ; r) C {x t} X V" for o < t < r. We define ft by ft(xo,y) = (xt,y) for y E U"(x o; r). It is clear that the family of isometries ft, 0 < t ::;;: r, has all the properties (1) and (2). The family ft can be extended easily for o < t < 1 and for a suitable r > 0 by covering the curve T' = X t by a finite number of neighborhoods of the form V = V' X V" and using the above argument for each neighborhood. LEMMA

4. Let T' = X t, 0 < t < 1, be a T'-curve and let T" = yS, o < S < so, be a T"-geodesic with yO = XO, where s is the arc length. Then there exists a homotopy z;, 0 ~ t ~ 1, 0 ~ s < so, with the following properties: (1) z~O) = X t and z(O) = yS ; (2) z; has properties (a), (b) and (c) of Lemma 3. The homotopy z; is uniquely determined. In fact, if Y t is the result of parallel displacement of the initial tangent vector Yo = jO of the geodesic T" along the curve T', then z; = exp sY t. Proof of Lemma 4. We first prove the uniqueness. By (a) and (c) and by the fact that T" is a geodesic, it follows that z(t) is parallel to Y t along the curve z(t). This means that z(t) is a LEMMA

190

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

geodesic with initial tangent vector Y t • Thus, zt = exp sY t , proving the uniqueness. I t remains therefore to prove that zt = exp sY t actually satisfies conditions (1) and (2). Condition (1) is obvious. To prove (2), we may assume that 7' is a differentiable curve so that zl is differentiable in (t, s). Let ft be the family of isometries as in Lemma 3. It is obvious that there exists a number b > 0 such that zl = ft(Y S ) for 0 < t < 1 and 0 ~ s < b. Thus, zl satisfies condition (2) for 0 < t < 1 and 0 < S < b. Let a be the supremum of such b. In order to prove a = So, assume a < So. First we show that zl satisfies (2) for 0 < t ~ 1 and 0 < S ~ a. Since zl is differentiable in (t, s), the parallel displacement along the curve z~a) is the limit of the parallel displacement along the curve z~S) as s t a (cf. Lemma for Theorem 4.2 of Chapter II). Thus condition (a) is satisfied. We have also ift) = lim ift) and sia

i~a) =

lim

i}s).

Combined with the above limit argument, this

sia

gives conditions (b) and (c) for 0 < t < 1 and s = a. In order to show that zl has property (2) beyond the value a, we apply Lemma 3 to the T'-curve 7(a) = z~a) and the T" -geodesic y U , where u = s - a. We see then that there exist a number r > 0 and a homotopy wf, 0 < t < 1, -r < u < r, satisfying a condition similar to (2), such that w~O) = z~a) and w~ = yS. Since w~O) is parallel to ja along the curve w~O) = z~a), it follows that zt = w~-a for 0 < t < 1 and a - r < s < a + r. This proves that z~ satisfies condition (2) for 0 < t < 1 and 0 < S < a + r, contradicting the assumption that a < So. LEMMA 5. Keeping the notation of Lemma 4, the projection of the curve 7' . 7"-1 in M' (ySo) coincides with 7(sO) = z~so), 0 < t < 1. Proof of Lemma 5. Since 7"-1 is a T"-curve, its projection in M' (ySo) is trivial, that is, reduces to the point ySo. Conditions (a) and (b) imply that, for each t, the parallel displacement of it along 7" . 7'-1 to y80 is the same as the parallel displacement of i~so) along z~so) to y·'1 o• This means that 7" . 7'-1 projects on 7(so). We now come to the main step for the proof of Theorem 6.1.

6. If two curves 71 and 7 2 from 0 to a point z in M, are homotopic to each other, then their projections in M' = M' (0) have the same end point. LEMMA

IV.

RIEMANNIAN CONNECTIONS

191

Proof of Lemma 6. We first remark that T2 is obtained from Tl by a finite succession of small deformations. Here a small deformation of a curve Zt means that, for a certain small neighborhood V, we replace a portion Zt, t 1 < t < t 2 , of the curve lying in V by a curve W t , t 1 < t < t 2, with W t 1 = Zt 1 and W t 2 = Zt 2 , lying in V. As a neighborhood V, we shall ahvays take a neighborhood of the form V' X V" as in Lemma 2. I t suffices therefore to prove the following assertion. Let T be a curve from 0 to ZI' fl a curve from ZI to Z2 which lies in a small neighborhood V = V' X V" and K a curve from Z2 to z. Let v be another curve from ZI to Z2 which lies in V. Then the projections of K • fl . T and K· V • T in M' have the same end point. To prove this, we may first replace the curve K by its projection in M' (Z2) by Lemma 1. Thus we shall assume that K is a T'curve. Let fl be represented by the pair (fl', fl") in V = V' X V". By Lemma 2, the projection of fl in M'(ZI) is fl'. Let fl* be a Til-geodesic in V joining Z2 and the end point of fl'. The parallel displacement of T' -vectors at Z2 along fl- 1 is the same as the parallel displacement along fl'-I. fl*, because fl" and fl* give the same parallel displacement for T'-vectors. By Lemma 5, we see that the projection of K • fl in M' (ZI) is the curve fl' followed by the curve K' obtained by using the homotopy zf constructed from the T"-geodesic fl* and the T'-curve K. The homotopy zf depends only on fl* and K and not on fl. Thus if we replace fl by v in the above argument, we see that the projection of K • v is equal to v' followed by K', where v = (v', v") in V = V' X V". We now divide T into a finite number of arcs, say, Tl' T2' • • • , Tk' such that each Ti lies in a small neighborhood Vi of the form V~ X V7. We show that the projections of the curves K' • fl' . T k and K' • v' . T k have the same end point in the maximal integral manifold of T' through the initial point of Tk. Again, let T k = (T~, T~) in Vk = V~ X V~ and let TZ be the geodesic in Vk joining the end point of T k to the end point of T~. As before, the projection of K' • fl' . Tk is the curve T~ followed by the curve obtained by the homotopy which is constructed from the T" -geodesic TZ and the T' -curve K' • fl'. Similarly for the projection of K' • v' . Tk. Each homotopy was constructed by the parallel displacement of the initial tangent vector of the geodesic TZ along K' • p/ or along

192 K' • V'.

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Since

1"-1 •

fl' is a curve in V', the parallel displacement

along 1"-1 • fl' is trivial for T" -vectors. This means that the parallel displacements of the initial tangent vector of TZ along fl' and v' are the same so that the two homotopies produce the curves flk and V k starting at the end point of T~ and ending at the same point, where a curve K k starts in such a way that K k ' flk • T~ and .. K k • V k • Tk, are t h e projectIons 0 f" K • fl • T k an d" K • V • T k respectively. We also remark that the parallel displacements of every T" -vector along flk and V k are the same; this indeed follows from property (a) of the homotopy in Lemma 4. We continue to the next stage of projecting the curves K k • flk • T~ • T k - 1 and K k ' V k • T~ • T k - l by the same method. As a result of the above remark, we have two curves ending at the same point. Now it is obvious that this process can be continued, thus completing the proof of Lemma 6. Now we are in position to complete the proof of Theorem 6.1. Lemma 6 allows us to define a mapping p' of Minto M'. Similarly, we define a mapping p" of Minto M". These mappings are differentiable. As we indicated before Lemma 1, we have only to show that the mapping p = (P', p") of Minto M' X M" is isometric at each point. Let z be any point of M and let T be a curve from 0 to z. For any tangent vector Z E Tz(M), let Z = X + Y, where X E T~ and YET;. By definition of the projection, it is clear that p' (Z) is the same as the vector obtained by the parallel displacement of X from z to 0 along T and then from 0 to p'(z) along p'(T). Therefore, p'(Z) and X have the same length. Similarly, p" (Z) and Y have the same length. I t follows that Z and p( Z) = (p' (Z), p" (Z)) have the same length, proving that p is isometric at z. QED. Combining Theorem 5.4 and Theorem 6.1, we obtain the decomposition theorem of de Rham.·

6.2. A connected, simply connected and complete Riemannian manifold M is isometric to the direct product M o X M 1 X . . . X M k , where M o is a Euclidean space (possibly of dimension 0) and M v ... , M k are all simply connected, complete, irreducible Riemannian manifolds. Such a decomposition is unique up to an order. THEOREM

Theorems 6.1 and 6.2 are due to de Rham [1]. The proof of Theorem 6.1 is new; it was inspired by the work of Reinhart [1].

7. Affine holonomy groups Let M be a connected Riemannian manifold. Fixing a point x of M, we denote the affine holonomy group <1>(x) and the linear holonomy group 'Y(x) simply by <1> and 'Y, respectively. We know (cf. Theorem 5.5) that the restricted linear holonomy group 'YO is a closed subgroup of SO(n), where n = dim M. <1> is a group of Euclidean motions of the affine (or rather Euclidean) tangent space T", (M). We first prove the following result. 7.1. If 'YO is irreducible, then either (1) <1>0 contains all translations of T",(M) ,

THEOREM

or (2) <1>0 fixes a point of T ro (M). Proof. Let K be the kernel of the homomorphism of <1>0 onto 'YO (cf. Proposition 3.5 of Chapter III). Since K is a normal subgroup of <1>0 and since every element a of <1>0 is of the form a = ~ . Ii where ii € 'Yo and ~ is a pure translation, 'YO normalizes K, that is, ii-1Kii = K for every ii € 'Yo. Consider first the case where K is not discrete. Since 'Yo is connected, it normalizes the identity component KO of K. Let V be the orbit of the origin of Tro(M) by KO. It is a non-trivial linear subspace of Tro(M) invariant by 'Yo; the invariance by 'Yo is a consequence of the fact that 'Yo normalizes KO. Since 'Yo is irreducible by assumption, we have V = Tro(M). This means that <1>0 contains all translations of Tro(M). Consider next the case where K is discrete. Since 'Yo is connected, 'Yo commutes with K elementwise. Hence, for every ~ € K, ~(O) is invariant by 'Yo (where 0 denotes the origin of Tro(M)). Since 'Yo is irreducible, ~(O) = 0 for every ~ € K. This means that K consists of the identity element only and hence that <1>0 is isomorphic to 'Yo in a natural manner. In particular, <1>0 is compact. On the other hand, any compact group of affine transformations of Tro(M) has a fixed point. Although we shall prove a more general statement in Volume II, we shall give here a direct proof of this fact. Let f be the mapping from <1>0 into T",(M) defined by f(a) = a(O) for a € <1>0.

194

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Let da be a bi-invariant Haar nleasure on <:Do and define

Xo

=

J

f(a) da.

It is easy to verify that X o is a fixed point of <:Do. QED. We now investigate the second case of Theorem 7.1 (without assuming the irreducibility of M).

7.2.

Let M be a connected, simply connected and complete Riemannian manifold. If the (restricted) affine holonomy group <:DO at a point x fixes a point of the Euclidean tangent space Tm(M), then M is isometric to a Euclidean space. Proof. Assuming that X o € Tm(M) is a point fixed by <:Do, let 7 be the geodesic from x to a pointy which is developed upon the t 1. We observe that the affine holonomy line segment tXo, 0 group <:DO(y) aty fixes the origin of Ty(M). In fact, for any closed curve ft at y, the affine parallel displacement along 7- 1 . ft . 7 maps X o into itself, that is, (7- 1 . ft . 7)XO = X o. Hence the origin of Ty(M) given by 7(XO) is left fixed by p. This shows that we may assume that <:Do fixes the origin of Tm(M). Since M is complete, the exponential mapping Tm( M) ---+ M is surjective. We show that it is 1: 1. Assume that two geodesics 7 and ft issuing from x meet at a pointy x. The affine parallel displacement ft- 1 . 7 maps the origin Om of Tm(M) into itself and hence we have THEOREM

where Oy denotes the origin of Ty(M). Since 7 -1(Oy) and ft-1(Oy) are the end points of the developments of 7 and ft in Tm(M) respectively, these developments which are line segments coincide with each other. Thus 7 - ft, contradicting the assumption that x #- y. This proves that the exponential mapping Tm(M) ---+ M is 1: 1. Assume that expx is a diffeomorphism of N(x; r) - {X € T x ( M); IIXII < r} onto U(x; r) = {y € M; d(x,y) r}, and let xl, ... , x n be a normal coordinate system on U(x; r). We set X = -~i=l Xi (ajax i ) and let p be the corresponding point field (cf. §4 of Chapter III). We show that p is a parallel point field. Since <:DO fixes the origin of Tx(M), it is sufficient to

IV.

195

RIEMANNIAN CONNECTIONS

prove that p is parallel along each geodesic through x. Our assertion follows therefore from 1. Let T = X t , 0 < t < 1, be a curve in a Riemannian manifold M and let f~ (resp. T~) denote the affine (resp. linear) parallel displacement along T from X t to X S • Then LEMMA

where Ct , 0 < t < 1, is the development of T = X t into T x (M). Proof of Lemma 1. Given Y € Txt(M), let p (resp. q) be the point field along T defined by the affine parallel displacement of Y (resp. the origin of Txt(M)) and let y* be the vector field along T defined by the linear parallel displacement of Y. Then p = q + y* at each point of T, that is, y* is the vector with initial point q and end point p at each point of T. At the point Xo, this means precisely f~( Y) = T~( Y) + Ct. Going back to the proof of Theorem 7.2, we assert that for any vector field V. This follows from

2.

Let p be a point field along a curve T = X t, 0 < t ::;;: 1, in a Riemannian manifold M and let X be the corresponding vector field along T. Then p is a parallel point field if and only if LEMMA

V.X Xt

+ xt

Proof of Lemma 2.

for 0 < t < 1.

= 0

From Lemma 1, we obtain

ft+h(p t Xt+h ) --

~+h(X

t

Xt+h )

+ Ct,h'

where Ct ,h (for a fixed t and with parameter h) is the development of T into Tx/M). Since f~+h(PTt+h) is independent of h (and depends only on t) if and only if P is parallel, we have

o=

lim -hI h~O

[T~+h(Xx t+h )

- X ] Xt

+ lim h~O

-hI

Ct,h

= V.X Xt + xt for 0 < t < 1 if and only if p is parallel, completing the proof of Lemma 2.

196

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Let Yand Z be arbitrary vector fields on M. From V yX + Y = oand VzX + Z 0, we obtain (cf. Theorem 5.1 of Chapter III) VxY = VyX

[X, Y] = -Y

and

VxZ

VzX

[X, Z] = -Z

+ [X,

Y]

+ [X, ZJ.

Hence, X(g(Y, Z))

g(VxY, Z) 2g(Y, Z)

+ g(Y, V xZ) + g([X,

Y], Z)

+ g(Y,

[X, Z]).

Let Yo/ax; and Z = %xk for any fixed} and k. Then we have X . gjk = -2gik

+ gjk + gjk

= O.

This means that the functions gik are invariant by the local 1parameter group of transformations CPt generated by X. But CPt is of the form Thus the functions gjk are constant along each geodesic through x. Hence, gik = g jk(X) = fJ ik at every point of U(x; r). This shows that expx is an isometric mapping of N(x; r) with Euclidean metric onto U(x; r). Let ro be the supremum of r > 0 such that expx is a diffeomorphism of N(x; r) onto U(x; r). Since the differential (expx)* is non-singular at every point of N(x; ro), expxis a diffeomorphism, hence an isometry, by the argument above, of N(x; ro) onto U(x; ro). If ro 00, it follows that (expx)* is isometric at every pointy of the boundary of N(x; ro) and hence nonsingular in a neighborhood of such y. Since the boundary of N(x; ro) is compact, we see that there exists c > 0 such that expx is a diffeomorphism of N(x; ro c) onto U(x; ro + c), contradicting the definition of roo This shows that expx is a diffeomorphism of Tx(M) onto M. By choosing a normal coordinate system Xl, . . . , x n on the whole M, we conclude that gik = fJ jk at every point of M, that is, M is a Euclidean space. QED. As a consequence we obtain the following corollary due to Goto and Sasaki [1 J.

IV.

RIEMANNIAN CONNECTIONS

197

7.3. Let M be a connected and complete Riemannian manifold. If the restricted affine holonomy group °(x) fixes a point of the Euclidean tangent space T x ( M) for some x E M, then M is locally Euclidean (that is, every point of M has a neighborhood which is isometric to an open subset of a Euclidean space). Proof. Apply Theorem 7.2 to the universal covering space of M. QED. COROLLARY

7.4. If M is a complete Riemannian manifold of dimension> 1 and if the restricted linear holonomy group ,¥O(x) is irreducible, then the restricted affine holonomy group °(x) contains all translations of T x (JlVf) . Proof. Since ,¥O(x) is irreducible, M is not locally Euclidean. Our assertion now follows from Theorem 7.1 and Corollary 7.3. COROLLARY

QED.

CHAPTER V

Curvature and Space Forms

1. Algebraic preliminaries Let V be an n-dimensional real vector space and R: V x V x V x V -+ R a quadrilinear mapping with the following three properties:

(a) R(v I , V2 , V3 , v4 ) = -R(v z, VI' V3 , v4 ) (b) R(v I , Vz, V3 , v4 ) = -R(v I , Vz' V4 , v3 ) (c) R(v I , vz, V3 , v4 ) + R(v I , V3 , V4 , v2 ) + R(v I , V4 , V2 , va)

=

O.

1.1. If R possesses the above three properties, then it possesses also the following fourth property : (d) R(v I , V2 , v3 , v4 ) = R(v 3 , V4 , VI' Vz). Proof. We denote by S(v I , VZ, Va' v4 ) the left hand side of (c). By a straightforward computation, we obtain PROPOSITION

o=

S(v I , Vz, Va, v4 )

S(v z, Va, V4 ,

-

S(v a, V4 ,

VI) -

VI'

V2 )

+ S(V4 , VI' V2 , Va) R(v I , V2 , Va' V4 )

R(v z,

-

VI'

V3 , V4 )

-

R(v3 , V4 ,

VI'

Vz)

R(V4 , Va,

VI'

Vz).

By applying (a) and (b), we see that

2R(Vb V2 , V3 , V4 )

2R(Va, V4 ,

-

VI'

Vz)

=

O.

QED.

1.2. Let Rand T be two quadrilinear mappings with the above properties (a), (b) and (c). If PROPOSITION

R(v b V2 , Vb V2 ) then R

=

T(v I , V2 ,

VI'

T. 198

v2 )

V.

199

CURVATURE AND SPACE FORMS

Proof. We may assume that T = 0; consider R - T and 0 instead of Rand T. We assume therefore that R(v I , V 2 , VI' V 2 ) = 0 for all VI' V 2 E V. We have

+ V4 , VI' V 2 + V4 ) R(v I , V 2 , VI' V4 ) + R(v I , V4 Vb V 2 )

o=

R(v I ,

=

V2

= 2R(v I ,

V 2 , VI' V4 )·

Hence,

(1) From (I) we obtain

o= =

R(v i

+ vs, V

R(v I ,

V 2,

+ vs, V4) + R(v s, V 2, VI' v4 )·

VI

2,

vs, V4 )

Now, by applying (d) and then (b), we obtain

o= =

+ R( VI' VM vs, V 2 )

R( VI'

V 2,

vs,

V4 )

R( VI'

V2 ,

vs,

V4 ) -

R (VI'

V4 , V 2 , V

s).

Hence,

(2)

R(v I ,

Replacing

(3)

V 2,

vs, v4 ) = R(v I ,

V 2 , V S , V4

R(v I ,

V 2,

by

V S , V4 , V 2 ,

vs, v4 ) = R(v I ,

V4 , V 2,

vs)

respectively, we obtain V S , V4 ,

v2 )

for all VI'

V 2 , V S , V4 E

+ R(v I , VS, V4 , v2 ) + R(v

vs),

V.

From (2) and (3), we obtain

3R(v b

V 2,

vs, v4 ) = R(v I ,

V 2,

vs, v4 )

I , V4 , V 2 ,

where the right hand side vanishes by (c). Hence,

R(v I ,

V 2,

vs, v4 ) = 0

for all

VI' V 2 , VS, V4 E

V.

QED. Besides a quadrilinear mapping R, we consider an inner product (i.e., a positive definite symmetric bilinear form) on V, which will be denoted by (,). Let p be a plane, that is, a 2dimensional subspace, in Vand let VI and V 2 be an orthonormal basis for p. We set

200

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

As the notation suggests, K(P) is independent of the choice of an orthonormal basis for p. In fact, if WI and W 2 form another orthonormal basis of p, then WI

= aV I

+ bv 2 ,

W 2

=

bV I

aV 2

(or bVI - av 2 ),

where a and b are real numbers such that a2 + b2 = 1. Using (a) and (b), we easily obtain R(v I, V2 , VI' V2 ) = R(w I, W 2 , WI' W 2 ). \

1.3. of a plane p in V, then K(p) PROPOSITION

If VI' =

v2 is a basis (not necessarily orthonormal)

R(v I , V 2, VI' V 2 ) 2 (Vl' VI) (V 2' V2) - (V l' V2)

•

Proof. We obtain the formula making use of the following orthonormal basis for p: VI ---...".,

(VI,V I)

1 a

- [(VI'

VI)V 2

(Vb V2 )V I] .

-

QED.

where a = [(VI' VI) ((VI' VI) (V 2, V2) - (VI' V2)2)]! We set

R I (VI' V2 , V3 , V4 ) = (VI' V3 ) (V 2 , V4 )

-

(V 2 , V3 ) (V4 , VI) for

VI'

V2 , V3 , V4

€

V.

It is a trivial matter to verify that R I is a quadrilinear mapping having the properties (a), (b) and (c) and that, for any plane p in V, we have KI(p) = RI(V b V2 , VI' V2 ) 1, where

VI'

v2 is an orthonormal basis for p.

1.4. Let R be a quadrilinear mapping with properties (a), (b) and (c). If K(p) = cfor all planes p, then R = cR I. Proof. By Proposition 1.3, we have PROPOSITION

Applying Proposition 1.2 to Rand cR I, we conclude R

=

cR I.

QED.

Let el' . . . , en be an orthonormal basis for V with respect to the inner product (,). To each quadrilinear mapping R having

V.

201

CURVATURE AND SPACE FORMS

properties (a), (b) and (c), we associate a symmetric bilinear form S on V as follows:

S(v l , v2)

=

R(e l ,

VI'

el , V 2)

+ R(e 2, VI' e2, V 2) + ... + R(en, VI' en' v2),

V. It can be easily verified that S is independent of the choice of an orthonormal basis el , ... , en. From the definition of S, we obtain VI' V 2 €

Let V € V be a unit vector and let v, e2, ... , en be an orthonormal basis for V. Then PROPOSITION

1.5.

S(v, v)

K(P2)

=

where Pi is the plane spanned by

V

+ ... + K(Pn),

and ei.

2. Sectional curvature Let M be an n-dimensional Riemannian manifold with metric tensor g. Let R (X, Y) denote the curvature transformation of Tx(M) determined by X, Y € T()J(M) (cf. §5 of Chapter III). The Riemannian curvature tensor (field) of M, denoted also by R, is the tensor field of covariant degree 4 defined by

R(Xl , X 2, X 3 , X 4 ) = g(R(X3 , X 4 )X2, Xl), Xi € T()J(M) , i

=

1, ... , 4.

The Riemannian curvature tensor, considered as a quadrilinear mapping T()J(M) X Tx(M) X Tx(M) X Tx(M) ~ R at each x € M, possesses properties (a), (b), (c) and hence (d) of §1. Proof. Let u be any point of the bundle O(M) of orthonormal frames such that 1T(U) = x. Let Xl, X: € Tit (O(M)) with 1T(Xl) = X 3 and 1T(X:) = X 4 • From the definition of the curvature transformation R(X3 , X 4 ) given in §5 of Chapter III, we obtain g(R(X , X 4 )X2, Xl) = g(u[20(Xl, XX) (U- l X 2)], Xl) PROPOSITION

2.1.

3

=

((20(X:, XX)) (U- l X 2), U-lX l ),

where (, ) is the natural inner product in Rn. Now we see that property (a) is a consequence of the fact that O(Xl, X:) € o(n) is a skew-symmetric matrix. (b) follows from R(X3 , X 4 ) = -R(X4 , X 3 ). Finally, (c) is a consequence of Bianchi's first identity given in Theorem 5.3 of Chapter III. QED.

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FOUNDATIONS OF DIFFERENTIAL GEOMETRY

For each plane p in the tangent space Tro(M) , the sectional curvature K(p) for p is defined by

K(P) = R(XI, X 2 , Xl' X 2 ) = g(R(XI, X 2 )X2 , Xl), where Xl' X 2 is an orthonormal basis for p. As we saw in §1, K(p) is independent of the choice of an orthonormal basis Xl' X 2 • Proposition 1.2 implies that the set of values of K(p) for all planes p in Tro(M) determines the Riemannian curvature tensor at x. If K(p) is a constant for all planes p in Tro(M) and for all points x E M, then M is called a space of constant curvature. The following theorem is due to F. Schur [1J. 2.2. Let M be a connected Riemannian manifold if dimension > 3. If the sectional curvature K(p), where p is a plane in T ro (M), depends only on x, then M is a space of constant curvature. Proof. We define a covariant tensor field R I of degree 4 as follows: THEOREM

RI(W, Z, X, Y)

=

g(W, X)g(Z, Y) - g(Z, X)g(Y, W), W, Z, X, Y

By Proposition 1.4, we have R

=

E

Tro(M).

kR 1,

where k is a function on M. Since g is parallel, so is R I • Hence,

(VuR) (W, Z, X, Y)

=

(V u k)R 1 (W, Z, X, Y)

This means that, for any X, Y, Z, U E

[(VuR) (X, Y)]Z

=

for any U Tro(M), we have

(Uk) (g(Z, Y)X

E

T ro (M).

g(Z, X)Y).

Consider the cyclic sum of the above identity with respect to (U, X, Y). The left hand side vanishes by Bianchi's second identity (Theorem 5.3 of Chapter III). Thus we have

o=

(Uk) (g(Z, Y)X

g(Z, X) Y)

+ (Xk) (g(Z, U) Y + (Yk) (g(Z, X) U -

g(Z, Y) U) g(Z, U)X).

V.

203

CURVATURE AND SPACE FORMS

For an arbitrary X, we choose Y, Z and U in such a way that X, Y and Z are mutually orthogonal and that U = Z with g(Z, Z) = 1. This is possible since dim M > 3. Then we obtain (Xk) Y - (Yk)X = O.

Since X and Yare linearly independent, we have Xk This shows that k is a constant. COROLLARY

2.3.

For a space

R(X, Y) Z

=

of constant curvature k,

=

Yk = O.

QED.

we have

k(g(Z, Y)X - g(Z, X) Y).

This was established in the course of proof for Theorem 2.2. If k is a positive (resp. negative) constant, M is called a space of constant positive (resp. negative) curvature. If R}kl and gij are the components of the curvature tensor and the metric tensor with respect to a local coordinate system (cf. §7 of Chapter III), then the components Rijk~ of the Riemannian curvature tensor are given by

If M is a space of constant curvature with K(P)

=

k, then

As in §7 of Chapter III, we define a set of functions R}kl on L(M) by where 0 (OJ) is the curvature form of the Riemannian connection. For an arbitrary point u of O(M), we choose a local coordinate system xl, ... , xn with origin x = 1T(U) such that u is the frame given by (oj ox1 ) x, ••• , (oj ox n ) x. With respect to this coordinate system, we have I

and hence R}k~ = Riik~ = k( bikb H

-

b jkb~i)

at x.

Let (J be the local cross section of L (M) given by the field of linear frames oj ox\ ... , oj ox n • As we have shown in §7 of

204

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Chapter III, we have a *RjH = RjH. Hence, ~

RjH

=

k(~ik~jZ - ~ik~U)

OiJ = k(Ji A (Ji

at u,

at u.

Since u is an arbitrary point of O(M), we have

If

M is a space of constant curvature with sectional curvature k, then the curvature form 0 = (OJ) is given by PROPOSITION

2.4.

OJ = k(Ji where (J

=

A

(Ji

on 0 (M) ,

((Ji) is the canonical form on 0 (M) .

3. Spaces of constant curvature In this section, we shall construct, for each constant k, a simply connected, complete space of constant curvature with sectional curvature k. Namely, we prove

Let (x\ ... , x n , t) be the coordinate system and M the hypersurface of R n+l defined by (X1)2 + ... + (X n )2 + rt 2 = r (r: a nonzero constant).

THEOREM

R n+l

3.1.

Let g be the Riemannian metric form to M:

of M

of

obtained by restricting the following

Then (1) M is a space of constant curvature with sectional curvature 1Jr. (2) The group G of linear transformations of R n+l leaving the quadratic form (X 1)2 + ... + (X n )2 + rt 2 invariant acts transitively on M as a group of isometries of M. (3) If r > 0, then M is isometric to a sphere of a radius rt . If r < 0, then M consists of two mutually isometric connected manifolds each of which is diffeomorphic with R n. Proof. First we observe that M is a closed submanifold of R n+l (cf. Example 1.1 of Chapter I) ; we leave the verification to the reader. We begin with the proof of (3). Ifr > 0, then we set xn +1 = rit. Then M is given by (X 1)2 + ... + (X n +1)2 = r,

V.

CURVATURE AND SPACE FORMS

205

and the metric g is the restriction of (dx l ) 2 + ... + (dx n +l ) 2 to M. This means that M is isometric with a sphere of radius rt. If r < 0, then t2 > 1 at every point of M. Let M' (resp. M") be the set of points of M with t > 1 (resp. t < -1). The mapping (xl, ... , xn, t) -+ (yl, ... ,yn) defined by

yi = xilt,

i = 1, ... , n,

is a diffeomorphism of M' (and M") onto the open subset of Rn given by ~i=l

(yi)2

+r

<

o.

In fact, the inverse mapping is given by

i t

=

=

1, ... , n,

±C +;, (y.)S·

A straightforward computation shows that the metric g is expressed in terms ofyl, ... ,yn as follows:

r[ (r

+ ~i (yi) 2) (~i (dyi) 2) (r + ~i (y~)2)2

(~i

yi dyi) 2]

To prove (2), we first consider G as a group acting on Rn+l. Since G is a linear group leaving (X I)2 + ... + (X n )2 + rt 2 invariant, it leaves the form (dX I )2 + ... + (dX n )2 + r dt 2 invariant. Thus, considered as a group acting on M, G is a group of isometries of the Riemannian manifold M. The transitivity of G on M is a consequence of Witt's theorem, which may be stated as follows. Let Q be a nondegenerate quadratic form on a vector space V. If f is a linear mapping of a subspace U of V into V such that Q(f(x)) = Q(x) for all x E U, thenf can be extended to a linear isomorphism of V onto itself such that Q(f(x)) = Q(x) for all x E V. In particular, if Xo and Xl are elements of V with Q(x o) = Q(x l ), there is a linear isomorphism f of V onto itself which leaves Q invariant and which maps X o into Xl. For the proof of Witt's theorem, see, for example, Artin [1, p. 121]. Finally, we shall prove (1). Let H be the subgroup of G which consists of transformations leaving the point 0 with coordinates'

206

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(0, ... , 0, 1) fixed. We define a mapping f: G -+ O(M) as follows. Let U o E 0 (M) be the frame at the point 0 = (0, ... , 0, 1) EM given by (ajax1)o, ... , (ajaxn)o. Every element a E G, being an isometric transformation of M, maps each orthonormal frame of M into an orthonormal frame. In particular, a(u o) is an orthonormal frame of M at the point a( 0). We define

f(a) = a(uoL

a E G.

1. The mapping f: G -+ 0 ( M) is an isomorphism of the principal fibre bundle G (Gj H, H) onto the bundle 0 (M) (M, 0 (n) ). Proof of Lemma 1. If we consider G as a group of (n 1) X (n 1)-matrices in a natural manner, then H is naturally isomorphic with O(n): LEMMA

+

+

It is easy to verify that f: G -+ O(M) commutes with the right translation R a for every a E H = 0 (n) :

f(ba) = f(b) . a

for bEG and a E H = 0 (n) .

The transitivity of G on M implies that the induced mapping f: GjH -+ M is a diffeomorphism and hence that f: G -+ O(M) is a bundle isomorphism. The quadratic form defining M is given by the following (n + 1) X (n + 1) -rnatrix :

Q=

(~n ~).

An (n + 1) X (n + I)-matrix a is an element ofG if and only if taQa = Q, where ta is the transpose of a. Let

a

=

(X wy), t

z

where X is an n X n-matrix, y and z are elements of R nand w is a real number. Then the condition for a to be in G is expressed by

txx

+ r· z tz

= In,

tXy

+ r . zw

=

0,

7Y + r· w 2

= r.

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CURVATURE AND SPACE FORMS

207

It follows that the Lie algebra of G is formed by the matrices of the form

where A is an n X n-matrix with tA + A elements ofRn satisfying b + rc = O. Let 1

I

IXI ••• IXn

=

0 and band care

RI

P

. .. .. .. ... . IX I ••• IXn

Rn P

YI ••. Yn

0

n

n

be the (left invariant) canonical I-form on G (cf. §4 of Chapter I). We have IX} + IX1 = 0, {3i + ry i = 0, i, J = 1, . . . , n. The Maurer-Cartan equation of G is expressed by

d{3i

=

dIXji

-

- ~k IXl A -

""

i

k-lk IXk

{3k,

A IXjk

-

{3 i A Y j,

J = 1, ... , n.

..

1"

2. Let () = (()i) and w = (w}) be the canonical form and the connection form on 0 (M). Then LEMMA

f*()i = (3i

and f*wj =

IX],

i,J = 1, ... , n.

Proof of Lemma 2. As we remarked earlier, every element a € G induces a transformation of O(M); this transformation corresponds to the left translation by a in G under the isomorphism f: G -+ O(M). From the definition of (), we see easily that () = (()i) is invariant by the transformation induced by each a € G. On the other hand, ({3i) is invariant by the left translation by each a € G. To prove f*()i = (3i, it is therefore sufficient to show that (f*()i)(X*) = (3i(X*) for all X* € Te(G). Set Xi = (ajaxi)o so that the frame U o is given by (X b ••• , X n ). The composite mapping 7r f: G -+ 0 (M) -+ M maps an element of Te(G) (identified with the Lie algebra of G) of the form 0

208

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

into the vector ~i biXi E To(M), where b\ ... , bn are the components of b. Therefore, if x* E Te(G), then 7T f(X*) = ~i fJt(X*)Xi and hence 0

(f*(P(X*), ... ,f*()n(X*)) = U 0 1 (7T of(X*)) = (~l(X*), ... , fJn(x*)), which proves the first assertion of the lemma. Let 9 and 1) be the Lie algebras of G and H, respectively. Let m be the linear subspace of 9 consisting of matrices of the form

It is easy to verify that m is stable under ad H, that is, ad (a) (m) m for every a E H. Applying Theorem 11.1 of Chapter II, we see that (aJ) defines a connection in the bundle G(GjH, H). Now, the second assertion of Lemma 2 follows from the following three facts: (1) (fJt) corresponds to (()i) under the isomorphism f: G ~ O(M); (2) the Riemannian connection form (w;) is characterized by the property that the torsion is zero (Theorem 2.2 of Chapter IV), that is, d()i = -~kw1A()k; (3) the connection form (aD satisfies the equality: dfJi = -~k A fJk. We shall now complete the proof of Theorem 3.1. Lemma 2, together with

at

and implies

showing that the curvature form of the Riemannian connection is given by

!r

()i A ()j.

By Proposition 2.4, M is a space of constant

curvature with sectional curvature

Ifr.

QED.

Remark. The group G is actually the group of all isometries of M. To see this, let ~(M) be the group of isometries of M and define a mapping f: ~(.Al) ~ O(M) in the same way as we defined f: G ~ O(M). Then G c ,3(M) and f: ,3(M) ~ O(M)

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CURVATURE AND SPACE FORMS

209

is an extension of f: G -+ O(M). Since f maps ~(M) 1: 1 into OeM) and sincef(G) = O(M), we must have G = ~(M). In the course of the proof of Theorem 3.1, we obtained (1) Let M be the sphere in Rn+l defined by (X 1)2 + ... + (X n +1)2 a2• Let g be the restriction of (dx1) 2 + ... + (dx n +1)2 to M. Then, with respect to this Riemannian metric g, M is a space ofconstant curvature with sectional curvature 1/a2 • (2) Let M be the open set in R n defined by (X1)2 + ... + (X n )2 < a2• THEOREM

3.2.

Then, with respect to the Riemannian metric given by a2[(a 2 - ~i (yi)2) (~i (4JJi)2) - (~i yi 4JJi)2] (a 2 - ~i (y~)2)2 ' M is a space

of constant curvature with sectional curvature -1/ a2 •

The spaces M constructed in Theorem 3.2 are all simply connected, homogeneous and hence complete by Theorem 4.5 of Chapter IV. The space Rn with the Euclidean metric (dX 1)2 + . . . + (dx n ) 2 gives a simply connected, complete space of zero curvature. A Riemannian -manifold of constant curvature is said to be elliptic, hyperbolic or flat (or localfy Euclidean) according as the sectional curvature is positive, negative or zero. These spaces are also called .space forms (cf. Theorem 7.10 of Chapter VI).

4. Flat affine and Riemannian connections Throughout this section, M will be a connected, paracompact manifold of dimension n. Let A(M) be the bundle of affine frames over M; it is a principal fibre bundle with structure group G = A(n; R) (cf. §3 of Chapter III). An affine connection of M is said to be flat if every point of Mhas an open neighborhood U and an isomorphism 1p: A(M) -+ U X G which maps the horizontal space at each u E A( U) into the horizontal space at 1p(u) of the canonical flat connection of U X G. A manifold with a flat affine connection is

210

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

said to be locally ajjine. A Riemannian manifold is flat (or locally Euclidean) if the Riemannian connection is a flat affine connection. 4.1. For an ajjine connection of M, the following conditions are mutually equivalent: (1) It is flat; (2) The torsion and the curvature of the corresponding linear connection vanish identically; (3) The affine holonomy group is discrete. Proof. By Theorem 9.1 of Chapter II, an affine connection is flat if and only if its curvature form on A(M) vanishes identically. The equivalence of (1) and (2) follows from Proposition 3.4 of Chapter III. The equivalence of (2) and (3) follows from Theorems 4.2 and 8.1 of Chapter II. QED. THEOREM

n

Remark. Similarly, for a linear connection of M, the following conditions are mutually equivalent: (1) It is flat, i.e., the connection in L(M) is flat; (2) Its curvature vanishes identically; (3) The linear (or homogeneous) holonomy group is discrete. When we say that the affine holonomy group and the linear holonomy group are discrete, we mean that they are O-dimensional Lie groups. Later (cf. Theorem 4.2) we shall see that the affine holonomy group of a complete flat affine connection is discrete in the affine group A(n; R). But the linear holonomy group need not be discrete in GL(n; R) (cf. Example 4.3). It will be shown that the linear holonomy group of a compact flat Riemannian manifold is discrete in 0 (n) (cf. the proofof (4) of Theorem 4.2 and the remark following Theorem 4.2). Example 4.1. Let ~l' ••• , ~k be linearly independent elements ofRn, k < n. Let G be the subgroup ofRn generated by ~l' . . . , ~k: G

=

{~ mi~ i; m i

integers}.

The action of G on Rn is properly. discontinuous, and Rn is the universal covering manifold of RnjG. The Euclidean metric (dx 1 ) 2 + ... + (dx n ) 2 of R n is invariant by G and hence induces a flat Riemannian metric on RnjG. The manifold RnjG with the Riemannian metric thus defined will be called a Euclidean cylinder. It is called a Euclidean torus if ~l' . • • , ~k form a basis of Rn, i.e., k = n. Every connected abelian Lie group with an invariant

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CURVATURE AND SPACE FORMS

211

Riemannian metric is a Euclidean cylinder, and ifit is, moreover, compact, then it is a Euclidean torus. In fact, the universal covering group of such a Lie group is isomorphic with a vector group Rn and its invariant Riemannian metric is given by (dx 1)2 + ... + (dx n )2 by a proper choice of basis in Rn. Our assertion is now clear. The following example shows that a torus can admit a flat affine connection which is not Riemannian. This was taken from Kuiper [1]. Example 4.2. The set G of transformations

(x,y) ~ (x

+ ny + m,y + n), n,m = 0, ± 1, ±2, ... ,

of R2 with coordinate sytem (x,y) forms a discrete subgroup of the group of affine transformations; it acts properly discontinuously on R2 and the quotient space R2jG is diffeonl0rphic with a torus. The flat affine connection of R2 induces a flat affine connection on R2jG. This flat affine connection of R2jG is not Riemannian. In fact, ifit is Riemannian, the induced Riemannian metric on the universal covering space R2 must be of the form a dx dx + 2b dx dy + c dy dy, where a, band c are constants, since the metric must be parallel. On the other hand, G is not a group of isometries of R2 with respect to this metric, thus proving our assertion. Let M be locally affine and choose a linear frame Uo E L (M) C A(M). LetM* be the holonomy bundle through Uo of the flat affine connection and M' the ho10nomy bundle through Uo of the corresponding flat linear connection. Then M* (resp. M') is a principal fibre bundle over M whose structure group is the affine holonomy group (u o) (resp. the linear holonomy group 'f(u o)). Since (u o) and 'Y (u o) are discrete, both M * and M' are covering manifolds of M. The homomorphism (3: A(M) ~L(M) defined in §3 of Chapter III maps M* onto M' (cf. Proposition 3.5 of Chapter III). Hence M* is a covering manifold of M'. 4.2. Let M be a manifold with a complete, flat ajJine connection. Let U o E L(M) c A(M). Let M* be the holonomy bundle through U o of the flat affine connection and M' the holonomy bundle through U o of the corresponding flat linear connection. Then THEOREM

212

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(1) M* is the universal covering space of M and, with respect to the flat affine connection induced on M*, it is isomorphic to the ordinary affine space An. (2) With respect to the flat affine connection induced on M', M' is a Euclidean cylinder, and the first homotopy group of M' is isomorphic to the kernel of the homomorphism
'.

dfP = -~i wj A fJ',

i _ dWj -

-~k

i

wk

k

A Wj,

i,j = 1, ... , n,

be the structure equations on L(M') of the flat affine connection of M'. Let N be the kernel of the homomorphism
i,j

=

1, ... , n.

Since a is horizontal, that is, a(M') is horizontal, we have w} = O. The structure equations imply that dO i = O. We assert that, for an arbitrarily chosen point 0 of M', there exists a unique abelian group structure on M' such that the point 0 is the identity element and that the forms Oi are invariant. Our assertion follows from the following three facts: (a) 01, ••• , form a basis for the space of covectors at every point of M'; (b) dO i = 0 for i = 1, ... , n; (c) Let X be a vector field on M' such that Oi(X) = ci (c i : constant) for i = 1, ... , n. Then X is complete in the sense that it generates a I-parameter group of global transformations of M'. The completeness of the connection implies (c) as follows. Let X* be the horizontal vector field on L(M') defined by Oi(X*) = ci , i = 1, ... ,n. Under the diffeomorphism a: M' ~ a(M'), X corresponds to X*. Since x* is complete (cf. Proposition 6.5 of

on

V.

CURVATURE AND SPACE FORMS

213

Chapter III), so is X. Note that (b) implies that the group is abelian. It is clear that 0101 + ... + is an invariant Riemannian metric on the abelian Lie group M'. As we have seen in Example 4.1, M' is a Euclidean cylinder.

onon

1. Let Rn/G, G = {~~f=l mt~i; mi integers}, be a Euclidean cylinder as defined in Example 4.1. Then the qtJine holonomy group of R n/G is a group of translations isomorphic with G. Proof of Lemma 1. We identify the tangent space Ta(Rn) at each point a ERn with R by the following correspondence: LEMMA

n

Ta(Rn)

3

~i=l Ai(a/axi)a~

(AI, ... ,An) ERn.

The linear parallel displacement from 0 ERn to a ERn sends (AI, , An) E To(Rn) into the vector with the same components (AI, , An) E Ta(Rn). The affine parallel displacement from 0 to a = (aI, ... , an) sends (AI, ... , An), considered as an element of the tangent affine space Ao(Rn), into (AI + aI, ... , An + an) E Aa(Rn). Let T* = xi, 0 ~ t ~ 1, be a line from 0 to ~f=l mi~i E G and let T = X t, 0 < t ~ 1, be the image of T * by the proj ection Rn ~ Rn/G. Then T is a closed curve in Rn/G. Let ~f=lmt~i =

(at, ... , an) ERn.

Then the affine parallel displacement along

(AI, ... , An)

~

(AI

T

yields the translation

+ aI, ... , An + an).

This completes the proof of Lemma 1. Being a covering space of M', M* is also a Euclidean cylinder. By Proposition 9.3 of Chapter II, the affine holonomy group of M* is trivial. By Lemma 1, M* must be the ordinary affine space An, proving (1). Since M' = M* /N, the first homotopy group of M' is isomorphic with N. This completes the proof of (2). Let Mil be a covering space of M. Since M* is the universal covering space of M, we can write Mil = M*/H, where H is a subgroup of
214

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(u o) ~ \}J'(u o). Since M' = M*/N, we may conclude that Mil is a covering space of M', thus proving (3). Suppose M' is a Euclidean torus. It follows that M is compact and the linear holonomy group (u o) of M is a finite group. This implies that the flat affine connection of M is Riemannian. In fact, we choose an inner product in Txo(M), X o = 7T(U O)' invariant by the linear holonomy group with reference point X o, and then we extend it to a Riemannian metric by parallel displacement. The fla t affine connection of M is the Riemannian connection with respect to the Riemannian metric thus constructed. Conversely, suppose M is a compact, connected, flat Rieman.. nian manifold. By virtue of (1), identifying M* with Rn, we may write M = Rn/G, where G is a discrete subgroup of the group of Euclidean motions acting on Rn. Let N be the subgroup of G consisting of pure translations. In view of (2) and (3) our problem is to prove that Rn/N is a Euclidean torus. We first prove several lemmas. LEMMA

2. Let A and B be unitary matrices of degree n such that A

commutes with ABA -1B-1. If the characteristic roots of B have positive real parts, then A commutes with B. Proof of Lemma 2. Since AABA-IB-l = ABA-IB-IA, we have ABA-IB-l = BA-IB-IA. Without loss of generality, we may assume that B is diagonal with diagonal elements bk = cos ~k + .J -1 sin ~k' k = 1, ... , n. Since A-I = t.J and B-1 = tB = B, we have Comparing the (i,i) -th entries, we have where A = (a;). Comparing the imaginary parts, we obtain for i = 1, ... , n. We may also assume that PI = P2 = ... = PPl < PPl +1 = ... = PPl +P2 < ... < Pn < PI + 7T. Since all the bk's have positive real parts, we have for i < PI

and j > PI"

V.

215

CURVATURE AND SPACE FORMS

Hence we must have a~J

=

a~~

and j > Pl.

for i ~ PI

= 0

Similarly, we have

aJ = a{ = 0 Continuing this argument we have

o

o

,

, B=

A=

o

o B I = bI]I'

B 2 = bp1 -d

]2, ••• ,

where A1} A 2, ... are unitary matrices of degree PI' P2' II, ]2' ... are the identity matrices of degree PI' P2' shows clearly that A and B commute. For any matrix A = (aj) of type (r, s) we set

, and This



+ B)

~


+



Every Euclidean motion of R n is given by x

---+

Ax

+ P,

X €

R n,

where A is an orthogonal matrix (called the rotation part of the motion) and P is an element of R n (called the translation part of the motion). This motion will be denoted by (A, p). LEMMA

3.

Given any two Euclidean motions (A,p) and (B, q), set (AI,PI) = (A,p)(B, q)(A,p)-I(B, q)-I.

216

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Let I be the identity matrix of degree n. cp(B - I) < b, then we have (1) cp (A 1 - I) < 2ab ; (2) cp (p 1) < b . cp (p) + a . cp (q) . Proof of Lemma 3.

Al - I

= =

If

cp (A - I) < a and

We have

ABA-IB-l - I = (AB - BA)A-IB-l ((A - 1)(B - I) - (B - I) (A - 1))A-IB-l.

Since A-IB-l is an orthogonal matrix, we have

cp(A I - I) < cp(A - I) . cp(B - I)

+ cp(B -

I) . cp(A - I) < 2ab.

By a simple calculation, we obtain

PI

=

A(1 - B)A- 1p

+ AB(1 -

A-l)B-l q.

By the same reasoning as above, we obtain

CP(Pl) < cp(1 - B) . cp(p) LEMMA

4.

+ cp(1 -

tA) . cp( q) < bcp(p)

+ acp(q).

With the same notation as in Lemma 3, set

(A k, Pk) = (A, p) (A k- 1,Pk-l) (A,p) -1(A k_1, Pk-l)-\ Then, for k = 1, 2, 3, ... , we have (1) cp(A k -I) <2 kakb; (2) CP(Pk) < (2 k - 1)ak - 1b· cp(p) + ak • cp(q).

k = 2, 3, ....

Proof of Lemma 4. A simple induction using Lemma 3 establishes the inequalities.

Let G be a discrete subgroup of the group of Euclidean motions ofRn. Let a < t and G(a) = {(A, p) € G; cp(A - I) < a}. LEMMA

5.

Then any two elements (A, p) and (B, q) of G(a) commute. Proof of Lemma 5. By Lemma 4, cp(A k - I) and CP(Pk) approach zero as k tends to infinity. Since G is discrete in A(n, R), there exists an integer k such that A k = I andpk = O. We show that the characteristic roots aI' ... , an of an orthogonal matrix A with cp(A - I) < t have positive real parts. If U is a unitary matrix such that UAU-l is diagonal, then cp(A - I) = cp( U(A - I) U-l) = cp( UAU-l - I) = (!a 1 - 11 2 + ... + Ian - 11 2 )!- < t,

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217

CURVATURE AND SPACE FORMS

which proves our assertion. By applying Lemma 2 to A k = AAk_1A-IA k_ 1, we see that A k- 1 = I. Continuing this argument, we obtain Al = I. Thus A and B commute. Hence,

PI P3

= =

(I - B)p - (I - A) q, P2 (A - I)P2 = (A - I)2Pl'

Pk = (A - I)Pk-l

=

=

(A - I)Pl'

(A - I)k-1Pl·

Since Pk = 0, we have

(A - I)k-1Pl =

o.

Changing the roles of (A, p) and (B, q) and noting that

(B, q) (A, p) (B, q) -l(A, p) -1 = (I, -PI)' we obtain for son1e integer m. Since A and B commute, there exists a unitary matrix U such that UAU-l and UBU-l are both diagonal. Set

o UAU-l

o , UBU-l =

=

o

,

o , Uq =

Then, from (A -- I)k-1Pl 0, we obtain

=

(A - I)k-l( (I - B)p - (I - A) q)

(a i - l)k-l{(l - bi)ri - (1 - ai)si} = 0, Similarly, from

i = 1, ... , n,

(B - I)m-1Pl = (B - I)m-l{(I - B)p - (I - A)q} = 0,

218

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

we obtain

(b i - l)m-I{(l - bi)ri - (1 - ai)si} = 0, Hence we have (1 -bi)ri - (1 -ai)si =0, In other words, we have

i = 1, ... , n.

i = l, ... ,n.

PI = (I - B)p - (I - A)q = 0, which completes the proof of Lemma 5. If (A,p) E G(a) and (B, q) E G, then (B, q)(A,p)(B, q)-I E G(a). Indeed, cp(BAB-I - I) = cp(B(A - I)B-I) = cp(A - I) < a. This shows that the group generated by G( a) is an invariant subgroup of G. By Lemma 5, it is moreover abelian if a < t. A subset V of R n is called a Euclidean subspace if there exist an element X o ERn and a vector subspace S of R n such that V = {x + xo; XES}. We say that a group G of Euclidean motions ofRn is irreducible if Rn is the only Euclidean subspace invariant by G.

6. If H is an abelian normal subgroup of an irreducible group G ofEuclidean motions ofRn, then H contains pure translations only. Proof of Lemma 6. Since H is abelian, we may assume, by applying an orthogonal change of basis of R n if necessary, that the elements (A, p) of H are simultaneously reduced to the following form: Al PI LEMMA

°

, p=

A=

, A.= t

( COS IX, •

-SIn

Ak

°

1n - 2k

Pk

(i

=

(Xi

sin IX,) cos

(Xi

'

1, ... , k)

P*

where 1n - 2k is the identity matrix of degree n - 2k, each Pi is a vector with 2 components and P* is a vector with n - 2k components. Moreover, for each i, there exists an element (A, P) of H such that Ai is different from the identity matrix 1 2 so that Ai - 1 2 is non-singular.

V.

219

CURVATURE AND SPACE FORMS

Our task is now to prove that k = 0, i.e., A = In for all (A, P) € H. Assuming k? I, we shall derive a contradiction. For each i, choose (A, p) € H such that Ai - 12 is non-singular and define a vector ti with 2 components by We shall show tha t for all (B, q)

€

H.

Since (A, p) and (B, q) commute, we have

Aiqi

+ Pi

= BiPi

+ qi

or Hence we have

(B i

-

12 )ti = (B i - 12 ) (Ai - 12 ) -IPi = (Ai - 12 ) -1(Bi - 12 )Pi =

(Ai - 12 )-I(A i - 12 )qi = qi'

thus proving our assertion. We define a vector t € Rn by

t1

t=

We have now

(1m t) (A, p) (1m t) -1 = (1m t) (A, p) (1m -t) =

(A, P - (A - 1n )t),

(A, P)

where

o •

• •

•

•

•

•

•

o P*

•

€

H,

220

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

By translating the origin of Rn to t, we may now assume that the elements (A, p) of H are of the form

Al

0

0

, p=

A=

o P*

o

Let V be the vector subspace of Rn consisting of all vectors whose first 2k components are zero. Then V is invariant by all elements (A, p) of H. We shall show that V is also invariant by G. First we observe that V is precisely the set of all vectors which are left fixed by all A where (A, p) E H. Let (C, r) E G. Since H is a normal subgroup of G, for each (A, p) E H, there exists an element (B, q) E H such that

(A,p)(C, r) = (C, rHE, q). If v E V, then ACv = CBv = Cv. Since Cv is left fixed by all A, it lies in V. Hence C is of the form

C= ( C'

o

0) C"

where C' and C" are of degree 2k and n - 2k, respectively. To prove that the first 2k components of r are zero, write

r=

For each i, let (A, p) be an element of f! such that Ai - 1 2 is nonsingular. Applying the equality (A, p) (C, r) = (C, rHE, q) to the zero vector ofRn and comparing the (2i - 1) -th and 2i-th components of the both sides, we have

V.

CURVATURE AND SPACE FORMS

Since (Ai - 12 ) is non-singular, we obtain r i element (G, r) of G is of the form

=

221

O. Thus .every

o

0)

GI Gr- ( 0 Gil' -

o r*

This shows that V is invariant by G, thus contradicting the irreducibility of G. This completes the proof of Lemma 6.

7. A group G of Euclidean motions of Rn is irreducible if RnjG is compact. Proof of Lemma 7. Assuming that G is not irreducible, let V be a proper Euclidean subspace of Rn which is invariant by G. Let Xo be any point of V and let L be a line through X o perpendicular to V. Let Xl' X 2 , • • • , X m , • • • be a sequence of points on L such that the distance between X o and X m is equal to m. Let G(x o) denote the orbit of G through X o• Since G(x o) is in V, the distance between G(x o) and X m is at least m and, hence, is equal to m. Therefore the distance between the images of X o and X m in RnjG by the projection Rn ~ RnjG is equal to m. This means that RnjG is not compact. We are now in position to complete the proof of (4). Let G and G(a) be as in Lemma 5 and assume a < t. Let H be the group generated by G(a); it is an abelian normal subgroup of G. Assume that RnjG is compact. Lemmas 6 and 7 imply that H contains nothing but pure translations. On the other hand, since G is discrete, GjH is finite by construction of G(a). Hence RnjH is also compact and hence is a Euclidean torus. Let N be the subgroup of G consisting of all pure translations of G. Since G(a) contains N, we have N = H. This proves that RnjN is a Euclidean torus. QED. LEMMA

Remark. (4) means that the linear holonomy group of a compact flat Riemannian manifold M = RnjG is isomorphic to GjN and hence is finite. Although (1), (2) and (3) are essentially in Auslander-Markus [1], we laid emphasis on affine holonomy groups. (4) was originally

222

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

proved by Bieberbach [1]. The proof given here was taken from Frobenius [1] and Zassenhaus [1]. Example 4.3. The linear holonomy group of a non-compact flat Riemannian manifold may not be finite. Indeed, fix an arbitrary irrational real number }L. For each integer m, we set

A(m)

cos Am7T

sin Am7T

0

-sin Am7T

cos Am7T

0 , p(m)

o o

1

m

o

o

Then we set G = {(A(m),p(m)); m = 0, ±l, ±2, ... }. It is easy to see that G is a discrete subgroup of the group of Euclidean motions of R3 and acts freely on R3. The linear holonomy group ofR3jG is isomorphic to the group {A(m); m = 0, ± 1, ±2, ...}. COROLLARY 4.3. A manifold M with a flat affine connection admits a Euclidean torus as a covering space if and only if M is a compact flat Riemannian manifold. Proof. Let M" be a Euclidean torus which is a covering space of M. By (3) of Theorem 4.2, M" is a covering space of M'. Thus, M' is a compact, Euclidean cylinder and hence is a Euclidean torus. By (3) of Theorem 4.2, M is a compact flat Riemannian manifold. The converse is contained in (4) of Theorem 4.2. QED.

Example 4.4. In Example 4.2, set M = R2jG. Let N be the subgroup of G consisting of translations: (x,Y)

----+

(x

+ m,Y),

m

=

0, ±l, ±2, ....

Then the covering space M' defined in Theorem 4.2 is given by R2j N in this case. Clearly, M' is an ordinary cylinder, that is, the direct product of a circle with a line. The determination of the 2-dimensional complete flat Riemannian manifolds is due to Killing [1,2], Klein [1,2] and H. Hopf [ll- We shall present here their results with an indication of the proof. There are four types of two-dimensional complete flat Riemannian manifolds other than the Euclidean plane. We give the fundamental group (the first homotopy group) for each type, describing its action on the universal covering space R2 in terms of the Cartesian coordinate system (x,y).

\

V.

CURVATURE AND SPACE FORMS

223

(1) Ordinary cylinder (orientable) (x,y) ---+ (x + n,y), n = 0, ± 1, ±2, .... (2) Ordinary torus (orientable) (x,y) ---+ (x + ma + n,y + mb)

m,n=0,±1,±2,... , a, b: real numbers, b

i=- 0.

(3) Mobius band with infinite width or twistedcylinder (non-'Orientable) (x,y) ---+ (x + n, (_1)ny), n =

0, ± 1, ±2, ....

(4) Klein bottle or twisted torus (non-orientable) (x,y) ---+ (x + n, (-l)ny + bm), n, m = 0, ± 1, ±2, ... , b: non-zero real number. Any two-dimensional complete flat Riemannian manifold Mis isometric, up to a constant factor, to one of the above four types of surfaces. The proof goes roughly as follows. By Theorem 4.2, the problem reduces to the determination of the discrete groups of motions acting freely on R2. Let G be such a discrete group. We first prove that every element of G which preserves the orientation of R2 is necessarily a translation. Set z = x + b. Then every orientation preserving motion of R2 is of the form z

---+

sz

+ w,

where s is a complex number of absolute value 1 and w is a complex number. If we iterate the transformation z ---+ sz + w r times, then we obtain the transformation

z

---+

sr z

+ (sr-l + sr-2 + ... + l)w.

We see easily that, if s i=- 1, then the point wj(l - s) is left fixed by the transformation z ---+ sz + w, in contradiction to the assumption that G acts freely on R2. Hence s = 1, which proves our assertion. Ifjis an element of G which reverses the orientation of R2, then j2 is an orientation preserving transformation and hence is a translation. We thus proved that every element of G is a transformation of the type

224

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Z -+ Z

+w

or

Z -+ Z

+ w,

where z is the complex conjugate of z. It is now easy to conclude that M must be one of the above four types of surfaces. The detail is left to the reader.

CHAPTER VI

Transformations

1. Affine mappings and affine transformations Let M and M' be manifolds provided with linear connections rand r' respectively. Throughout this section, we denote by P(M, G) and P'(M', G') the bundles of linear frames L(M) and L(M') of M and M', respectively, so that G = GL(n; R) and G' = GL(n'; R), where n = dim M and n' = dim M'. A differentiable mapping j: M ---+ M' of class Cl induces a continuous mapping j: T( M) ---+ T(M'), where T( M) and T( M') are the tangent bundles of M and M', respectively. We call j: M ---+ M' an affine mapping if the induced mapping j: T(M) ---+ T(M') maps every horizontal curve into a horizontal curve, that is, ifj maps each parallel vector field along each curve T of M into a parallel vector field along the curve j( T).

An affine mapping j: M ---+ M' maps every geodesic of M into a geodesic of M' (together with its affine parameter). Consequently, j commutes with the exponential mappings, that is, PROPOSITION

1.1.

jo exp X Proof.

=

exp oj(X),

This is obvious from the definition of an affine mapping. QED.

Proposition 1.1 implies that an affine mapping is necessarily of class Coo provided that the connections rand r' are of class Coo. We recall that a vector field X of M isj-related to a vector field X' of M' ifj(XaJ = X;(x) for all x E M (cf. §l of Chapter I).

1.2. Let j: M ---+ M' be an affine mapping. Let X, Y and Z be vector fields on M which are j-related to vector fields X', Y' and Z' on M', respectively. Then PROPOSITION

225

226

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(1) V x Y is J-related to V x' y', where V denotes covariant differentiation both in M and M' ;

(2) T(X, Y) is J-related to T'(X', Y'), where T and T' are the torsion tensor fields oj M and M', respectively; (3) R(X, Y)ZisJ-relatedto R'(X', Y')Z', where Rand R' are the curvature tensor fields of M and M', respectively. Proof. (1) Let x t be an integral curve of X such that x = X o and let Tb be the parallel displacement along this curve from x t to x = X o• Then (cf. §l of Chapter III) (V xY)x = lim ~ (TbYxt - Y x). t-+O t Set x; = J(x t ) and let T~t be the parallel displacement along this image curve from x; to x' = x~. Since J commutes with parallel displacement, we have

J((V xY)x) = lim ~ [j( TbYxJ -J(Yx)] t-+O t

= lim ~ t-+O t

(T~t Y~'t - y~,)

= (Vx'Y')x'.

(2) and (3) follow from (1) and Theorem 5.1 of Chapter III.

QED. A diffeomorphism J of M onto itself is called an affine transformation of M if it is an affine mapping. Any transformationJ of M induces in a natural manner an automorphism J of the bundle P(M, G); j maps a frame u = (Xl' ... ,Xn ) at x € M into the frame J(u) = (jX I , • • . ,JXn ) at J(x) € M. Since J is an automorphism of the bundle P, it leaves every fundamental vector field of P invariant. 1.3. (1) For every transformation J of M, the induced automorphism of the bundle P oj linear Jrames leaves the canonical Jorm e invariant. Conversely, every fibre-preserving transformation of P leaving e invariant is induced by a transformation of M. (2) IfJ is an affine transformation of M, then the induced automorphism J oj P leaves both the canonicalJorm e and the connectionJorm OJ invariant. Conversely, every fibre-preserving transformation oj P leaving both eand OJ invariant is induced by an affine transformation oj M. PROPOSITION

J

VI.

227

TRANSFORMATIONS

Proof. (1) Let X* E Tu(P) and set X = 7T(X*) so that X where x = 7T(U). Then (cf. §2 of Chapter III) ~

8(X*) = u- (X) 1

and

E

Tx(M),

~

8(fX*) = f(u) -1 (fX) ,

where the frames u andf(u) are considered as linear mappings of Rn onto Tx(M) and Tf(x)(M), respectively. It follows from the definition off that the following diagram is commutative: Rn

~j(U) Tx(M) 7 Tf(X)(M). ;/

Hence, u- 1 (X) = j(u)-I(fX), thus proving that 8 is invariant by]. Conversely, let F be a fibre-preserving transformation of P leaving 8 invariant. Let f be the transformation of the base M induced by F. We prove thatJ = F. We set J = 0 F. Then J is a fibre-preserving transformation of P leaving 8 invariant. Moreover, J induces the identity transformation on the base M. Therefore, we have

J-l

u- 1 (X) = 8(X*) = 8(JX*) = J(U)-I(X)

for X*

E

Tu(P).

This implies that J(u) = u, that is,f(u) = F(u). (2) Letfbe an affine transformation of M. The automorphism jof P maps the connection r into a connection, say,j(r), and the form j*w is the connection form ofJ(r) (cf. Proposition 6.1 of Chapter II). From the definition of an affine transformation, we see thatf maps, for each u E P, the horizontal subspace of Tu(P) onto the horizontal subspace of Tj(u)(P). This means thatj(r) = rand hencef*w = w. Conversely, let F be a fibre-preserving transformation of P leaving 8 and w invariant. By (1), there exists a transformation f • of M such that F = j Since maps every horizontal curve of P into a horizontal curve of P, the transformation f: T(M) ---+ T(M) maps every horizontal curve of T(M) into a horizontal curve of T(M). This means thatf: M ---+ M is an affine mapping, thus completing the proof. QED. "-

J

Remark. Assume that M is orientable. Then the bundle P consists of two principal fibre bundles, say P+(M, GO) and

228

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

P-(M, GO), where GO is the connected component of the identity of G = GL (n; R). Then any transformation F of P+ or Pleaving 0 invariant is fibre-preserving and hence is induced by a transformation f of the base M. In fact, every vertical vector X* of P+ or P- is mapped into a vertical vector by F since O(FX*) = O(X*) = O. Any curve in any fibre of P+ or P- is therefore mapped into a curve in a fibre by F. Since the fibres of P+ or pare connected, F is fibre-preserving. 1.4. Let r be a linear connection on M. For a transformation f of M, the following conditions are mutually equivalent: (1) f is an affine transformation of M; (2) f*w = w, where w is the connectionform of r andf is the transformation of P induced by f; (3) j leaves every standard horizontal vector field B(~) invariant; (4) f( V y Z) = V!y (fZ) for any vector fields Y and Z on M. Proof. (i) The equivalence of (1) and (2) is contained in Proposition 1.3. (ii) (2) ~ (3). By Proposition 1.3, we have PROPOSITION

~

~

~

~

Since

=

w(B(~))

o=

O(B(~))

~

= (f *0) (B(~)) =

O(f-lB(~)).

= 0, (2) implies ~

w(B(~))

=

(f*w)(B(~))

~

=

W(f-l·B(~)).

This means thatf-l . B(~) = B(~). (iii) (3) ~ (2). The horizontal subspace at u is given by the set of B(~)u. Hence (3) implies thatfmaps every horizontal subspace into a horizontal subspace. This means thatj(r) = r and hence ~

f*w = w. (iv) (1) ~ (4). This follows from Proposition 1.2. (y) (4) ~ (1). Let Z be a parallel vector field along a curve T = X t • Let Y be the vector field along T tangent to T, that is, Y Xt = Xt. We extend Yand Z to vector fields defined on M, which will be denoted by the same letters Y and Z respectively. (4) implies that fZ is parallel alongf( T). This means thatf is an affine transformation. QED. The set of affine transformations of M, denoted by 'l1 (M) or 'l1( r), forms a group. The set ofall fibre-preserving transformations

VI.

229

TRANSFORMATIONS

of P leaving fJ and w invariant, denoted by S21(P), forms a group which is canonically isomorphic with S21(M). We prove that S21(M) is a Lie group by establishing that S21(P) is a Lie group with respect to the compact-open topology in P.

1.5. Let r be a linear connection on a manifold M with a finite number of connected components. Then the group S21(M) of affine transformations of M is a Lie transformation group with respect to the compact-open topology in P. Proof. Let fJ = (fJ t ) and w = (wt) be the canonical form and the connection form on P. We set THEOREM

g(X*, y*) = ~i fJi(X*)fJi(Y*)

+ ~j,k wi(X*)wi(Y*), X*, y*

€

Tu(P).

Since the n2 + n I-forms fJi, wi, i, j, k = 1, ... , n, form a basis of the space of covectors at every point u of P (cf. Proposition 2.6 of Chapter III), g is a Riemannian metric on P which is invariant by S21(P) by Proposition 1.3. The group of isometries of P is a Lie transformation group of P with respect to the compact-open topology by Theorem 4.6 and Corollary 4.9 of Chapter I (cf. also Theorem 3.10 of Chapter IV). Since S21(P) is clearly a closed subgroup of the group of isometries of P, S21(P) is also a Lie transformation group of P. QED.

2. Infinitesilnal affine transformations Throughout this section, P(M, G) denotes the bundle of linear frames over a manifold M so that G = GL(n; R), where n = dimM. Every transformation cp of M induces a transformation of P in a natural manner. Correspondingly, every vector field X on M induces a vector field X on P in a natural manner. More precisely, we prove

2.1. For each vector field X on M, there exists a unique vector field g on P such that (1) g is invariant by R a for every a E G; (2) LgfJ = 0; (3) g is 7T-related to X, that is, 7T(XJ = X1T(u)for every u E P. PROPOSITION

230

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Conversely, given a vector field X on P satisfying (1) and (2), there exists a unique vector field X on M satisfying (3). We shall call X the natural lift of X. Proof. Given a vector field X on M and a point x E M, let CPt be a local I-parameter group of local transformations generated by X in a neighborhood U of x. For each t, CPt induces a transformation fPt of 7T- l ( U) onto 7T- l ( CPt( U)) in a natural manner. Thus we obtain a local I-parameter group of local transformations fPt: 7T- l ( U) ---+ P and hence the induced vector field on P, which will be denoted by X. Since fPt commutes with R a for every a E G, X satisfies (1) (cf. Corollary 1.8 of Chapter 1). Since fPt preserves the form e, X satisfies (2). Finally, 7T fPt = fPt 7T in1plies (3). To prove the uniqueness of X, let Xl be another vector field on Psatisfying (1), (2) and (3). Let i[Jt be a local I-parameter group of local transformations generated by Xl. Then i[Jt commutes with every R a , a E G, and preserves the canonical form e. By (1) of Proposition 1.3, it follows that i[Jt is induced by a local I-parameter group of local transformations "Pt of M. Because of (3), "Pt induces the vector field X on M. Thus "Pt = CPt and hence i[Jt = fPt, which implies that X = Xl. Conversely, let g be a vector field on P satisfying (1) and (2). For each x E M, choose a point u E P such that 7T(U) = x. We then set Xx = 7T(Xu). Since X satisfies (1), Xx is independent of the choice of u and thus we obtain a vector field X which satisfies (3). The uniqueness of X is evident. QED. 0

0

Let r be a linear connection on M. A vector field X on M is called an infinitesimal affine transformation of M if, for each x E M, a local I-parameter group of local transformations CPt of a neighborhood U of x into M preserves the connection r, more precisely, if each CPt: U ---+ M is an affine mapping, where U is provided with the affine connection r I U which is the restriction of r to U.

Let r be a linear connection on M. For a vector field X on M, the following conditions are mutually equivalent: (1) X is an infinitesimal affine transformation of M,. (2) LgOJ = 0, where OJ is the connection form of r and X is the natural lift of X,. (3) [X, B(~)] = 0 for every ~ ERn, where B(~) is the standard horizontal vector field corresponding to ~,. PROPOSITION

2.2.

VI.

231

TRANSFORMATIONS

(4) Lx V y - V y Lx = V[X,y]!or every vector field Y on M. Proof. Let f{Jt be a local I-parameter group of local transformations of M generated by X and let, for each t, iftt be a local transformation of P induced by f{Jt. (i) (1) ---+ (2). By Proposition 1.4, iftt preserves w. Hence we have (2). (ii) (2) ---+ (3). For every vector field X, we have (Proposition 2.1 ) 0

o=

0

X( O(B(~)))

(LxO) (B(~))

=

+ 0([X, B(~)])

which means that [X, B(~)] is vertical. If Lxw

o=

X(w(B(~))) = (Lxw) (B(~))

=

+ w([X, B(~)])

0([X, B(~)]),

=

0, then =

w([X, B(~)]),

which means that [X, B(~)] is horizontal. Hence, [X, B(~)] = o. (iii) (3) ---+ (1). If [X, B(~)] = 0, then iftt leaves B(;j invariant and thus maps the horizontal subspace at u into the horizontal subspace at iftt(U) , whenever iftt(u) is defined. Therefore t preserves the connection r and X is an infinitesimal affine transformation of M. (iv) (1) ---+ (4). By Proposition 1.4, we have f{Jt(V yZ) = VIpt Y(f{JtZ)

for any vector fields Yand Z on M.

From the definition of Lie differentiation given in §3 of Chapter I, we obtain Lx V y Z = lim ~ [V y Z - f{J t (V y Z) ] t~O t 0

=

VLxyZ

+ Vy

0

LxZ = V[X,y]Z

+ Vy

0

LxZ.

We thus verified the formula: Lx

0

VyK - V y

0

LxK

=

V[x,y]K,

when K is a vector field. If K is a function, the above formula is evidently true. By the lemma for Proposition 3.3 of Chapter I, the formula holds for any tensor field K. (v) (4) ---+ (1). We fix a point x E M. We set

V(t) = (f{Jt(VyZ))x

and

W(t) = (VlptY(f{JtZ))x'

232

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

For each t, both Vet) and Wet) are elements of Tx(M). In .view of Proposition 1.4, it is sufficient to prove that Vet) = W(t). As in (iv), we obtain

dV(t)/dt

=

cpt((L x

0

V y Z)cp;l{X))'

dW(t) /dt = CPt( (V[X,y]Z

+ Vy

0

L XZ)cp;l{X')'

From our assumption we obtain dV(t)/dt = dW(t)/dt. On the other hand, we have evidently V(O) = W(O). Hence, Vet) = Wet).

QED. Let oeM) be the set of infinitesimal affine transformations of M. Then oeM) forms a subalgebra of the Lie algebra X(M) of all vector fields on M. In fact, the correspondence X ----+ X defined in Proposition 2.1 is an isomorphism of the Lie algebra X(M) of vector fields on M into the Lie algebra X(P) of vector fields on P. Let o(P) be the set of vector fields X on P satisfying (1) and (2) of Proposition 2.1 and also (2) of Proposition 2.2. Since L[x,x'] = Lx Lx' - Lx' Lx (cf. Proposition 3.4 of Chapter I), o(P) forms a subalgebra of the Lie algebra X(P). It follows that oeM) is a subalgebra of X(M) isomorphic with o(P) under the correspondence X ----+ X defined in Proposition 2.1. 0

0

2.3. If M is a connected manifold with an affine connection r, the Lie algebra oeM) of infinitesimal affine transformations of M is of dimension at most n2 + n, where n = dim M. If dim o( M) = n2 + n, then r is flat, that is, both the torsion and the curvature of r vanish identically. Proof. To prove the first statement it is sufficient to show that o(P) is of dimension at most n 2 + n, since oeM) is isomorphic with o(P). Let u be an arbitrary point of P. The following lemma implies that the linear mapping f: o(P) ----+ T u (P) defined by X) = Xu is injective so that dim o(P) < dim Tu(P) = n 2 + n. THEOREM

fe

If an

element X of o(P) vanishes at some point of P, then it vanishes identically on P. Proof of Lemma. If Xu = 0, then Xua = 0 for every a E G as X is invariant by R a (cf. Proposition 2.1). Let F be the set of points x = 7T(U) E M such that Xu = O. Then F is closed in M. Since M is connected, it suffices to show that F is open. Assume Xu = O. Let bt be a local I-parameter group of local transformations LEMMA.

VI.

233

TRANSFORMATIONS

generated by a standard horizontal vector field B(~) in a neighborhood of u. Since [X, B(~)] = 0 by Proposition 2.2, X is invariant by bt and hence XbtU = O. In the definition of a normal coordinate system (cf. §8 of Chapter III), we saw that the points of the form 7T(b t u) cover a neighborhood of x = 7T(U) when ~ and t vary. This proves that F is open. To prove the second statement, we assume that dim a(M) = dim a(P) = n2 + n. Let u be an arbitrary point of P. Then the linear mapping f: f( X) = Xu, maps a(P) onto T u(P). In particular, given any element A E g, there exists a (unique) element X E a(P) such that Xu = A~, where A * denotes the fundamental vector field corresponding to A. Let B = B(~) and B' = B(~') be the standard horizontal vector fields corresponding to ~ and ~', respectively. Then / X u (8(B, B'))

=

A~(8(B,

B')).

We compute both sides of the equality separately. From L x 8 = Lx(dfJ + w A fJ) = 0 and from (3) of Proposition 2.2, we obtain X(8(B,B'))

(L x 8)(B,B') +8([X,B],B') +8(B, [X,B']) =0.

=

To compute the right hand side, we first observe that the exterior differentiation d applied to the first structure equation yields

o=

-0

fJ

A

+ w A 8 + d8.

Hence we have L A*8

=

(d

=

{,A*

{,A*

0

0

+ {,A*

d8

=

0

d)8

(,A*(O

A

fJ - w

A

8)

=

-w(A *) . 8

and (L A*8) (B, B') = -A . 8(B, B').

Therefore, A*(8(B, B')) = -A· 8(B, B')

+ 8([A *, B], B') + 8(B, [A *, B']).

If we take as A the identity matrix of 9 = gI(n; R), then, by Proposition 2.3 of Chapter III, we have [A *, B] = Band

[A *, B'] = B'.

234

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Thus we have

o=

X u (C3(B, B')) = A:(0(B, B')) = -0 u (B, B') + 0 u (B, B') 0 u (B, B') = 0 u (B, B'),

showing that the torsion form vanishes. Similarly, comparing the both sides of the equality: Xu (O(B, B'))

=

A: (O(B, B'))

and letting A equal the identity matrix of g that the curvature form vanishes identically.

=

gl(n; R), we see QED.

We now prove the following result due to Kobayashi [2].

2.4. Let r be a complete linear connection on M. Then every infinitesimal affine transformation X of M is complete, that is, generates a global I-parameter group of ajjine transformations of M. Proof. It suffices to show that every element g of a(P) is complete under the assumption that M is connected. Let Uo be an arbitrary point of P and let fPt: U ---* P, It I < b, be a local 1parameter group of local transformations generated by g (cf. Proposition 1.5 of Chapter I). We shall prove that fPt(u) is defined for every u € P and It I b. Then it follows that g is complete. By Proposition 6.5 of Chapter III, every standard horizontal vector field B(~) is complete since the connection is complete. Given any point u of P, there exist standard horizontal vector fields B(~ 1), ... , B(~k) and an element a € G such that THEOREM

u

(b:l a b;2 a • • • a brkuo)a, where each b; is the I-parameter group of transformations of P generated by B(~ i). In fact, the existence of normal coordinate neighborhoods (cf. Proposition 8.2 of Chapter III) and the 1T(U) can be joined connectedness of M imply that the point x to the point Xo = 1T( uo) by a finite succession of geodesics. By Proposition 6.3 of Chapter III, every geodesic is the projection of an integral curve of a certain standard horizontal vector field. This means that by taking suitable B(~l)' ... , B(~k)' we obtain a point v = btl a b;2 a • • • a b~uo which lies in the same fibre as u. Then u = va for a suitable a € G, thus proving our assertion. We then define fP t ( u) by fPt(u)

=

(btl a b~o ... a brk(fPt(uO)))a,

It I

<1.

VI.

235

TRANSFORMATIONS

The fact that ~t(u) is independent of the choice of bi, a _ 1 ... , br, k and that ~t is generated by X follows from (1) of Proposition 2.1 and (3) of Proposition 2.2; note that (3) of Proposition 2.2 implies that bs 0 ~t(u) = ~t 0 bs(u) whenever they are both defined.

QED.

In general, every element of the Lie algebra of the group 'lI(M) of affine transformations of M gives rise to an element of a(M) which is complete, and conversely. In other words, the Lie algebra of 'lI(M) can be identified with the subalgebra of a(M) consisting of complete vector fields. Theorem 2.4 means that if the connection is complete, then a( M) can be considered as the Lie algebra of 'lI( M). For any vector field X on M, the derivation A x Lx - Vx is induced by a tensor field of type (1, 1) because it is zero on the function algebra 3'(M) (cf. the proof of Proposition 3.3 of Chapter I). This fact may be derived also from the following PROPOSITION

2.5.

For any vector fields X and Y on M, we have

AxY = -VyX - T(X, Y), where T is the torsion. Proof. By Theorem 5.1 of Chapter III, we have AxY = LxY VxY [X, Y] = -VyX - T (X, Y).

(VyX

+ [X, Y] + T(X, Y)) QED.

We conclude this section by

2.6. (1) A vector field X on M is an infinitesimal ojJine transformation if and only if PROPOSITION

Vy(A x)

=

R(X, Y)

for every vector field Y on M.

(2) If both X and Yare infinitesimal affine transformations of M, then A[x,Y]

=

[A x' A y ]

;¥ R(X, Y),

where R denotes the curvature. Proof. (1) By Theorem 5.1 of Chapter III, we have R(X, Y) = [V x, Vy ] - V[X,Y] [L x, V y] - V[X, Y]

= -

[Lx - Ax, Vy ] [A x, V y ] .

V[X,Y]

236

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

By Proposition 2.2, X is an infinitesimal affine transformation if and only if R(X, Y) = - [A x, Vy] for every Y, that is, if and only if R(X, Y)Z = Vy(AxZ) - Ax(VyZ) = (Vy(Ax))Z for all Y and Z. (2) By Theorem 5.1 of Chapter III and Proposition 2.2, we have [Ax, A y ] - A[x,Y]

= =

[Lx - V x, L y - V y ] - (L[x,Y] - V[X,Y]) [Lx, L y ] - [V x, L y ] - [Lx, Vy]

+ [V x, Vy]

- L[x,Y]

+ V[X,Y]

=

R(X, Y).

QED.

3. Isometries and infinitesimal isometries Let M be a manifold with a Riemannian metric g and the corresponding Riemannian connection r. An isometry of M is a transformation of M which leaves the metric g invariant. We know from Proposition 2.5 of Chapter IV that an isometry of M is necessarily an affine transformation of M with respect to r. Consider the bundle 0 (M) of orthonormal frames over M which is a subbundle of the bundle L(M) of linear frames over M. We have

3.1. (1) A transformation f of M is an isometry if the induced transformation j of L(M) maps O(M) into

PROPOSITION

and only if itself; (2) A fibre-preserving transformation F of O(M) which leaves the canonical form 0 on 0 (M) invariant is induced by an isometry of M. Proof. (1) This follows from the fact that a transformation f of M is an isometry if and only if it maps each orthonormal frame at an arbitrary point x into an orthonormal frame atf(x). (2) Letfbe the transformation of the base M induced by F. We set J = ]-1 F. Then J is a fibre-preserving mapping of O(M) into L (M) which preserves O. Moreover, J induces the identity transformation on the base M, Therefore we have 0

u- 1 (X) = O(X*) = O(JX*) = J(U)-I(X),

X*

E

Tu(O(M)),

X = 7T(X*).

VI.

237

TRANSFORMATIONS ~

This implies that J(u) = u, that is, f(u) = F(u). By (1), f is an isometry of M. QED. A vector field X on M is called an infinitesimal isometry (or, a Killing vector field) if the local I-parameter group of local transformations generated by X in a neighborhood of each point of M consists of local isometries. An infinitesimal isometry is necessarily an infinitesimal affine transformation.

3.2. For a vector field X on a Riemannian manifold M, the following conditions are mutually equivalent: (1) X is an infinitesimal isometry; (2) The natural lift X of X to L(M) is tangent to OeM) at every point of O(M); (3) Lxg = 0, where g is the metric tensor field of M; (4) The tensorfield A x = Lx - Vx oftype (1, 1) is skew-symmetric with respect to g everywhere on M, that is, g(A xY, Z) = -g(A xZ, Y) for arbitrary vector fields Y and Z. Proof. (i) To prove the equivalence of (1) and (2), let CPt and fPt be the local I-parameter groups of local transformations generated by X and X respectively. If X is an infinitesimal isometry, then CPt are local isometries and hence fPt map O(M) into itself. Thus X is tangent to O(M) at every point of O(M). Conversely, if X is tangent to O(M) at every point of O(M), the integral curve of X through each point of O(M) is contained in O(M) and hence each fPt maps O(M) into itself. This means, by Proposition 3.1, that each CPt is a local isometry and hence X is an infinitesimal isometry. (ii) The equivalence of (1) and (3) follows from Corollary 3.7 of Chapter 1. (iii) Since V xg = 0 for any vector field X, L xg = 0 is equivalent to A xg = o. Since A x is a derivation of the algebra of tensor fields, we have PROPOSITION

Ax(g(Y, Z))

=

(Axg) (Y, Z)

+ g(AxY, Z) + g(Y, AxZ) for Y,Z

E

X(M).

Since Ax maps every function into zero, Ax(g(Y, Z)) = O. Hence Axg = 0 if and only if g(AxY, Z) + g(Y, AxZ) = 0 for all Yand Z, thus proving the equivalence of (3) and (4). QED.

238

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

The set of all infinitesimal isometries of M, denoted by i(M), forms a Lie algebra. In fact, if X and Yare infinitesimal isometries of M, then L[x,y]g = Lx Lyg - L y Lxg = 0 0

0

by Proposition 3.2. By the same proposition, [X, Y] is an infinitesimal isometry of lv/.

3.3. The Lie algebra i(M) of infinitesimal isometries of a connected Riemannian manifold M is ofdimension at most ·~n (n + 1), where n = dim M. if dim i(M) = In(n + 1), then M is a space of constant curvature. Proof. To prove the first assertion, it is sufficient to show that, for any point u of O(M), the linear mapping X ---+ Xu maps i(M) 1:1 into Tu(O(M)). By Proposition 3.2, Xu is certainly an element of Tu(O(M)). If Xu = 0, then the proof of Theorem 2.3 shows that X = O. We now prove the second assertion. Let X, X' be an orthonormal basis of a plane p in Tx(M) and let u be a point of O(M) such that 7T(U) = x. We set ~ = u-I(X), ~' = u-I(X'), B = B(~) and B' = B(~'), where B(~) and B(~') are the restrictions to O(M) of the standard horizontal vector fields corresponding to ~ and ~', respectively. From the definition of the curvature transformation given in §5 of Chapter III we see that the sectional curvature K(p) (cf. §2 of Chapter V) is given by K(P) = ((20(B u, B~))~', ~), THEOREM

where ( , ) denotes the natural inner product in Rn. To prove that K (p) is independent of p, let Y, Y' be an orthonormal basis of another plane q in T x ( M) and set 17 = u- l (Y) and 17' = u- l (Y'). Let a be an element of SO(n) such that a~ = 17 and a~' = 17'. By Proposition 2.2 of Chapter III, we have

O(B(17)u, B(17')u) = O(B(a~)u, B(a~')u) = o (R a-l(B ua ), Ra-l(B~a)) = ad (a) (O(B ua , B~a)) = a . O(B ua , B~a) . a-I. Hence the sectional curvature K(q) is given by

K(q)

((20(B(17)u,B(17')u)17', 17) = ((a· 20(B ua , B~a)· a-l)a~', = ((20(B ua , B~a)) e, ~). =

a~)

VI.

To

prove that K(p)

239

TRANSFORMATIONS

show that O(B ua , B~a) = O(B u , B~). Given any vertical vector x* E Tv(O(J..vl)) with 7T(V) = x, there exists an element X E i(M) such that Xv = x* if dim i(M) = !n(n + 1), since the mapping X ~X v maps i(M) onto Tv(O(M)). We have =

K(q), it is sufficient to

X(O(B, B')) = (LxO) (B, B')

+ O([X, B], B') + O(B, [X, B'])

= O.

This implies that O(B u , B~) = o (B ua , B~a) for rvery a E SO(n). We thus proved that K(p) depends only on the point x. We prove that K(p) does not depend even on x. Given any vector y* E Tu(O(M)), let Y be an element ofi(M) such that Y u = Y*. We have again Y(O(B, B')) = O. Hence, for fixed ~ and ~', the function ((20(B~ B'))~',~) is constant in a neighborhood of u. This means that K(P), considered as a function on M, is locally constant. Since it is continuous and M is connected, it must be constant on M. (If dim M > 3, the fact that K(p) is independent of x follows also from Theorem 2.2 of Chapter V. ) QED.

-

3.4. (1) For a Riemannian manifold M with a finite number of connected components, the group ~(M) of isometries of M is a Lie transformation group with respect to the compact-open topology in M,(2) The Lie algebra of ~(M) is naturally isomorphic with the Lie algebra of all complete infinitesimal isometries; (3) The isotropy subgroup ~AM) of ~(M) at an arbitrary point x is compact; (4) If M is complete, the Lie algebra of ~ (M) is naturally isomorphic with the Lie algebra i (M) of all infinitesimal isometries of M,(5) If M is compact, then the group ~(M) is compact. Proof. (1) As we indicated in the proof of Theorem 1.5, this follows from Theorem 4.6 and Corollary 4.9 of Chapter I and Theorem 3.10 of Chapter IV. (2) Every I-parameter subgroup of ~(M) induces an infinitesimal isometry X which is complete on M and, conversely, every complete infinitesimal isometry X generates a I-parameter subgroup of ~(M). (3) This follows from Corollary 4.8 of Chapter I. (4) This follows from (2) and Theorem 2.4. (5) This follows from Corollary 4.10 of Chapter 1. QED. THEOREM

Clearly,

~(M)

is a closed subgroup of '21(M). We shall see that,

240

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

in many instances, the identity component 3°(M) of 3(M) coincides with the identity component m:O(M) of m:(M). We first prove a result by Hano [1].

3.5. If M = M o X M 1 X . . . X M k is the de Rham decomposition of a complete, simply connected Riemannian manifold M, then X m:O(Mk ) , m:O(M) ~ m:O(Mo) X m:O(M 1 ) X THEOREM

3°(M) Proof.

~

3°(Mo)

X

3°(M1 )

X

X

3°(Mk ).

We need the following two lemmas.

Let Tx(M) = ~f=o T~i) be the canonical decomposition. (1) If cP E A(M), then cp( T~O») = T~~~) and for each i, 1 :::;: i < k, cp(T~i») = T<J<~)for some}, 1 <} < k; (2) If cP E AO(M), then cp( T~») = T~(~) for every i, 0 < i :::;: k. Proof of Lemma 1. Let T be any loop at x and set T' = cp( T) so that T' is a loop at cp(x). Ifwe denote by the same letter T and T' LEMMA

1.

the parallel displacements along cP

0

T(X)

=

T'

0

T

cp(X)

and

T'

respectively, then

for X

E

Tw(M).

It follows easily that cp( T~O») is invariant elementwise and every cp( T~»), 1 < i < k, is irreducible by the linear holonomy group 'F( cp(x)). Hence, cp( T~O») c T~~~) and, their dimensions being the same, cp( T~O») = T~~~). Thus we obtain the canonical decomposition Trp(x) = ~r=o cp( T~») which should coincide with the canonical decomposition T rp(x) = ~r =0 T~i(~) up to an order by (4) of Theorem 5.4 of Chapter IV. This means precisely the statement (1). Let CPt be a I-parameter subgroup of SllO(M) and let X be a non-zero element of T~i). Let T = X t = CPt(x). Since g(CPO(X) , X) = g(X, X) *- 0, we have g( CPt(X), T~X) *- 0 for It I < b for some b > 0, where T7 denotes the parallel displacement from X o to X t along T. This means that CPt( T~i») = T~i)(X) for It I < b' for . . numb . lact, r ' fcor T(i) some pOSItive er b' ; In I'f X 1, ••• , X r t.IS a b aSIs x, then g(CPt(Xj), T7 Xj) *- 0 for 1 <} < rand It I < b' for some positive number b' and hence CPt(Xj) E T~)(x) for It I < b', which implies CPt( T~i») = T~i!<X) for It I < b' beca~se of the linearity of CPt. This concludes the proof of the statement (2), since m:O(M) is generated by I-parameter subgroups. Lemma 1 is due to Nomizu [3].

VI.

241

TRANSFORMATIONS

2. Let CPi be an arbitrary transformation of M i for every i, o < i < k. Let cP be the transformation of M = M o X MIX ... X M k defined by LEMMA

cp(x)

=

(CPo (x o), CPI(X I ), ••• , CPk(X k))

for x

=

(x o, Xl"'" Xk) EM.

Then (I) cP is an afJine transformation of M if and only if every cP i is an affine transformation of Mi' (2) cP is an isometry of M if and only if every CPi is an isometry of Mi' The proof is trivial. The correspondence (CPo, CPI, , CPk) ~ cP defined in Lemma 2 maps 'lI(Mo) X 'lI(M I ) X X 'lI(Mk) isomorphically into \21(M). To complete the proof of Theorem 3.5, it suffices to show that, for every cP E 'lI0 ( M), there exist transformations CPi: M i ~ M i, 0 ::::: i < k, such that cp(X) = (CPo(xo), CPI(X I ), ... , CPk(X k)) for X = (x o, Xl' ... , Xk) EM. We prove that, ifPi: M ~ M i denotes the natural projection, then Piecp(x)) depends only on Xi = Pi(X), Given any point Y = (Yo, ,Yi-l, Xi,Yi+l, ... ,Yk), let, for each j = 0, 1, ... , i-I, i + 1, , k, 7 j = xj(t), 0 < t < 1, be a curve from x j tOYj in M j so that xj(O) = Xj and x j(l) = Yj' Let 7 = x(t), 0 < t < 1, be the curve from X to Y in M defined by

x(t)

=

(XO(t),xl(t), ... ,Xi-l(t),Xi,Xi+l(t), ... ,xk(t)), 0
For each t, the tangent vector x(t) to 7 at x(t) is in the distribution T(O) + ... + T( i-I) + T(i+l) + ... + T(k). By Lemma 1, cp(x(t)) lies also in the same distribution. Hence Piecp(x(t))) is independent of t (cf. Lemma 2 for Theorem 7.2 of Chapter II). In particular, Pi(CP(X)) = pz(cp(y)), thus proving our assertion. We then define a transformation CPi: M i ~ M i by Clearly, we have

cp(x) = (CPo(x o), CPI(X I), ... , CPk(X k)).

QED.

It is therefore important to study 'lI(M) when M is irreducible. The following result is due to Kobayashi [4].

242

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

3.6. If M is a complete, irreducible Riemannian manifold, then m(M) = 3(M) except when M is a I-dimensional Euclidean space. Proof. A transformation cp of a Riemannian manifold is said to be homothetic if there is a positive constant c such that g(cp(X), cp( Y)) = c2g(X, Y) for all X, Y E Tx(M) and x E M. Consider the Riemannian metric g* defined by g*(X, Y) = g(cp(X), cp(Y)). From the proof (B) of Theorem 2.2 of Chapter III, we see that the Riemannian connection defined by g* coincides with the one defined by g. This n1eans that every homothetic transformation of a Riemannian manifold M is an affine transformation of M. THEOREM

1. If M is an irreducible Riemannian manifold, then every affine transformation cp of M is homothetic. ProofofLemma 1. Since p is an affine transformation, the two Riemannian metrics g and g* (defined above) determine the same Riemannian connection, say r. Let '¥(x) be the linear holonomy group of r with reference point x. Since it is irreducible and leaves both g and g* invariant, there exists a positive constant Cx such that g*(X, Y) = c~ . g(X, Y) for all X, Y E Tx(M), that is, g; = c~ . gx (cf. Theorem 1 of Appendix 5). Since both g * and g are parallel tensor fields with respect to r, Cx is constant. LEMMA

LEMMA

2. If M is a complete Riemannian manifold which is not

locally Euclidean, then every homothetic transformation cp of M is an isometry. Proofof Lemma 2. Assume that cp is a non-isometric homothetic transformation of M. Considering the inverse transformation if necessary, we may assume that the constant c associated with cp is less than 1. Take an arbitrary point x of M. If the distance between x and cp(x) is less than (j, then the distance between cpm(x) and cpm+l(x) is less than cm(j. It follows that {cpm(x); m = 1, 2, ... } is a Cauchy sequence and hence converges to some point, say x*, since M is complete. It is easy to see that the point x * is left fixed by cpo Let U be a neighborhood of x* such that 0 is compact. Let K* be a positive number such that Ig(R(Y 1 , Y 2 )Y 2 , Y1)1 < K* for any unit vectors Y1 and Y 2 at y E U, where R denotes the curvature tensor field. Let Z E M and q any plane in Tz(M). Let X, Y be an orthonormal basis for q. Since cp is an affine

VI.

243

TRANSFORMATIONS

transformation, (3) of Proposition 1.2 implies that

R(cpmX, cpmy) (cpmy)

=

cpm(R(X, Y)Y).

Hence we have

g(R(cpmX, cpmy) (cpmy), cpmX) = g(cpm(R(X, Y)Y), cpmX) = c2mg(R(X, Y) Y, X) c2r.t,K(q). ::::;t

On the other hand, the distance between x* = cpm(x*) and cpm(z) approaches 0 as m tends to infinity. In other words, there exists an integer mo such that cpm(z) E U for every m > mo. Since the lengths of the vectors cpmX and cpm Yare equal to cm, we have for m > mo. Thus we obtain

c2mK* > IK(q) I

for m > mo.

Letting m tend to infinity, we have K(q) locally Euclidean.

=

O. This shows that Mis QED.

Let X be an infinitesimal affine transformation on a complete Riemannian manifold M. Using Theorems 3.5 and 3.6, we shall find a number of sufficient conditions for X to be an infinitesimal isometry. Assuming that M is connected, let AI be the universal covering manifold with the naturally induced Riemannian metric g = p*(g), where p: M ~ M is the natural projection. Let g be the vector field on M induced by X; g is p-related to X. Then g is an infinitesimal affine transformation of M. Clearly, g is an infinitesimal isometry of M if and only if X is an infinitesimal isometry of M. Let M = M o X M I X . . . X M k be the de Rham decomposition of the complete simply connected Riemannian manifold M. By Theorem 3.5, the Lie algebra oeM) is isomorphic with o(Mo) + o(MI ) -t- ... + o(Mk )· Let (Xo, X b • • • ,Xk ) be the element of o(Mo) + o(MI ) + ... + o(Mk ) corresponding to g E o(M). Since Xl' ... ,Xk are all infinitesimal isometries by Theorem 3.6, X is an infinitesimal isometry if and only if X o is.

3.7. If M is a connected, complete Riemannian manifold whose restricted linear holonomy group tyO(x) leaves no non-zero vector at x fixed, then Sl{O(M) = ~O(M). Proof. The linear holonomy group ofMis naturally isomorphic COROLLARY

244

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

with the restricted linear holonomy group 'f"O(x) of M (cf. Example 2.1 of Chapter IV). This means that M o reduces to a point and hence Xo 0 in the above notations. QED. 3.8. If X is an infinitesimal affine transformation of a complete Riemannian manifold and if the length of X is bounded, then X is an infinitesimal isometry. Proof. We may assume M to be connected. If the length of X is bounded on M, the length of Xo is also bounded on Mo. Let xl, ... , x'l' be the Euclidean coordinate system in M o and set COROLLARY

Applying the formula (L xo V y V y LxJ Z = V[Xo,y]Z (cf. fJ ojox and Z = ojoxY , we see that Proposition 2.2) to Y 0

0

02~IX

oxfJ oxY =

O.

This means that X o is of the form ~'l'

~IX=l

(~'l'

~(3=1

a{3 xfi IX

It is easy to see that length of Xo is bounded on M o ifand only if acp - 0 for a, f3 = 1, ... , r. Thus if Xo is of bounded length, then Xo is an infinitesimal isometry of Mo. QED. Corollary 3.8, obtained by Hano [1], implies the following result of Yano [1] which was originally proved by a completely different method. COROLLARY

3.9.

On a compact Riemannian manifold M, we have

mO(M) ~O(M). Proo£ On a compact manifold M, every vector field is of bounded length. By Corollary 3.8, every infinitesimal affine transformation X is an infinitesimal isometry. QED.

4. Holonomy and infinitesiaml isometries Let M be a differentiable manifold with a linear connection r. For an infinitesimal affine transformation X of M, we give a geometric interpretation of the tensor field A x = Lx - Vx introduced in §2.

VI.

245

TRANSFORMATIONS

Let x be an arbitrary point of M and let CPt be a local I-parameter group of affine transformations generated by X in a neighborhood of x. Let T be the orbit X t = CPt(x) of x. We denote by Tf the parallel displacement along the curve T from X s to x t. For each t, we consider a linear transformation Ct = Tb (cpt) * of Tx(M). 0

4.1. Ct is a local I-parameter group of linear transformations of Tx(M) : Ct+s = Ct Cs, and Ct = exp (-t(Az)x). ~ Proof. Since CPt maps the portion of T from X o to X s into the portion of T from Xt to Xt+s and since CPt is compatible with parallel displacement, we have PROPOSITION

0

CPt

8

0

TO

= Tit +8 CPt. 0

Hence

This proves the first assertion. Thus there is a linear endomorphism, say, A of Tx(M) such that Ct = exp tAo The second assertion says that A = - (A x)x. To prove this, we show that

First, consider the case where Xx =I=- O. Then x has a coordinate neighborhood with local coordinate system xl, ... , x n such that the curve T = X t is given by Xl = t, x 2 = . . . = x n = 0 for small values of t. We may therefore extend Y x to a vector field Y on Min such a way that CPt( Y x) = Y Xt for small values of t. Evidently, (LxY)x = O. We have

-(Ax)xYx = (\7 xY)x - (LxY)x = (\7 xY)x

= lim ~ (rbYXt - Y x) = lim ~ t~O t t~O t

(T~

0

CPtYx -

Y x)

Second, consider the case where Xx = O. In this case, CPt is a local I-parameter group of local transformations leaving x fixed and the

246

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

parallel displacement Tb reduces to the identity transformation of Tx(M). Thus (V xY)x = O. We have

-(Ax)xYx = (V xY)x - (LxY)x = -(LxY)x lim! (Yx - CPtYx) t~O t

=' -

=

lim! (CtYx - Y x). t~O t

This completes the proof of the second assertion.

QED.

Remark. Proposition 4.1 is indeed a special case of Proposition 11.2 of Chapter II and can be derived from it. 4.2. Let N(,¥(x)) and N (,¥O(x)) be the normalizors of the linear holonomy group '¥(x) and the restricted linear holonomy group ,¥O(x) in the group of linear transformations of Tx(M). Then Ct is contained in N(,¥(x)) as well as in N(,¥O(x)). Proof. Let CPt and be as before. For any loop fl at x, we set fl; = CPt(fl) so that fl; is a loop at X t = CPt(x). We denote by fl and fl; the parallel displacements along fl and fl;, respectively. Then CPt fl = fl; CPt. We have ' T 0. Cto fl Ct-l = TOt CPt fl CPt-1 T t0 = TOIflt' CPt CPt- lT tO = TOtflt t This shows that Ct fl Ct- 1 is an element of,¥(x). It is in ,¥O(x) if fl is in ,¥O(x). (Note that N('¥(x)) c N('¥O(x)) since ,¥O(x) is the identity component of ~('¥(x).) QED. PROPOSITION

T:

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

4.3. If X is an infinitesimal affine transformation ofM, then, at each point x E },,{, (Ax)x belongs to the normalizor N(g(x)) of the Lie algebra g(x) of '¥(x) in the Lie algebra of endomorphisms of Tro(M) . COROLLARY

We recall that N(g(x)) is by definition the set of linear endomorphisms A of Tx(M) such that [A, B] E g(x) for every BEg(X). If X is an infinitesimal isometry of a Riemannian manifold M, then A x is skew-symmetric (cf. Proposition 3.2) and, for each t, Ct is an orthogonal transformation of Tx(M). We have then

4.4. Let M be a Riemannian manifold and g(x) the Lie algebra of '¥(x). If X is an infinitesimal isometry of M, then, for each x E M, (A x)x is in the normalizor N (g(x)) of g(x) in the Lie algebra E(x) of skew-symmetric linear endomorphisms of Tx(M). THEOREM

VI.

247

TRANSFORMATIONS

The following theorem is due to Kostant [1].

4.5. If X is an infinitesimal isometry of a compact Riemannian manifold M, then,for each x E M, (Ax)x belongs to the Lie algebra g(x) of the linear holonomy group lY(x). Proof. In the Lie algebra E(x) of skew-symmetric endomorphisms of Tx(M), we introduce a positive definite inner product ( , ) by setting (A, B) = -trace (AB). THEOREM

Let B(x) be the orthogonal conlplement of g(x) in E(x) with respect this inner product. For the given infinitesimal isometry X of M, we set Ax=Sx+Bx, where Sx E g(x),

Bx

E

B(x),

x

E

M.

The tensor field B x of type (1, 1) is parallel. Proof of Lemma. Let 7' be an arbitrary curve from a point x to another point y. The parallel displacement 7' gives an isomorphism of E(x) onto E(y) which maps g(x) onto g(y). Since the inner products in E(x) and in E(y) are preserved by 7', 7' maps B(x) onto B(y). This means that, for any vector field Y on M, Vy(S x) is in g(x) whereas Vy(B x) is in B(x) at each point x E M. On the other hand, the formula Vy(A x) = R(X, Y) (cf. Proposition 2.6) implies that Vy (A x) belongs to 9 (x) at each x E llVf (cf. Theorem 9.1 of Chapter III). By comparing the 9 (x) -component and the B(x)-component of the equality Vy(A x ) = Vy(B x ) + Vy(Sx), we see that Vy(B x ) belongs to g(x) also. Hence Vy(B x) = 0, concluding the proof of the lemma. We shall show that B x = O. We set Y = B xX. By Green's theorem (cf. Appendix 6), we have (assuming that llVf is orientable for the moment) LEMMA.

fM div Y dv =

0

(dv: the volume element).

Since div Y is equal to the trace of the linear mapping V at each point x, we have (Lemma and Proposition 2.5)

---+

VvY

div Y = trace (V ---+ Vv(B xX)) = trace (V ---+ B x(VvX)) = -trace (B x A x) = -trace (B x B x) - trace (B x S x) = -trace (B xB x) > o.

248

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Thus

fM trace (B x B x) dv =

0,

which implies trace B xB x = 0 and hence B x = O. If M is not orientable, we lift X to an infinitesimal isometry X* of the twofold orientable covering space M* of M. Then B x* = 0 implies B x = O. QED. As an application of Theorem 4.5, we prove a result of H. C. Wang [1].

4.6. If M is a compact Riemannian manifold, then (1) Every parallel tensor field K on M is invariant by the identity component ~O(M) of the group of isometries of M; (2) At each point x, the linear isotropy group of ~O(M) is contained in the linear holonomy group 'Y(x). Proof. (1) Let X be an arbitrary infinitesimal isometry of M. By Proposition 4.1 and Theorem 4.5, the I-parameter group C t of linear transformations of Tx(M) is contained in 'Y(x). When C t is extended to a I-parameter group of automorphisms of the tensor algebra over Tx(M), it leaves K invariant. Thus cpt(Kx) = T~Kx = K Xt for every t, where CPt is the I-parameter group of isometries generated by X. Since ~O(M) is connected, it leaves K invariant. (2) Let cP be any element of ~O(M) such that cp(x) = x. Since ~O(M) is a compact connected Lie group, there exists a I-parameter subgroup CPt such that cP = CPt o for some to. In the proof of (1), we saw that C t (obtained from CPt) is in 'Y(x). On the other hand, since CPto(x) = x, T~ is also in 'Y(x). Hence CPto = T~o C to belongs to 'Y(x). QED. THEOREM

0

5. Ricci tensor and infinitesimal isometries Let M be a manifold with a linear connection r. The Ricci tensor field S is the covariant tensor field of degree 2 defined as follows:

S(X, Y)

=

trace of the map V

~

R( V, X) Y of Tx(M), 1

VI.

249

TRANSFORMATIONS

where X, Y, V € T x ( M). If M is a Riemannian manifold and if VI' ... , Vn is an orthonormal basis of Tx(M), then

SeX, Y) = ~i=I g(R( Vi' X) Y, Vi) = ~i=I R(Vi, Y, Vi' X), X,Y € Tx(M), where R in the last equation denotes the Riemannian curvature tensor (cf. §2 of Chapter V). Property (d) of the Riemannian curvature tensor (cf. §1 of Chapter V) implies SeX, Y) = S( Y, X), that is, S is symmetric.

5.1. If X is an infinitesimal affine transformation of a Riemannian manifold M, then PROPOSITION

div (AxY)

= -SeX,

Y) - trace (AXA y )

for every vector field Y on M. In particular, div (AxX) = -SeX, X) - trace (AxA x ). Proof. By Proposition 2.6, we have R( V, X) - Vv(A x ) for any vector field Von M. Hence

R(V,X)Y= -(Vv(Ax))Y= -Vv(AxY) = -Vv(AxY) - AXAyV.

=

-R(X, V)

+ Ax(VvY)

Our proposition follows from the fact that SeX, Y) is the trace of V ~ R(V, X) Yand that div (AxY) is the trace of V ~ Vv(AxY).

QED. 5.2. For an infinitesimal isometry X of a Riemannian manifold M, consider the function f = tg( X, X) on M. Then (1) Vf = g( V, A xX) for every tangent vector V; (2) V2f = g( V, Vv(A xX)) for every vector field V such that VvV = 0; (3) div (AxX) > 0 at any point wheref attains a relative minimum; div (A xX) < 0 at any point where f attains a relative maximum. Proof. Since g is parallel, we have PROPOSITION

Z(g(X, Y))

=

Vz(g(X, Y))

=

g(VzX, Y)

+ g(X,

VzY)

for arbitrary vector fields X, Yand Z on M. Applying this formula to the case where X = Y and Z = V, we obtain

Vf=g(VvX,X) = -g(AxV,X) = g(V, AxX)

250

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

by virtue of Proposition 2.5 and the skew-symmetry of Ax (cf. Proposition 3.2). This proves (1). If V is a vector field such that VvV = 0, then

V2f = V(g(V, AxX))

=

g(VvV, AxX)

+ g(V, Vv(AxX))

= g( V, Vv(AxX)), proving (2). To prove (3), let V l' . . . , Vn be an orthonormal basis for Tro(M). For each i, let 'T i = xi(t) be the geodesic with the initial condition (x, Vi) so that Vi = Xi (0). We extend each Vi to a vector field which coincides with xi(t) at xi(t) for small values of t. Then we have

d2f(x i (t))fdt 2

=

Vf f

=

g(V v z Vi' AxX)

+ g(Vi, Vv (AxX)) i

= g(Vi, Vvz(AxX)). Since div (AxX) is the trace of the linear mapping V we have

~

Vv(AxX),

Now (3) follows from the fact that, iff attains a relative minimum (resp. maximum) at x, then (Vrf)ro > 0 (resp. <0). QED. As an application of these two propositions, we prove the following result of Bochner [1].

5.3. Let M be a connected Riemannian manifold whose Ricci tensor field S is negative definite everywhere on M. If the length of an infinitesimal isometry X attains a relative maximum at some point of M, then X vanishes identically on M. Proof. Assume the length of X attains a relative maximum at x. By Proposition 5.2, we have div (A xX) < 0 at x. By Proposition 5.1, we obtain -SeX, X) - trace (AxA x ) ~ O. But SeX, X) < 0 by assumption,-and trace (AxA x ) < 0 since Ax is skew-symmetric. Thus we have SeX, X) = 0 and Ax = 0 at x. Since S is negative definite, X = 0 at x. Since the length of X attains a relative maximum at x, X vanishes in a neighborhood x. If u is any point of OeM) such that 7T(U) = x, then the natural lift X of X vanishes in a neighborhood of u. As we have seen in the proof of Theoren1 '1.3, X vanishes identically on M. QED. THEOREM

VI.

251

TRANSFORMATIONS

5.4. If M is a compact Riemannian manifold with negative definite Ricci tensor field, then the group ~ (M) of isometries of 11111 is finite. Proof. By Theorem 5.3, ~O(M) reduces to the identity. Since ~(M) is compact (cf. Theorem 3.4), it is finite. QED. COROLLARY

Remark. Corollary 5.4 can be derived from Proposition 5.1 by means of Green's theorem in the following way. We may assume that M is orientable; otherwise, we have only to consider the orientable twofold covering space of M. From Proposition 5.1 and Green's theorem, we obtain LCS(X, X)

+ trace (AxA x )] dv

= O.

Since S(X, X) < 0 and trace (AxA x ) ~ 0, we must have S(X, X) = 0 and trace (AxA x ) = 0 everywhere on M. Since Sis negative definite, we have X = 0 everywhere on M. This proof gives also 5.5. If M is a compact Riemannian manifold with vanishing Ricci tensor field, then every infinitesimal isometry of M is a parallel vector field. Proof. By Proposition 2.5, we have 0 = Ax V = -VvX for every vector field Von M. QED. COROLLARY

From Corollary 5.5, we obtain the following result of Lichnerowicz [1].

5.6. If a connected compact homogeneous Riemannian manifold M has zero Ricci tensor, then M is a Euclidean torus. Proof. By Theorem 5.1 of Chapter III and Corollary 5.5, we have [X, Y] = V xY - VyX = 0 COROLLARY

for any infinitesimal isometries X, Y. Thus ~O(M) is a compact abelian group. Since ~O( M) acts effectively on M, the isotropy subgroup of ~O(M) at every point M reduces to the identity element. As we have seen in Example 4.1 of Chapter V, M is a Euclidean torus. QED. As another application of Proposition 5.2, we prove

252

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

5.7. Let f{Jt be the I-parameter group of isometries generated by an infinitesimal isometry X of a Riemannian manifold M. If x is a critical point of the length function g(X, X)t, then the orbit f{Jt(x) is a geodesic. Proof. If x is a critical point of g(X, X)t, it is a critical point of the function f = tg(X, X) also. By (1) of Proposition 5.2, we have g( V, A xX) = 0 for every vector V at x. Hence A xX = 0 at x, that is, V xX = 0 at x. Since f{Jt( XaJ = Xlpt(X) by (1) of Proposition 1.2, we have V xX = 0 along the orbit f{Jt(x). This shows that the orbit f{Jt(x) is a geodesic. QED. PROPOSITION

6. Extension of local isomorphisms Let M be a real analytic manifold with an analytic linear connection r. The bundle L (M) of linear frames is an analytic manifold and the connection form co is analytic. The distribution Q which assigns the horizontal subspace Qu to each point u E L(M) is analytic in the sense that each point u has a neighborhood and a local basis for the distribution Q consisting of analytic vector fields. The same is true for the distribution on the tangent bundle T(M) which defines the notion of parallel displacement in the bundle T(M) (for the notion of horizontal subspaces in an associated fibre bundle, see §7 of Chapter II). The main object of this section is to prove the following theorem. 6.1. Let M be a connected, simply connected analytic manifold with an analytic linear connection. Let M' be an analytic manifold with a complete analytic linear connection. Then every affine mapping f u of a connected open subset U of Minto M' can be uniquely extended to an affine mapping f of Minto M'. The proof is preceded by several lemmas. THEOREM

1. Let f and g be analytic mappings of a connected analytic manifold M into an analytic manifold M' . Iff and g coincide on a non-empty open subset of M, then they coincide on M. Proof of Lemma 1. Let x be any point of M and let Xl, . . . , x n be an analytic local coordinate system in a neighborhood of x. Let yl, ... ,ym be an analytic local coordinate system in a neighborhood of the point f(x). The mapping f can be expressed by LEMMA

VI.

TRANSFORMATIONS

253

a set of analytic functions y~

= fi(xl, ... , xn ),

i = 1, ... , m.

These functions can be expanded at x into convergent power series of Xl, ••• , xn • Similarly for the mapping g. Let N be the set of points x E M such that f(x) = g(x) and that the power series expansions off and g at x coincide. Then N is clearly a closed subset of M. From the well known properties of power series, it follows that N is open in M. Since M is connected, N = M.

2.

Let Sand S' be analytic distributions on analytic manifolds M and M'. Let f be an analytic mapping of Minto N' such that LEMMA

(*)

f(S~J c

S;('l;)

or every point x of an open subset of M. If ltJ is connected, then (*) is satisfied at every point x of M. Proof of Lemma 2. Let N be the set of all points x E M such that (*) is satisfied in a neighborhood of x. Then N is clearly a non-empty open subset of M. Since M is connected, it suffices to show that N is closed. Let X k E Nand X k -»- X o' Letyl, ... ,ym be an analytic local coordinate system in a neighborhood V of f(x o)' Let Zl' , Zk be a local basis for the distribution S' in V. From oj oyl, , oj oym, choose m - h vector fields, say, Zk+l' ... , Zm such that Zb ... , Zk' Zk+1, ... , Zm are linearly independent at f(x o) and hence in a neighborhood V' of f(x o). Let U be a connected neighborhood of X o with an analytic local coordinate , x n such that f( U) c V' and that S has a local system xl, , X k consisting of analytic vector fields defined on U. basis X l' Since f is analytic, we have

f(Xi)x - I:.j=lf{(x) . Zj,

i = 1, ... , k,

wheref{(x) are analytic functions of xl, ... , xn • Since X k E Nand X k -»- X o, there exists a neighborhood U 1 of some X k such that U 1 c U and that (*) is satisfied at every point x of U 1 • In other words, f{{x) = 0 on U 1 for 1 i k and h + 1 s;: j < m. It follows thatf{ = 0 on U for the same i andj. This proves that (*) is satisfied at every point x of U. 3. Let M and M' be analytic manifolds with analytic linear connections and f an analytic mapping of Minto M'. If the LEMMA

254

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

restriction off to an open subset U of M is an affine mapping and if Mis connected, then f is an affine mapping of Minto M'. Proof of Lemma 3. Let F be the analytic mapping of the tangent bundle T(M) into T( llVf') induced by f. By assumption, F maps the horizontal subspace at each point of 7T- 1 ( U) into a horizontal subspace in T(M') (here, 7T denotes the projection of T( M) onto M) . Applying Lemma 2 to the mapping F, we see that fis an affine mapping of Minto M'. 4. Let M and M' be differentiable manifolds with linear connections and let f and g be affine mappings of Minto M'. If f( X) = g(X) for every X € Ta:(M) at some point x € M and if M is connected, then f and g coincide on M. Proof of Lemma 4. Let N be the set of all points x € M such thatf(X) = g(X) for X € Tx(M). Then N is clearly a non-empty closed subset of M. Since f and g commute with the exponential mappings (Proposition 1.1), x € N implies that a normal coordinate neighborhood of x is in N. Thus N is open. Since M is connected, we have N = M. We are now in position to prove Theorem 6.1. Under the assumptions in Theorem 6.1, let x(t), 0 :::;;: t :::;;: 1, be a curve in M such that x(O) € U. An analytic continuation offu along the curve x(t) is, by definition, a family of affine mappings ft, 0:::;;: t :::;;: 1, satisfying the following conditions: (1) For each t, ft is an affine mapping of a neighborhood U t of the point x(t) into M'; (2) For each t, there exists a positive number b such that if Is - tl < b, then xes) € U t andJ: coincides withft in a neighborhood of xes) ; (3) fo = fu· I t follows easily from Lemma 4 that an analytic continuation of f u along the curve x(t) is unique if it exists. We now show that it exists. Let to be the supremum of t 1 > 0 such that an analytic continuationft exists for 0 :::;;: t :::;;: ti . Let W be a convex neighborhood of the point x(to) as in Theorem 8.7 of Chapter III such that every point of W has a normal coordinate neighborhood containing W. Take t1 such that t l < to and that x(t 1 ) € W. Let V be a normal coordinate neighborhood of x(t I ) which contains W. Since there exists an analytic continuationft offu for 0 < t < t 1 , LEMMA

VI.

TRANSFORMATIONS

255

we have the affine mapping ft 1 of a neighborhood XUl) into M'. We extend ft 1 to an analytic mapping, say g, of V into M' as follows. Since the exponential mapping gives a diffeomorphism ofan open neighborhood V* of the origin in Tx,(t1)(M) onto V, each pointy E V determines a unique element X E V* c Tx,(t1)(M) such thaty = exp X. Set X' =h1(X) so that X' is a vector atftl(X(t1)). Since M' is complete, exp X' is well defined and we set g(y) = exp X'. The extension g of ft 1 thus defined commutes with the exponential mappings. Since the exponential mappings are analytic, g is also analytic. By Lemma 3, g is an affine mapping of V into M'. We can easily define the continuationft beyond t 1 by using this affine mapping g. VVe have thus proved the existence of an analytic continuationft along the whole curve x(t), 0 t l. To complete theproofofTheorem 6.1, let x be an arbitrarily fixed point of U. For each pointy of M, let xU), 0 < t ~ 1, be a curve from x to y. The affine mappingfu can be analytically continued along the curve x(t) and gives rise to an affine mapping g of a neighborhood ofy into M'. We show that g(y) is independent of the choice of a curve from x toy. For this, it is sufficient to observe that if x(t) is a closed curve, then the analytic continuation ft of f u along x( t) gives rise to the affine mapping fl which coincides with f u in a neighborhood x. Since M is simply connected, the curve x( t) is homotopic to zero and our assertion follows readily from the factorization lemma (cf. Appendix 7) and from the uniqueness of an analytic continuation we have already proved. I t follows that the given mapping f u can be extended to an affine mappingf of Minto M'. The uniqueness off follows from Lemma 4. QED.

6.2.

Let M and M' be connected and simply connected analytic manifolds with complete analytic linear connections. Then every affine isomorphism between connected open subsets of M and M' can be uniquely extended to an affine isomorphism between M and M'. COROLLARY

We have the corresponding results for analytic Riemannian manifolds. The Riemannian connection of an analytic Riemannian metric is analytic; this follows from Corollary 2.4 of Chapter

IV. 6.3. Let M and M' be analytic Riemannian manifolds. If M is connected and simply connected and if M' is complete, then every THEOREM

256

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

isometric immersion f u of a connected open subset U of Minto M' can be uniquely extended to an isometric immersion f of Minto M'. Proof. The proof is quite similar to that of Theorem 6.l. We indicate only the necessary changes. Lemma 1 can be used without any change. Lemma 2 was necessary only to derive Lemma 3. In the present case, we prove the following Lemma 3' directly.

3'. Let M and M' be analytic manifolds with analytic Riemannian metrics g and g', respectively, and letf be an analytic mapping of Minto M'. If the restriction off to an open subset U of M is an isometric immersion and if M is connected, then f is an isometric immersion of Minto M'. Proof of Lemma 3'. Compare g andf* (g'). Since they coincide on U, the argument similar to the one used in the proof of Lemma 1 shows that they coincide on the whole of J..vf. In Lemma 4, we replace "affine mappings" by "isometric immersions." Since an isometric immersion maps every geodesic into a geodesic and hence commutes with the exponential mappings, the proof of Lemma 4 is still valid. In the rest of the proof of Theorem 6.1, we replace "affine mapping" by "isometric immersion." Then the proof goes through without any other change. QED. LEMMA

Remark. Since an isometric immersion f: M ---+ M' is not necessarily an affine mapping, Theorem 6.3 does not follow from Theorem 6.1. If dim M = dim M', then every isometric immersion f: M ---+ M' is an affine mapping (cf. Proposition 2.6 of Chapter IV;. Hence the following corollary follows from Corollary 6.2 as well as from Theorem 6.3.

6.4. Let M and M' be connected and simply connected, complete analytic Riemannian manifolds. Then every isometry between connected open subsets of M and M' can be uniquely extended to an isometry between M and M'. COROLLARY

7. Equivalence problem Let M be a manifold with a linear connection. Let xl, ... , x n be a normal coordinate system at a point X o and let U be a neighborhood of X o given by Ixil < b, i = 1, ... , n. Let Uo be the linear

VI.

257

TRANSFORMATIONS

•

frame at the origin Xo given by (aj ax\ ... , aj ax n). We define a cross section a: U -+ L(M) as follows. If x is a point of U with coordinates (a\ ... , an), then a(x) is the frame obtained by the parallel displacement of U o along the geodesic given by Xi = ta i , o < t :::;;: 1. We call a the cross section adapted to the normal coordinate system Xl, . . . , x n. The first objective of this section is to prove the following theorem.

7.1. Let M and M' be manifolds with linear connections. Let U (resp. V) be a normal coordinate neighborhood of a point Xo E M (resp. Yo EM') with a normal coordinate system x\ ... , xn (resp. yl, ... ,yn) and let a: U -+ L(M) (resp. a': V -+ L(M')) be the cross section adapted to Xl, • . . , xn (resp. y\ . . . ,yn). A diffeomorphism f of U onto V is an affine isomorphism ifit satisfies thefollowing two conditions: (1) fmaps theframe a(x) into theframe a'(f(x))foreachpoint x E U; (2) f preserves the torsion and curvature tensor fields. Proof. Let () = (()i) and W = (w}) be the canonical form and the connection form on L(M) respectively. We set THEOREM

(1) Oi = a*()i = ~jA}dxj, -i -"" Bijk dx, k (2) Wj a *Wji -- £..Jk LEMMA

1.

i = 1, ... , n, .. Z,)

=

1,

,

n.

,0) wIth lail < b,

For any (a\ ... , an) ==I- (0,

we have

(3) ~i Aj(ta)a i

=

ai,

(4) ~k Bjk(ta)ak = 0,

0 :::;;: t < 1,

i

0

i,j = 1,

=::;;:

t :::;;: 1,

=

1,

, n, , n,

where ta stands for (tal, ... , tan). Proof of Lemma 1. For a fixed a = (a i ), consider the geodesic Xt given by Xi = ta i , 0 =::;;: t :::;;: 1, i = 1, ... ,n. Let U t = a(x t ), which is the horizontal lift of X t starting from Uo. Since the frames U t are parallel along x t , we have

Oi(X t ) = ()i(U t ) -- ai . On the other hand, we have

xt

= ~i ai (ajax i )

and

Oi(X t ) = ~i Aj(ta)a 5•

This proves (3). Similarly, (4) follows from the fact that w(u t )

=

O.

258

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

We set (cf. §7 of Chapter III)

(5) Eli = a*Eli = ~j,k l1j~ OJ

(6)

Q; =

LEMMA

A

(1j~ = a* 1j~),

Ok

a*Oj = ~k,l tR}kZOk A Ol

(RjkZ = a*RAz).

2. For an arbitrarily fixed (a!, ... , an), we set Aj(t) = tAj(ta), 1\

•

iJjk(t) = tB}k(ta),

_.

""'.

Tjk(t) = Tjk(ta),

_.

RJkZ(t) = RAz(ta).

Then the functions Aj(t) and iJjk (t) satisfy the following system ordinary linear differential equations:

of

""i ... i l " i ""m (7) dAj(t)/dt = bji + ~l Blj(t)a + ~l,m TZm(t)Aj (t)a l , (8) diJjk(t)/dt = ~Z,m RjkZ(t)Ar(t)a l.

with the initial conditions: (9) Aj(O) = 0, iJjk(O) =

o.

Proof of Lemma 2. We consider the open set Q of R n+l defined by Q = {(t, a!, ... , an); Itail < b for i = 1, ... , n}. Let p be the mapping of Q into U defined by

p(t, a!, ... ,an) = (tal, ... , tan). We set

ei =

p*Oi,

wj = p*wj,

Eli = p*Eli,

OJ = p*Qj.

From Lemma 1, we obtain

(10) (ji = ~ j tAj(ta) da j + ai dt, (11) wj = ~k tBjk(ta) da k. From (5) and (6), we obtain

(12) Eli

=

~j,lC lTA(ta) OJ

(13) OJ

=

~k,ztRjkZ(ta) 'Ok A 'O l.

A

Ok,

From (10) and (11), we obtain

(14)

iOi

-.

(15) dw]

= =

-~i [:t (tAj(ta)) -~k

-

~j ]

[a. at (tBjk(ta)) ] da k

da i A

dt

A

dt

+ ... ,

+ ... ,

where the dots denote the terms not involving dt.

VI.

259

TRANSFORMATIONS

From (10), (11), (12) and (13), we obtain

+ 8i = -I;j [I;z tBI/ta)a Z+ I;z Tz~(ta) (tAj(ta) )azJ da i A dt + ... , (17) -I;k wl A wf + OJ -I;k [I;z,m Rjzm(ta) (tAr(ta)) aZJ da k A dt + ... , (16)

I;i

W} A Oi

where the dots denote the terms not involving dt. Now (7) follows from (14), (16) and the first structure equation. Similarly, (8) follows from (15), (17) and the second structure equation. Finally, (9) is obvious from the definition of A}(t) and Bjk(t). This proves Lemma 2. From Lemma 2 and from the uniqueness theorem on systems of ordinary linear differential equations (cf. Appendix 1), it follows that the functions Aj(t) and Bjk(t) are uniquely determined by Tjk(t) and RjkZ(t). On the other hand, the functions Tjk(t) and Rjkz(t) are uniquely determined by the torsion tensor fields T and the curvature tensor fields R and also by the cross section (for each fixed (a l , . . . , an)). From (1) we see that the connection QED. form w is uniquely determined by T, R and G. A

A

A.

A.

A

A

A.

A

In the case of a real analytic linear connection, the torsion and curvature tensor fields and their successive covariant derivatives at a point determine the connection uniquely. More precisely, we have

7.2. Let M and M' be analytic manifolds with analytic linear connections. Let T, R and V (resp. T', R' and V') be the torsion, the curvature and the covariant differentiation of M (resp. M'). If a linear isomorphism F: Ta:o(M) -+ T'Yo(M') maps the tensors (vmT)ooo and (vmR)ooo into the tensors (V'mT')'Yo and (V'mR')'Yo' respectively, for m = 0, 1, 2, . . . , then there is an affine isomorphism f of a neighborhood U of X o onto a neighborhood V ofYo such that f(x o) = Yo and that the dijferential off at X o is F. Proof. Let Xl, . • . , xn , Ixil < 0, be a normal coordinate system in a neighborhood U of X o' Let yt, ... ,yn, Iyil < 0, be a normal coordinate system in a neighborhood V of Yo such that (01 oyi) 'Yo = F( ( 01 ox i)000)' i = 1, ... ~ n; such a normal coordinate THEOREM

260

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

system exists and is unique. Letfbe the analytic homeomorphism of U onto V defined by

i

=

1, ... , n.

Clearly the differential off at X o coincides with F. We shall show thatf is an affine isomorphism of U onto V. We use the same notation as in the proof of Theorem 7.1. It suffices to prove the following five statements. If the normal coordinate system xl, ... , x n is fixed, then (i) The tensors CvmT)x 0, m = 0, 1, 2, ... , determine the functions T}k(t), 0
A

A.

A

_

Let U t, 0 < t < 1, be a horizontal lift of a curve X t, o < t < 1, to L(M). Let T~ be the tensor space of type (r, s) over Rn. Given a tensor field K of type (r, s) along X t, let K be the T~-valued function defined along U t by LEMMA

1.

K (~t)

=

ut- 1 (Kx )

,

0 :::: t :::: 1,

where U t is considered as a linear mapping of T~ onto the tensor space T~(xt) at X t of type (r, s). Then we have dK(u t) dt

=

u- 1 (V K) t

0 -.:;: t :::: 1.

Xt'

Proof of Lemma 1. This is a special case of Proposition 1.3 of ,., Chapter III. The tensor field K and the function K here correspond to the cross section ffJ and the function f there. Although ffJ in Proposition 1.3 of Chapter III is defined on the whole of M, the proof goes through when ffJ is defined on a curve in M (cf. the lemma for Proposition 1.1 of Chapter III). To prove (i), we apply Lemma 1 to the torsion T, the geodesic X t given by Xi = ta i , i = 1, ... , n, and the horizontal lift U t of X t

VI.

261

TRANSFORMATIONS

with U o = ((ojox 1 )xo' ... ' (ojoxn)xo). Then Lemma 1 (applied m times) implies that, for each t, ut- 1 ((V i )mT) is the element of the t tensor space T~ with components dmTJk(t) jdt m. In particular, setting t = 0, we see that, once the coordinate system xl, ... ,xn and (a\ ... ,an) are fixed, (dmTJkjdtm)t=o, m = 0, 1,2, ... , are all determined by (vmT)xo. (Actually, it is not hard to see that A.

A.

all . . . alm , (dmT~Jk jdt m)t=o = ~ ll"."lm T~... Jk,ll"",lm (x) 0 where TJk;ll;'" ;lm are the components of Vm T with respect to xl, ... ,xn.) Since each Tjk(t) is an analytic function of t, it is determined by (vmT)xo' m = 0, 1, 2, .... This proves (i). The proof of (ii) is similar. Lemma 2 for Theorem 7.1 implies that the functions T]k( t) and Rjkl(t) determine the functions AJ(t) and Bjk(t). Now (iii) follows from the formula (1) and (2) in the proof of Theorem 7.1. (iv) follows from the following lemma. A

A.

A

A

LEMMA

A

2. Let a and a' be two cross sections of L(M) over an open

subset U of M. If a*O = a' *0 on U, then a = a'. Proof of Lemma 2. For each X E Tx(M), where x have (a*O) (X) = O( aX) = a(x) -1(7T( aX)) = a(x) -IX,

E

U, we

where a(x) E L(M) is considered as a linear isomorphism of Rn onto Tx(M). Using the same equation for a', we obtain a(x)-IX

=

a'(x)-IX.

Since this holds for every X in Tx(M), we obtain a(x) : : : :; a' (x). Finally, (v) is evident from the definition of wj. QED. 7.3. In Theorem 7.2, if M and M' are, moreover, connected, simply connected analytic manifolds with complete analytic linear connections, then there exists a unique affine isomorphism f of M onto M' whose differential at X o coincides with F. Proof. This is an immediate consequence of Corollary 6.2 and Theorem 7.2. 0 ED. COROLLARY

--

7.4. Let M and M' be differentiable manifolds with linear connections. Let T, R and V (resp. T', R' and V') be the torsion, the curvature and the covariant differentiation of M (resp. M'). Assume THEOREM

262

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

VT = 0, VR = 0, V'T' = 0 and V'R' = O. If F is a linear isomorphism of Txo(M) onto TYo(M') and maps the tensors To;o and R xo at X o into the tensors T~o and R~o at Yo respectively, then there is an affine isomorphism f of a neighborhood U of X o onto a neighborhood V ofYo such that f(x o) = Yo and that the differential off at X o coincides with F. Proof. We follow the notation and the argument in the proof of Theorem 7.2. By Lemma I in the proof of Theorem 7.2, the functions Tlk(t) and R.jkZ(t) are constant functions and hence are determined by T xo and R xo (and the coordinate system xl, ... , xn ). Our theorem now follows from (iii), (iv) and (v) in the proof of Theorem 7.2. QED. 7.5. Let M be a differentiable manifold with a linear connection such that V T = 0 and V R = o. Then, for any two points x andy of M, there exists an affine isomorphism of a neighborhood of x onto a neighborhood ofy. Proof. Let T be an arbitrary curve from x to y. Since V T = 0 and VR = 0, the parallel displacement T: Tx(M) ---+ Ty(M) maps the tensors T x and Rx at x into the tensors T y and R y at y. By Theorem 7.4, there exists a local affine isomorphism f such that f(x) = y and that the differential off at x coincides with T. QED. COROLLARY

Let M be a manifold with a linear connection r. The connection r is said to be invariant by parallelism if, for arbitrary points x andy of M and for an arbitrary curve T from x to y, there exists a (unique) local affine isomorphism j' such that f(x) = y and that the differential off at x coincides with the parallel displacement T: Tx(M) ---+ Ty(M). In the proof of Corollary 7.5, we saw that if V T = 0 and VR = 0, then the connection is invariant by parallelism. The converse is also true. Namely, we have 7.6. A linear connection is invariant by parallelism if and only ijVT = 0 and VR = o. Proof. Assuming that the connection is invariant by parallelism, let T be an arbitrary curve from x to y. Letfbe a local affine isomorphism such thatf(x) = y and that the differential off at x coincides with the parallel displacement T. Then f maps T x and Rx into T y and R y respectively. Hence the parallel displacement T maps T x and R x into T y and R y, respectively. This means that T and R are parallel tensor fields. QED. COROLLARY

VI.

263

TRANSFORMATIONS

7.7. Let M be a differentiable manifold with a linear connection such that VT = 0 and VR = O. With respect to the atlas consisting of normal coordinate systems, M is an analytic manifold and the connection is analytic. Proof. Let xl, ... , xn be a normal coordinate system in an open set U. We introduce a coordinate system (Xi, XDi,j,k=l, ... ,n in 7T- 1 ( U) c L(M) in a natural way as in §7 of Chapter III (cf. Example 5.2 of Chapter I). If we denote by (Ut) the inverse matrix of (X1)' then the canonical form and the connection form can be expressed as follows (cf. Propositions 7.1 and 7.2 of Chapter THEOREM

III) : (18) fJ i

=

~j

(19) wj

=

~k U1( dXff

Uj dx j ,

i

=

1, . . . , n;

+ ~l,m r~lXJdxm),

i,j

=

1, ... , n.

The forms fJi are analytic with respect to (Xi, Xi). We show that the forms wj are also analytic with respect to (xi, X{). Clearly it is sufficient to show that the components rjk of the connection are analytic in xl, ... , xn • We use the same notation as in the proof of Theorem 7.1. Since the functions Tjk( t) and Rjkl( t) are constants which do not depend on (a\ ... , an) by virtue of the assumption that V T = 0 and VR = 0, Lemma 2 in the proof of Theorem 7.1 implies (cf. Appendix 1) that the functions Aj(t) and BJkl(t) are analytic in t and depend analytically on (a\ ... , an). Hence the functions Aj and Bjk are analytic in xl, ... , xn • From (1) in the proof of Theorem 7.1, we see that the cross section (1: U -+ L(M) is given by A

A

(20) Uj = Aj,

A

A

i,j = 1, ... , n.

Let (CJ) be the inverse matrix of (AJ). From (19) and (20), we obtain

(21) (1*wJ = wJ = ~kAHdCff

+ ~l,m r~lCJdxm).

By comparing (21) with (2) in the proof of Theorem 7.1, we obtain (22) Bjm = ~k A1(aC:/axm + ~z r~lCJ). Transforming (22) we obtain (23) r~l= ~j (~i CfBJm - aCff/axm)A{,

264

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

which shows that the components rJk are analytic functions of I n X, ••• ,x. Since the n 2 + n I-forms ei and w~ are analytic with respect to (Xi, X~) and define an absolute parallelism (Proposition 2.5 of Chapter III), the following lemma implies that L(M) is an analytic manifold with respect to the atlas consisting of the coordinate system (Xi, Xj) induced from the normal coordinate systems (xl, ... , x n ) of M. LEMMA. Let wI, ... , wm be l-forms defining an absolute parallelism on a manifold P of dimension m. Let u\ ... , um (resp. vI, ... , vm) be a local coordinate system valid in an open set U (resp. V). If the forms wI, ... , wm are analytic with respect to both u\ ... , um and vI, ... , vm, then the functions i

=

1, ... , m,

which define the coordinate change are analytic. Proof of Lemma. We write Wi =

~.1

ai(u) du j = J

~ 1. b~(v) J

dv j ,

where the functions aj(u) (resp. bj(v)) are analytic in u\ ... , um (resp. v\ ... , vm). Let (cj(v)) be the inverse matrix of (bJ(v)). Then the system of functions Vi = fi (u\ ... , um), i = 1, ... , m, is a solution of the following system of linear partial differential equations:

avijau j = ~k c1(v)aT(u),

Z,)

= 1, ... , n.

Since the functions 4 (v) and aT (u) are analytic in VI, ••• , vm and u\ ... , um respectively, the functionsfi(u\ ... , um) are analytic in uI , • • • , um (cf. Appendix 1). This proves the lemma. Let x\ ... , xn and y\ ... ,yn be two normal coordinate systems in M. Let (Xi, Xl) and (yi, Yt) be the local coordinate systems in L (M) induced by these normal coordinate systems. By the lemma just proved, yI, ... ,yn are analytic functions of Xi and X{ Since y\ ... ,yn are clearly 'independent of xt, they are analytic functions of xl, ... , x n • This proves the first assertion of Theorem 7.7. Since we have already proved that the forms wj are analytic with respect to (xi, xt), the connection is analytic. QED. As an application of Theorem 7.7 we have

VI.

TRANSFORMATIONS

265

7.8. In Theorem 7.4, if M and M' are, moreover, connected, simply connected and complete then there exists a unique affine isomorphism f of M onto M' such that f(x o) = Yo and that the dijferential off at X o coincides with F. Proof. This is an immediate consequence of Corollary 6.2, Theorem 7.4, and Theorem 7.7. QED. THEOREM

7.9. Let M be a connected, simply connected manifold with a complete linear connection such that V'T = 0 and V'R = o. If F is a linear isomorphism of T xo (M) onto T yo (M) which maps the tensors T xo and R x0 into T y 0 and R y 0 , respectively, then there is a unique azifine transformation f of M such that f( xo) = Yo and that the dijferential of fat X o is F. In particular, the group S21(M) of affine transformations of M is transitive on M. Proof. The first assertion is clear. The second assertion follows from Corollary 7.5 and Theorem 7.8. QED. COROLLARY

In §3 of Chapter V, we constructed, for each real number k, a connected, simply connected complete Riemannian manifold of constant curvature k. Any connected, simply connected complete space of constant curvature k is isometric to the model we constructed. Namely, we have

7.10. Any two connected, simply connected complete Riemannian manifolds of constant curvature k are isometric to each other. Proof. By Corollary 2.3 of Chapter V, for a space of constant curvature, we have V'R = O. Our assertion now follows from Theorem 7.8 and from the fact that, if both M and M' have the same sectional curvature k, then any linear isomorphism F: Txo(M) ~ TYo(M') mapping the metric tensor gx o at Xo into the metric tensor g~o at Yo necessarily maps the curvature tensor R x o at Xo into the curvature tensor R y' 0 at Yo (cf. Proposition 1.2 of Chapter V). QED. THEOREM

APPENDIX 1

Ordinal:r linear differential equations

The purpose of this appendix is to state the fundamental theorem on ordinary linear differential equations in the form needed in the text. The proof will be found in various text books on differential equations. For the sake of simplicity, we use the following abbreviated notation:

y = (y\

,yn),

'Yj = ('Yj\

cp

,cpn),

S =

=

(cp\

(s\

, 'Yjn), f= (f\ ,sm),

x

=

(x\

,jn), ,xm).

Then we have

Letf(t,y, s) be afamily of nfunctions defined in It I < £5 and (y, s) € D, where D is an open set in Rn+m. Iff(t,y, s) is continuous in t and dijJerentiable of class Cl in y, then there exists a unique family cp(t, 'Yj, s) of n functions defined in It I < £5' and ('Yj, s) € D', where o < b' < band D' is an open subset of D, such that (1) cp(t, 'Yj, s) is differentiable of class Cl in t and 'Yj; (2) acp( t, 'Yj, s) I at = f( t, cp( t, 'Yj, s), s) ; (3) cp(O, 'Yj, s) = 'Yj. Iff(t,y, s) is differentiable of class CP, 0 ::;;: p < w, in t and of class cq, I < q ::;;: w, in y and s, then cp( t, 'Yj, s) is differentiable of class CP+l in t and of class Cq in 'Yj and s. THEOREM.

Consider the system of differential equations:

dyjdt = f(t,y, s) which depend on the parameters s. Theny = cp(t, 'Yj, s) is called the solution satisfying the initial condition:

y = 'Yj

when t = O. 267

268

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Consider now a system of partial differential equations:

ayijax J =fj(x,y),

i = 1, ... , n;j = 1, ... , m.

It follows from the theorem that if the functions fJ(x,y) are r w, then every solution differentiable of class C r , 0 y = 'ljJ(x) is differentiable of class Cr +1 • This fact is used in the proof of Theorem 7.7 of Chapter VI.

APPENDIX 2

A connected, locally compact metric space is separable

We recall that a topological space M is separable if there exists a dense subset D which contains at most countably many points. It is called locally separable if every point of M has a neighborhood which is separable. Note that, for a metric space, the separability is equivalent to the second axiom of countability (cf. Kelley [1, p. 120]). The proof of the statement in the title is divided into the following three lemmas. LEMMA

1.

A compact metric space is separable.

For the proof, see Kelley [1, p. 138]. LEMMA

2.

A local[Y compact metric space is locally separable.

This is a trivial consequence of Lemma 1. The following lemma is due to Sierpinski [1]. 3. A connected, locally separable metric space is separable. Proof of Lemma 3. Let d be the metric of a connected, locally separable metric space M. For every point x € M and every positive number r, let U(x; r) be the interior of the sphere of center x and radius r, that is, U(x; r) = {y € M; d(x,y) r}. We say that two points x and y of Mare R-related and write xRy, if there exist a separable U(x; r) containing y and a separable U(y; r') containing x. Evidently, xRx for every x € M. We have also xRy if and only ifyRx. For every subset A of M, we denote by SA the set of points which {y € M;yRx for some x € A}. are R-related to a point of A: SA Set snA = ssn-lA, n = 2, 3, .... If {x} is the set consisting of a single point x, we write Sx for S{x}. We see easily that y € Snx if LEMMA

269

270

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

and only if x € sry. We prove the following three statements: (a) Sx is open for every x € M; (b) If A is separable, so is SA; CXJ

(c) Set U(x) =

U Snx for each x € M. Then, for any x,y € M, n=1

either U(x) n U(y) is empty or U(x) = U(y). Proof of (a). Let y be a point of Sx. Since xRy, there exist positive numbers rand r' such that U(x; r) and U(y; r') are separable and thaty€ U(x; r) andx€ U(y; r'). Sinced(x,y) < r', there is a positive number r 1 such that

d(x,y) < r 1 < r'. Let ro be any positive number such that ro < r' - r 1, ro < r - d(x,y), ro < r 1 - d(x,y). It suffices to show that U(y, ro) is contained in Sx. If z € U(y; ro), then d(x, z) ~ d(x,y) + d(y, z) < d(x,y) + ro < min {r, r1 }. Hence z is in U( x; r) which is separable and x is in U( z; r1). To prove that U( z; r 1) is separable, we shall show that U( z; r 1) is contained in U(y; r') which is separable. Let w € U( z; r1) so that d(z, w) < r 1 • Then

d(y, w) < d(y, z)

+ d(z, w)

< d(y, z)

+r

1

< ro + r1 < r'.

Hence w € U(y; r'). This proves that zRx for every z € U(y; ro), that is, U(y; ro) c Sx. Proof of (b). Let A be a separable subset of M and D a countable dense subset of A. It suffices to prove that every x € SA is contained in a separable sphere whose center is a point of D and whose radius is a rational number, because there are only countably many such spheres and the union of these spheres is separable. Let x € SA. Then there is y € A such that xRy and there is a separable sphere U(y; r) containing x. Let ro be a positive rational number such that d(x,y) < ro < r. Since D is dense in A, there is z € D such that

d( z,y) < min {ro - d(x,y), r - ro}. It suffices to show that U(z; ro) contains x and is separable. From d(x, z) < d(x,y)

+ d(y,

z) <

To,

APPENDIX

2

271

it follows that x € U(z; ro). To prove that U(z; ro) is separable, we show that U(z; ro) is contained in U(y; r) which is separable. If w € U(z; ro), then

d(w,y) < d(w, z)

+ d(z,y)

< ro + d(z,y) < r,

and hence w € U(y; r). Proof of (c). Assume that U(x) n U(y) is non-empty and let z € U(x) n U(y). Then z € Smx and z € S~ for some m and n. From z € Smx, we obtain x € Smz. Hence x € Sm z C Sm+ny . This implies Sk X c Sk+m+ny for every k and hence U(x) c U(y). Similarly, we have U(y) c U(x), thus proving (c). By (a), SA = U Sx is open for any subset A of M. Hence U(x) XEA

is open for every x € M. By (b), Snx is separable for every n. Hence U(x) is separable. Since M is connected and since each U(x) is open, (c) implies M = U(x) for every x € M. Hence Mis separable, thus completing the proof of the statement in the title. We are now in position to prove

For a connected differentiable manifold M, the following conditions are mutually equivalent: (1) There exists a Riemannian metric on M; (2) M is metrizable; (3) M satisfies the second axiom of countability; (4) M is paracompact. Proof. The implication (1) ~ (2) was proved in Proposition 3.5 of Chapter IV. As we stated at the beginning, for a metric space, the second axiom of countability is equivalent to the separability. The implication (2) -+ (3) is therefore a consequence of the statement in the title. If (3) holds, then Mis metrizable by Urysohn's metrization theorem (cf. Kelley [1, p. 125]) and, hence, M is paracompact (cf. Kelley [1, p. 156]). This shows that (3) implies (4). The implication (4) ~ (1) follows from QED. Proposition 1.4 of Chapter III. THEOREM.

APPENDIX 3

Partition of unity

Let {Ui}iEl be a locally finite open covering of a differentiable n1anifold M, i.e., every point of M has a neighborhood which intersects only finitely many U/s. A family of differentiable functions {fJ on M is called a partition of unity subordinate to the covering {Ui}, if the following conditions are satisfied: ( 1) 0 <}: < 1 on M for every i E I; (2) The support of each }:, i.e., the closure of the set {x E M;}:(x) #- O}, is contained in the corresponding Ui; (3) ~i}:(X) ~ 1. Note that in (3), for each point x E M, h(X) ~ 0 except for a finite number of i's so that ~i}:(X) is a finite sum for each x. We first prove

1. Let {Ui } be a locally finite open covering of a paracompact manifold M such that each Ui has compact closure Vi. Then there exists a partition of unity {}:} subordinate to {Ui }. Proof. We first prove the following three lemmas. The first two are valid without the assumption that M is paracompact whereas the third holds for any paracompact topological space. THEOREM

1. For each point x E M and for each neighborhood U of x, there exists a differentiable junction j (of class COO) on M such that (1) o <j< 1 on M,. (2) j(x) ~ 1,. and (3) j ~ 0 outside U. Proof of Lemma 1. This can be easily reduced to the case where M ~ Rn, x ~ 0 and U ~ {(xl, ... , x n) ; Ixil < a}. Then, for each j,j ~ 1, ... , n, we letfj(x j) be a differentiable function such that jj(O) ~ 1 and thatjj(x j) ~ 0 for Ixjl > a. We set f(x\ ... ,xn) ~ fl(X 1 ) • • • fn(x n). This proves Lemma 1. LEMMA

2. For every compact subset K oj M and jor every neighborhood U oj K, there exists a differentiable function j on M such that (1) f > 0 on M,. (2) f > 0 on K,. and (3) f ~ 0 outside U. LEMMA

272

APPENDIX

3

273

Proof of Lemma 2. For each point x of K, letfx be a differentiable function on M with the property off in Lemma 1. Let Vx be the neighborhood of x defined by fx > t. Since K is compact, there exist a finite number of points Xl) ••• , X k of K such that V...wI u··· u V",""'k ::> K. Then we set

This completes the proof of Lemma 2. 3. Let {U,J be a locally finite open covering of M. Then there exists a locally finite open refinement {Vi} (with the same index set) of { Ui} such that Vi c Ui for every i. Proof of Lemma 3. For each point x E M, let Woo be an open neighborhood of x such that Woo is contained in some Ui. Let {W~} be a locally finite refinement of {Woo; x EM}. For each i, let Vi be the union of all W~ whose closures are contained in Ui. Since {W~} is locally finite, we have Vi = U W~, where the union is taken over all C( such that W~ CUi. We thus obtained an open covering {Vi} with the required property. We are now in position to complete the proof of Theorem 1. Let {Vi} be as in Lemma 3. For each i, let Wi be an open set such that Vi C Wi C Wi CUi. By Lemma 2, there exists, for each i, a differentiable function gi on M such that (1) gi > 0 on M; (2) gi > 0 on Vi; and (3) gi = 0 outside Wi. Since the support of each gi contains Vi and is contained in Ui and since {Ui} is locally finite, the sum g = ~i gi is defined and differentiable on M. Since {Vi} is an open covering of M, g > 0 on M. We set, for each i, LEMMA

Then {J:} is a partition of unity subordinate to {Ui }.

QED.

Let f be a function defined on a subset F of a manifold M. We say thatf is differentiable on F if, for each point x E F, there exists a differentiable functionft on an open neighborhood Vx of x such thatf=ftonFn Vx •

2. Let F be a closed subset of a paracompact manifold M. Then every dijJerentiable function f defined on F can be extended to a differentiable function on M. THEOREM

274

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Proof. For each x € F, letfa; be a differentiable function on an open neighborhood Va; of x such thatfa; = f on F n Va;. Let Ui be a locally finite open refinement of the covering of M consisting of M - F and Va;, x € F. For each i, we define a differentiable function gi on Ui as follows. If Ui is contained in some Va;, we choose such a Va; and set

gi = restriction

off~

to Ui ·

If there is no Va; which contains Ui , then we set

gi =

o.

Let {J:} be a partition of unity subordinate to {Ui }. We define

g = ~iJ:gi· Since {Ui } is locally finite, every point of M has a neighborhood in which ~iJ:gi is really a finite sum. Thus g is differentiable on M. It is easy to see that g is an extension off QED. In the terminologies of the sheaf theory, Theorem 2 means that the sheaf of germs of differentiable functions on a paracompact manifold M is soft ("mou" in Godement [1 J).

APPENDIX 4

On arcwise connected subgroups of a Lie group

Kuranishi and Yamabe proved that every arcwise connected subgroup ofa Lie group is a Lie subgroup (see Yamabe [1]). We shall prove here the following weaker theorem, which is sufficient for our purpose (cf. Theorem 4.2 of Chapter II). This result is essentially due to Freudenthal [1]. THEOREM. Let G be a Lie group and H a subgroup of G such that every element ofH can be joined to the identity e by a piecewise differentiable curve of class CI which is contained in H. Then H is a Lie subgroup of G. Proof. Let S be the set of vectors X € Te(G) which are tangent to differentiable curves of class CI contained in H. We identify Te(G) with the Lie algebra 9 of G. Then LEMMA. S is a subalgebra of g. Proof of Lemma. Given a curve X t in G, we denote by xt the vector tangent to the curve at the point X t • Let r be any real number and set Zt = X rt • Then 20 = r xO• This shows that if X € S, then rX € S. Let Xt and Yt be curves in G such that Xo = Yo = e. If we set Vt = xtYt, then Do = Xo + Yo (cf. Chevalley [1, pp. 120-122]). This shows that if X, Y € S, then X + Y € S. There exists a curve W t such that W t2 = xtYtXt-Yt-1 and we have Wo = [xo,Yo] (cf. Chevalley [1, pp. 120-122] or Pontrjagin [1, p. 238]). This shows that if X, Y € S, then [X, Y] € S, thus completing the proof of the lemma. Since S c Te(G) = 9 is a subalgebra of g, the distribution x ---+ LxS, x € G, is involutive (where Lx is the left translation by x) and its maximal integral manifold through e, denoted by K, is the Lie subgroup of G corresponding to the subalgebra S. We shall show that H = K. We first prove that K :::> H. Let a be any point of Hand 'T = X t , o < t < 1, a curve from e to a so that e = X o and a = Xl. We claim 275

276

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

that the vector xt is in LxtS for all t. In fact, for each fixed t, L;/(x t ) is the vector tangent to the curve L;/( 'T) at e and hence lies in S, thus proving our assertion. Since xt E LxtS for all t and X o = e, the curve X t lies in the maximal integral manifold K of the distribution x ---+ LxS (cf. Lemma 2 for Theorem 7.2 of Chapter II). Hence a E K, showing that K ::> H. To prove that H::> K, let e1, ••• ,ek be a basis for Sand x}· .. x~, 0 < t < 1, be curves in H such that = e and xb = ei for i = 1, ... , k. Consider the mapping f of a neighborhood U of the origin in Rk into K defined by f(t b . . . ,tk ) = xl1 ... ~,k (t 1 , • •• , t k ) E U. Since x~, ... , x~ form a basis for S, the differential off: U ---+ K at the origin is non-singular. Taking U sufficiently small, we may assume thatfis a diffeomorphism of U onto an open subset f( U) of K. From the definition of 1, we have f( U) c H. This shows that a neighborhood of e in K is contained in H. Since K is connected, K c H. QED.

xt

APPENDIX 5

Irreducible subgroups of O(n)

We prove the following two theorems.

1. Let G be a subgroup of O(n) which acts irreducibly on the n-dimensional real vector space R n. Then every symmetric bilinear form on R n which is invariant by G is a multiple of the standard inner product THEOREM

n

(x,y)

=

1 X~i.

i=1

THEOREM

2.

irreducibly on R n.

Let G be a connected Lie subgroup of SO(n) which acts Then G is closed in SO (n) .

We begin with the following lemmas. 1. Let G be a subgroup of GL(n; R) which acts irreducibly on R n. Let A be a linear transformation of R n which commutes with every element of G. Then (1) If A is nilpotent, then A = o. (2) The minimal polynomial of A is irreducible over R. (3) Either A = aIn (a: real number, In: the identity transformation of Rn), or A = aln + bJ, where a and b are real numbers, b =1= 0, J is a linear transformation such that J2 = -In, and n is even. Proof. (1) Let k be the smallest integer such that Ak = o. Assuming that k > 2, we derive a contradiction. Let W = {x E Rn; Ax = O}. Since W is invariant by G, we have either W = Rn or W = (0). In the first case, A = o. In the second case, A is non-singular and Ak-l = A-I. Ak = o. (2) If the minimal polynomialf(x) of A is a productfl(x) ·f2(x) with (fl,f2) = 1, then Rn = WI + W 2 (direct sum), where Wi = {x E Rn;J:(A)x = O}. Since every element of G commutes with A and hence withJ:(A), it follows that Wi are both invariant by G, contradicting the assumption of irreducibility. Thus LEMMA

277

278

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

f(x) = g(x)\ where g(x) is irreducible. Applying (1) to g(A), we see thatf(A) = g(A)k = 0 implies g(A) = o. Thusf = g. (3) By (2), the minimal polynomialf(x) of A is either (x - a) or (x - a)2 + b2 with b =F o. In the first case, A = aIn. In the second case, let J = (A - aIn)lb. Then J2 = -In and A = aln + bJ. We have (-I)n = det J2 = (det J)2 > 0 so that n is even. Let G be a subgroup of O(n) which acts irreducibly on Rn. Let A, B, ... be linear transformations ofRn which commute with G. (1) If A is symmetric, i.e., (Ax,y) = (x, Ay), then A = aIn(2) If A is skew-symmetric, i.e., (Ax,y) + (x, Ay) = 0, then A = 0 or A = bJ, where J2 = -In and n = 2m. (3) If A =F 0 and B are skew-symmetric and AB = BA, then B = cA. Proof. (1) By (3) of Lemma 1, A = aIn + bJ, possibly with b = o. If A is symmetric, so is bJ. If b =F 0, J is symmetric so that (Jx, Jx) = (x, J2 X ) = - (x, x), which is a contradiction for x =F o. (2) Since the eigenvalues of skew-symmetric A are 0 or purely imaginary, the minimal polynomial of A is either x or x2 + b2 , b =F o. In the first case, A = O. In the second case, A = -bJ with J2 = -1n . (3) Let A = bJ and B = b'K, where J2 = K2 = -In. We have JK = KJ. We show that Rn = WI + W2 (direct sum), where WI = {x ERn; Jx = Kx} and W2 = {x ERn; Jx = -Kx}. Clearly, WI n W 2 = (0). Every x ERn is of the form y + z wi'th y E WI and z E W 2, as we see by setting y = (x - JKx)/2 and z = (x + JKx)/2. WI and W 2 are invariant by G, because J and LEMMA

2.

K commute with every element of G. Since G is irreducible, we have either WI = Rn or W 2 = Rn, that is, either K = J or K = -J. This means that B = cA for some c. Proof of Theorem 1. For any symmetric bilinear form f(x, y) , there is a symmetric linear transformation A such that f(x,y) = (Ax,y). Iffis invariant by G, then A commutes with every element of G. By (1) of Lemma 2, A = aIn and hencef(x,y) = a(x,y). Proof of Theorem 2. We first show that the center 3 of the Lie algebra 9 ofG is at most I-dimensional. Let A =F 0 and BE 3. Since A, B are skew-symmetric linear transformations which commute with every element of G, (3) of Lemma 2 implies that B = cA for some c. Thus dim 3 < 1. If dim 3 = 1, then 3 = {cJ; c real}, where

APPENDIX

5

279

J is a certain skew-symmetric linear transformation wi.th J2 = -In. Now J is representable by a matrix which is a block form, each

0-1) (

block being 1

0 ' with respect to a certain orthonormal basis

ofRn. The I-parameter subgroup exp tJ consists of matrices of a COS sin block form, each block being . , and hence is iso( · WIt . h t h e circ . Ie group. sIn t cos t morp h IC Since 9 is the subalgebra of the Lie algebra of all skew-symmetric matrices, 9 has a positive definite inner product (A, B) = -trace (AB) which is invariant by ad (G). It follows that the orthogonal complement 5 of the center 3 in 9 with respect to this inner product is an ideal of 9 and 9 = 3 + 5 is the direct sum. If 5 contains a proper ideal, say, 51' then the orthogonal complement 5' of 51 in 5 is an ideal of 5 (in fact of g) and 5 = 51 + 5'. Thus we see that 5 is a direct sum of simple ideals: 5 = 51 + ... + 5k. We have already seen that the connected Lie subgroup generated by 3 is closed in SO(n). We now show that the connected Lie subgroup generated by 5 is closed in SO(n). This will finish the proof of Theorem 2. We first remark that Yosida [1 J proved the following result. Every connected semisimple Lie subgroup G of GL(n; C) is closed in GL(n; C). His proof, based on a theorem of Weyl that any representation of a semisimple Lie algebra is completely reducible, also works when we replace GL(n; C) by GL(n; R). In the case of a subgroup G of SO(n), we need not use the Weyl theorem. We now prove the following result by the same method as Yosida's.

t - t)

A connected semisimple Lie subgroup G of SO(n) is closed in SO(n). Proof. Since 9 is a direct sum of simple ideals gl' ... , gk of dimension > 1 and since gi = [gi' 9iJ for each i, it follows that 9 = [g, gJ. Now consider SO(n) and hence its subgroup G as acting on the complex vector space Cn with standard hermitian inner product which is left invariant by SO(n). Then Cn is the direct sum of complex subspaces VI' ... ' Vr which are invariant and irreducible by G. Assuming that G is not closed in SO(n), let G be its closure. Since G is a connected closed subgroup of SO(n), it is a Lie subgroup. Let 9 be its Lie algebra. Obviously, 9 c g. Since ad (G) 9 C g, we have ad (G) 9 c g, which implies that 9 is

280

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

an ideal of g. Since the Lie algebra of SO(n) has a positive definite inner product invariant by ad (SO(n)) as we already noted, it follows that g is the direct sum of g and the orthogonal complement u of g in g. Each summand Vi of Cn is also invariant by G and hence by 9 acting on Cn. For any A E g, denote by Ai its action on Vi for each i. For any A, BEg, we have obviously trace [Ai, B i ] = o. Since A ~ Ai is a representation of g on Vi and since g = [g, g], we have trace Ai = 0 for every A E g. Thus the restriction of a E G on each Vi has determinant 1 (cf. Corollary 1 of Chevalley [1; p.6]). By continuity, the restriction ofaEG on each Vi has determinant 1. This means that trace Ai = 0 for every A E g and for each i. Now let B E u. Its action B i on Vi commutes with the actions of {Ai; A E g}. By Schur's Lemma (which is an obvious consequence of Lemma 1, (2), which is valid for any field instead of R), we have B i = hiI, where I is the identity transformation of Vi. Since trace B i = 0, it follows that hi = 0, that is, B i = o. This being the case for each i, we have B = O. This means that u = (0) and g = g. This proves that G = G, that is, G is closed in SO (n).

/'

APPENDIX 6

Green's theorem

Let M be an oriented n-dimensional differentiable manifold. An n-form w on M is called a volume element, if w( 0/ ox\ ... , %xn ) > 0 for each oriented local coordinate system xl, ... , x n • With a fixed volume element w (which will be also denoted by a more intuitive notation dv), the integral

fM f

dv of any continuous

function f with compact support can be defined (cf. Chevalley [1, pp. 161-167]). For each vector field X on M with a fixed volume element w, the divergence of X, denoted by div X, is a function on M defined by (div X) w = Lxw, where Lx is the Lie differentiation in the direction of X. GREEN'S THEOREM. Let M be an oriented compact manifold with a fixed volume element w = dv. For every vector field X on M, we have

fM div X

dv = O.

Proof. Let CPt be the I-parameter group of transformations generated by X (cf. Proposition 1.6 of Chapter I). Since we have (cf. Chevalley [1, p. 165])

fM,/,,-l*W = fMW, fM,/"l*W, considered as a function of t, is a constant. By definition 281

282

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

of Lx, we have [dd (qJt-1*w)] = -Lxw. Hence t t=O

o =[ddf qJt-1*w] t M t=O = -

r Lxw =

JM

=J [dd (qJt-1*w)] M t t=O

-1

div X dv.

M

QED.

Remark 1. The above formula is valid for a non-compact manifold M as long as X has a compact support. Remark 2. The above formula follows also from Stokes' formula. In fact, since dw = 0, we have Lxw = do tx W + tx dw = d txW. We then have 0

0

r Lxw =IaM txW = O.

JM

Let M be an oriented manifold with a fixed volume element w = dv.1f r is an affine connection with no torsion on M such that w is parallel with respect to r, then,Jor every vector field X on M, we have PROPOSITION.

(div X) x = trace of the endomorphism V

~

VvX ,

V € T x ( M).

Proof. Let A x be the tensor field of type (1, 1) defined by Ax = Lx - V x as in §2 of Chapter VI. Let Xl' ... ,Xn be a basis of Tx(M). Since V XW = 0 and since A x, as a derivation, maps every function into zero, we have

(LxW) (Xl' ... ,Xn ) = AX(w(XI, =

-~i

W(X I,

=

(AXw) (Xl' ... , X n )

,Xn ))

-

~i

W(X I, ... , AXXi, ... , X n )

, A XXi' ... , X n )

= -(trace AX)xW(XI, ... , X n ). This shows that div X

=

-trace Ax.

Our assertion follows from the formula (cf. Proposition 2.5 of Chapter VI) : AxY = -VyX - T (X, Y) and from the assumption that T = O.

QED.

6

APPENDIX

283

Remark 3. The formula div X = -trace A x holds without the assumption T = O. Let M be an oriented Riemannian manifold. We define a natural volume element dv on M. At an arbitrary point x of M, let Xl' ... ,Xn be an orthonormal basis of Tx(M) compatible with the orientation of M. We define an n-form dv by

dv(X I ,

•••

,Xn )

=

1.

It is easy to verify that dv is defined independently of the frame Xl' ... ,Xn chosen. In terms of an allowable local coordinate system xl, ... , xn and the components gii of the metric tensor g, we have dv = VG dx l A dx 2 A ••• A dx n , where G = det (gii). In fact, let (oj oxi)x = ~k Cf X k so that gif = ~k CfC} and G = det (Cf)2 at x. Since oj ox\ ... , oj oxn and Xl' ... , X n have the same orientation, we have det (Cf) = VG > O. Hence, at x, we have

dv( oj ox\ ... , oj oxn )

=

~it, ... , in

B

Cft ... C:r dv(Xit , ... , Xi)

= det (Cf) = VG, where B is 1 or -1 according as (it, ... ,in) is an even or odd permutation of (1, ... , n). Since the parallel displacement along any curve 'T of M maps every orthonormal frame into an orthonormal frame and preserves the orientation, the volume element dv is parallel. Thus the proposition as well as Green's theorem is valid for the volume element dv of a Riemannian manifold.

APPENDIX 7

Factorization lemma

Let M be a differentiable manifold. Two continuous curves x(t) and y(t) defined on the unit interval I = [0, 1] with x(O) = yeO) and x( 1) = y( 1) are said to be homotopic to each other if there exists a continuous mapping j: (t, s) E I X I --+ j (t, s) E M such thatj(t, 0) = x(t),j(t, 1) = y(t),j(O, s) = x(O) = yeO) and j(l, s) = x(l) = y(l) for every t and s in 1. When x(t) andy(t) are piecewise differentiable curves of class Ck (briefly, piecewise Ck_ curves), they are piecewise Ck-homotopic, if the mapping j can be chosen in such a way that it is piecewise Ck on I X I, that is, for a r

certain subdivision I

=

:2

Ii,jis a differentiable mapping of class

i=l

Ck of Ii

X

LEMMA.

Ii into M for each (i,j).

If two piecewise Ck-curves x(t) andy(t) are homotopic to each

other, then they are piecewise Ck-homotopic. Proof.

We can take a suitable subdivision I =

r

:2 Ii

so that

i=l

j(Ii X Ii) is contained in some coordinate neighborhood for each pair (i,j). By modifying the mapping j in the small squares Ii X Ii we can obtain a piecewise Ck-homotopy between x(t) and yet) . Now let U be an arbitrary open covering. We shall say that a closed curve 7 at a point x is a U-lasso if it can be decomposed into three curves 7 = fl- 1 • (J • fl, where fl is a curve from x to a point y and (J is a closed curve aty which is contained in an open set of U. Two curves 7 and 7' are said to be equivalent, if 7' can be obtained from 7 by replacing a finite number of times a portion of the curve of the form fl- 1 • fl by a trivial curve consisting of a single point or vice versa. With these definitions, we prove 284

APPENDIX

7

285

Let U be an arbitrary open covering of M. (a) Any closed curve which is homotopic to zero is equivalent to a product of afinite number of U-lassos. (b) If the curve is moreover piecewise Ck, then each U-lasso in the product can be chosen to be of the form fl- 1 • a . fl, where fl is a piecewise Ck-curve and a is a Ck-curve. Proof. (a) Let 7 = x(t), t::;: 1, so that x = x(o) = x(l). Letfbe a homotopy I X I~Msuch thatf(t, O) =x(t~,f(t, 1) = x, f(O, s) = f(l, s) = x for every t and s in 1. We divide the FACTORIZATION LEMMA.

°::;:

square I x I into m2 equal squares so that the image of each small square by f lies in sonle open set of the covering U. For each pair of integers (i,j), 1 < i, j < m, let A(i,j) be the closed curve in the square I x I consisting of line segments joining lattice points in the following order: (0, 0)

~

(O,jjm)

~

((i - l)jm,jjm)

((i - l)jm, (j - l)jm) ((i - l)jm,jjm)

~

~

(O,jjm)

~

(ijm, (j - l)jm) ~

~

(ijm,jjm)

~

(0, 0).

Geometrically, A(i,j) looks like a lasso. Let 7(i,j) be the image of A( i,j) by the mapping): Then 7 is equivalent to the product of U-Iassos

7(m, m) ... 7(1, m) ... 7(m, 2) .. ·7(1,2) . 7(m, 1) ... 7(1, 1). (b) By the preceding lemma, we may assume that the homotopy mappingf is piecewise Ck. By choosing m larger if necessary, we may also assume thatf is Ck on each of the m2 small squares. Then each lasso 7(i,j) has the required property. QED. The factorization lemma is taken from Lichnerowicz [2, p. 51].

NOTES

Note 1. Connections and holon9my groups 1. Although differential geometry of surfaces in the 3-dimensional Euclidean space goes back to Gauss, the notion ofa Riemannian space originates with Riemann's Habilitationsschrift [1] in 1854. The Christoffel symbols were introduced by Christoffel [1] in 1869. Tensor calculus, founded and developed in a series of papers by Ricci, was given a systematic account in Levi-Civita and Ricci [1] in 1901. Covariant differentiation.which was formally introduced in this tensor calculus was given a geometric interpretation by Levi-Civita [1] who introduced in 1917 the notion of parallel displacement for the surfaces. This discovery led Weyl [1, 2] and E. Cartan [1, 2, 4, 5, 8, 9] to the introduction of affine, projective and conformal connections. Although the approach of Cartan is the most natural one and reveals best the geometric nature of the connections, it was not until 1950 that Ehresmann [2] clarified the general notion of connections from the point of view of contemporary mathematics. His paper was followed by Chern [1,2], Ambrose-Singer [1], Kobayashi [6], Nomizu [7], Lichnerowicz [2] and others. Ehresmann [2] defined, for the first time, a connection in an arbitrary fibre bundle as a field of horizontal subspaces and proved the existence of connections in any bundle. He introduced also a connection form wand defined the curvature form Q by means of the structure equation. The definition of Q given in this book is due to Ambrose and Singer [1] who proved also the structure equation (Theorem 5.2 of Chapter II). Chern [1, 2] defined a connection by means of a set of differential forms W a on Ua with values in the Lie algebra of the structure group, where {Ua } is an open covering of the base manifold (see Proposition 1.4 of Chapter II). Ehresmann [2] also defined the notion of a Cartan connection, whose examples include affine, projective and conformal connections. See also Kobayashi [6] and Takizawa [1]. We have 287

288

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

given in the text a detailed account of the relationship between linear and affine connections. 2. The notion of holonomy group is due to E. Cartan [1, 6]. The fact that the holonomy group is a Lie group was taken for granted even for a Riemannian connection until Borel and Lichnerowicz [1] proved it explicitly. The holonomy theorem (Theorem 8.1 of Chapter II) of E. Cartan was rigorously proved first by Ambrose-Singer [1]. The proof was simplified by Nomizu [7] and Kobayashi [6] by first proving the reduction theorem (Theorem 7.1 of Chapter II), which is essentially due to Cartan and Ehresmann. Kobayashi [6] showed that Theorem 8.1 is essentially equivalent to the following fact. For a principal fibre bundle P(M, G), consider the principal fibre bundle T(P) over T(M) with group T(G), where T( ) denotes the tangent bundle. For any connection r in P, there is a naturally induced connection T(r) in T(P) whose holonomy group is T(
NOTES

289

(2) For every n-dimensional manifold M andfor every reduced subbundle P of L (M) with group G, there is a unique torsion-free connection in P. The implication (1) -+ (2) is clear from Theorem 2.2 of Chapter IV (in which g can be an indefinite Riemannian metric); in fact, if G is such a group, any G-structure on M corresponds to an" indefinite Riemannian metric on M in a similar way to Example 5.7 of Chapter I. The implication (2) -+ (1) is nontrivial. See also Klingenberg [1]. Let G be the subgroup of GL(n; R) consisting of all matrices which leave the r-dimensional subspace Rr of Rn invariant. A Gstructure on an n-dimensional manifold M is nothing but an rdimensional distribution. Walker [3] proved that an r-dimensional distribution is parallel with respect to a certain torsion-free linear connection if and only if the distribution is integrable. See also Willmore [1, 2]. Let G be GL(n; C) regarded as a subgroup of GL(2n; R) in a natural manner. A G-structure on a 2n-dimensional manifold M is nothing but an almost complex structure on M. This structure will be treated in Volume II. 4. The notions of local and infinitesimal holonomy groups were introduced systematically by Nijenhuis [2]. The results in §10 of Chapter II were obtained by him in the case of a linear connection ( §9 of Chapter III). Nijenhuis' results were generalized by Ozeki [ 1] to the general case as presented in §10 of Chapter 11. See also Nijenhuis [3]. Chevalley also obtained Corollary 10.7 of Chapter II in the case of a linear connection (unpublished) and his result was used by Nomizu [2] who discussed invariant linear connections on homogeneous spaces. His results were generalized by Wang [1] as in §11 of Chapter II. 5. By making use of a connection, one can define characteristic classes of any principal fibre bundle. This will be treated in Volume II. See Chern [2], H. Cartan [2, 3]. We shall here state a result of Narasimhan and Ramanan [1] which is closely related to the notion of a universal bundle (cf. Steenrod [1, p. 101]).

Given a compact Lie group G and a positive integer n, there exists a principal bundle E (N, G) and a connection r 0 on E such that any connection r in any principal bundle P( M, G), dim M < n, can be obtained as the inverse image of r 0 by a certain homomorphism ofPinto E THEOREM.

290

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

(that is, w = f*w o, where wand Wo are the connectionforms ofr and r o, respectively, see Proposition 6.2 of Chapter II). The connection r 0 is therefore called a universal connection for G (and n). For example, the canonical connection in a Stiefel manifold with structure group O(k) is universal for O(k). For the canonical connections in the Stiefel manifolds, see also Kobayashi [5] who gave an interpretation of the Riemannian connections of manifolds imbedded in Euclidean spaces (see Volume II). 6. The holonomy groups of linear and Riemannian connections were studied in detail by Berger [1]. By a careful examination ofthe curvature tensor, he obtained a list of groups which can be restricted linear holonomy groups of irreducible Riemannian manifolds with non-parallel curvature tensor. His list coincides with the list of connected orthogonal groups acting transitively on spheres. Simons [1] proved directly that the linear holonomy group of an irreducible Riemannian manifold with non-parallel curvature tensor is transitive on the unit sphere in the tangent space. See Note 7 (symmetric spaces). 7. The local decomposition of a Riemannian manifold (Proposition 5.2 of Chapter IV) has been treated by a number of authors. The global decomposition (Theorem 6.2 of Chapter IV) was proved by de Rham [1]; the same problem was also treated by Walker [2]. A more general situation than the direct product has been studied by Reinhart [1], Nagano [2] and Hermann [1]. I t is worthwhile noting that even the local decomposition is a strongly metric property. Ozeki gave an example of a torsion-free linear connection with the following property. The linear holonomy group is completely reducible (that is, the tangent space is the direct sum of invariant irreducible subspaces) but the linear connection is not a direct product even locally. His example is as follows: On R2 with coordinates (xl, x 2 ), take the linear connection given by the Christoffel symbols rtl (xl, x2 ) = x 2 and other

rjk = O. The holonomy group is { (~

D;

a

> O}.

8. The restricted linear holonomy group of an arbitrary Riemannian manifold is a closed subgroup of the orthogonal group. Hano and Ozeki [1] gave an example of a torsion-free linear connection whose restricted linear holonomy group is not

291

NOTES

closed in the general linear group. The linear holonomy group of an arbitrary Riemannian manifold is not in general compact, as Exanlple 4.3 of Chapter V shows. For a compact flat Riemannian manifold, it is compact (Theorem 4.2 of Chapter V). Recently, Wolf [6] proved that this is also the case for a compact locally symmetric Riemannian manifold.

Note 2. Complete affine and Riemannian connections Hopf and Rinow [1] proved Theorem 4.1 (the equivalence of (1), (2) and (3)), Theorem 4.2 and Theorem 4.4 of Chapter IV. Theorem 4.2 goes back to Hilbert [1]; his proof can be also found in E. Cartan's book [8]. In §4 of Chapter IV, we followed the appendix of de Rham [1]. Condition (4) of Theorem 4.1 of Chapter IV was given as the definition of completeness by Ehresmann [1, 2]. For a complete affine connection, it does not hold in general that every pair of points can be joined by a geodesic. To construct counterexamples, consider an affine connection on a connected Lie group G such that the geodesics emanating from the identity are precisely the I-parameter groups of G. Such connections will be studied in Volume II. For our present purpose, it suffices to consider the affine connection which makes every left invariant vector field parallel; the existence and the uniqueness of such a connection is easy to see. Then the question is whether every element of G is on a I-parameter subgroup. The answer is yes, if G is compact (well known) or ifG is nilpotent (cf. Matsushima [1]). For a solvable group G, this is no longer true in general; Saito [1] gave a necessary and sufficient condition in terms of the Lie algebra of G for the answer to be affirmative when G is a simply connected solvable group. For some linear real algebraic groups, this question was studied by Sibuya [1]. Even for a simple group, the answer is not affirmative in general. For instance, a direct computation shows that an element

a (e db)

(ad - be

=

1)

of SL(2; R) lies on some I-parameter subgroup if and only if

292

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

either a + d > -2 or a = d = -1 and b = c = O. This means that, for every element A of SL(2; R), either A or -A (possibly both) lies on a I-parameter subgroup. Thus the answer to our question is negative for SL(2; R) and is affirmative for SL(2; R) modulo its center. Smith [1] also constructed a Lorentz metric, i.e., an indefinite Riemannian metric, on a 2-dimensional manifold such that the (Riemannian) connection is complete and that not every pair of points can be connected by a geodesic. It is not known whether every pair of points of a compact, connected manifold with a complete affine connection can be joined by a geodesic. An affine connection on a compact manifold is not necessarily complete as the following example of Auslander and Markus [1] shows. Consider the Riemannian connection on the real line Rl defined by the metric ds 2 = eX dx 2 , where x is the natural coordinate system in R 1 ; it is flat. It is not complete as the length of the geodesic from x = 0 to x = - 00 is equal to 2. The translation x -)- x + 1 is an affine transformation as it sends the metric eX dx 2 into e ex dx 2 • Thus the real line modulo 1, i.e., a circle, has a non-complete flat affine connection. This furnishes anon-complete, compact, homogeneous affinely connected manifold. An example ofa non-complete affine connection on a simply connected compact manifold is obtained by defining the above affine connection on the equator of a sphere and extending it on the whole sphere so that the equator is a geodesic. I t is known that every metrizable space admits a complete uniform structure (compatible with the topology) (Dieudonne [1]). N omizu and Ozeki [1] proved that, given a Riemannian metric g on a manifold M, there exists a positive functionj on M such that j . g is a complete Riemannian metric.

Note 3. Ricci tensor and scalar curvature Analogous to the theorem of Schur (Theorem 2.2 of Chapter V), we have the following classical result.

1. Let M be a connected Riemannian manifold with metric tensor g and Ricci tensor S. If S = Ag, where Ais ajunction on M, then A is necessarily a constant provided that n = dim M > 3. Proof. The simplest proof is probably by means of the THEOREM

293

NOTES

classical tensor calculus. Let gij, R ijkl and R ij be the components of the metric tensor g, the Riemannian curvature tensor R and the Ricci tensor S, respectively, with respect to a local coordinate system xl, ... , xn • Then Bianchi's second identity (Theorem 5.3 of Chapter III) is expressed by Rijlrl;m

+ Rijlm;k + Rijmk;l

=

O.

Multiplying by gik and gil, summing with respect to i, j, k and l and finally using the following formulas R ijkl

=

-

R jikl

= -

R ijlk ,

~ i, k gikR ijkl

we obtain

(n - 2)

A;m =

=

R

jl

=

Ag jl,

o.

Hence A is a constant.

QED.

A Riemannian manifold is called an Einstein manifold if S = Ag, where A is a constant. The following proposition is due to Schouten and Struik [lJ. PROPOSITION

2. If M is a 3-dimensional Einstein manifold, then i

is a space of constant curvature. Proof. Let P be any plane in Tx(M) and let Xl' X 2 , X 3 be an orthonormal basis for Tx(M) such that P is spanned by Xl' X 2 • LetPii be the plane spanned by Xi and X j (i =I=- j) so thatPii = PH. Then SeX!} Xl) K(PI2) + K(PI3)

S(X2, X 2) = K(p21) S(X3, X 3) = K(P31)

+ + K(P32),

K(P23)

where K(Pij) denotes the sectional curvature determined by the plane PH. Hence we have

S(X I, Xl)

+ S(X2, X 2) -

S(X3, X 3) = 2K(P12) = 2K(p).

Since S(Xi, Xi) = A, we have K(p) =

tAo

QED.

Remark. The above formula implies also that, if 0 < c < SeX, X) < 2c for all unit vectors X E Tx(M), then K(p) > 0 for all planes P in Tx(M). Similarly, if 2c < SeX, X) < c < 0 for all unit vectors X E Tx(M), then K(p) < 0 for all planesp in Tx(M).

294

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Going back to the general case where n = dim M is arbitrary, let Xl' ... ' X n be an orthonormal basis for Tx(M). Then S(X I , Xl) + ... + S(Xn , X n ) is independent of the choice of orthonormal basis and is called the scalar curvature at x. In terms of the components R ii and gii of Sand g, respectively, the scalar curvature is given by ~i,3' g'tj R i j •

Note 4. Spaces of constant positive curvature Let M be an n-dimensional, connected, complete Riemannian manifold of constant curvature 1/a 2 • Then, by Theorem 3.2 of Chapter V and Theorem 7.10 of Chapter VI, the universal covering manifold of M is isometric to the sphere Sn of radius a in Rn+l given by (X I)2 + ... + (X n+I)2 = a 2, that is, M = sn/G, where G is a finite subgroup of O(n + 1) which acts freely on Sn. In the case where n is even, the determination of these groups G is extremely simple. Let X(M) denote the Euler number of M. Then we have (cf. Hu [1; p. 277J) 2 = X(Sn) = X(M)

X

order of G

(if n is even).

Hence, G consists of either the identity I only or I and another element A of O(n + 1) such that A2 = 1. Clearly, the eigen-values of A are ± 1. Since A can not have any fixed point on sn, the eigenvalues of A are all equal to -1. Hence, A = -1. We thus obtained 1. Every connected, complete Riemannian manifold M of even dimension n with constant curvature 1/a2 is isometric either to the sphere Sn of radius a or to the real projective space sn/{±I}. THEOREM

The case where n is odd has not been solved completely. The most general result in this direction is due to Zassenhaus [2J.

2. Let G be a finite subgroup of O(n + 1) which acts freely on Sn. Then, any subgroup of G of order pq (where p and q are prime numbers, not necessarily distinct) is cyclic. Proof. It suffices to prove that if G is order pq, then G is cyclic. First, consider the case G is of order p2. Then, G is either cyclic or a direct product of two cyclic groups GI and G2 of order p (cf. Hall [1, p. 49J). Assuming the latter, let A and B be generators of THEOREM

295

NOTES

G 1 and G2, respectively. Since every element T =1= I of G is oforder p, we have T(~f~J Tiy) = ~f~J Tiy for each y E Rn+l. Since T has no fixed point on Sn, we have ~f~J

Ty

= 0

for y

E Rn+l.

By setting T = AiB andy = x, we obtain ~r~J

(AiB)ix

=

0

for

xERn+l

and

i = O,l, ... p - 1,

and hence

o=

~f~J ~r~J

(AiB)ix

=

~r==-J ~f==-J

AiiBiX

for x

E Rn+l.

On the other hand, by setting T = Ai andy = Bix, we obtain ~f==-J

AiiBix = 0

for x

E Rn+l

and j = 1, 2, ... ,p - 1.

Hence, we have

o=

~p-l ~J?-l

J=O

~=o

AiiBix =

~J?-l ~=o

AOBox = px

for x

E Rn+l ,

which is obviously a contradiction. Thus, G must be cyclic. Second, consider the case where p < q. Then G is either cyclic or non-abelian. Assuming that G is non-abelian, let S and A be elements of order p and q, respectively. Then, we have (c£ Hall [1, p. 51J) SAS-l = At, where 1 < t < q and t P - 1 mod q, and every element of G can be written uniquely as AiS\ where 0 ::;;: i ::;;: q - 1 and 0 ::;;: k ~ P - 1. For each integer k, define an integer f(k) by f(k) = 1 + t + t 2 + ... + t k - 1 • We then have (a) f(p) 0 mod q; (b) f(k) - 1 mod q, if k - 1 modp; (c) (AiS)k = Ai"!(k)Sk.

=

Indeed, (a) follows from t P - 1 mod q, and (c) follows from SAS-l = At. For each i, 0 < i < q - 1, let Gi be the cyclic subgroup of G generated by AiS. Since (AiS)P = Ai'!(p)SP = I, Gi is of order p. Hence we have either Gi n Gi = {I} or Gi = G i for o < i, j < q - 1. We prove that Gin G j = {I} if i =1= j. If Gi = G j, there exists an integer k such that (AiS)k = AiS. By (c), we have Ai "!(k)Sk = AiS and, hence, Sk = S. This implies k - 1 mod p andj(k) = 1 mod q. Hence, we have A~Sk = AjS, which

296

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

implies i = j. Let N be the normal subgroup of G generated by A. Since N is of order q and since Gi is of order p, we have Gi n N = {I} for each i, 0 ~ i < q - 1. By counting the orders of N, Go, G1} ... , Gq - 1 , we see that G is a disjoint union of N, Go - {I}, G1 - {I}, ... , Gq- 1 - {I}. Therefore we have ~T €N

Tx

+ ~T€Go

Tx

+ ... + ~T€Gq_l

On the other hand, for every To TO(~T€N

Tx) =

~T€N

E

ToTx =

Tx

=

Tx + qx for x E Rn+l.

~T€G

N, we have ~T€N

Tx

for x E Rn+l.

Since G acts freely on sn, we have ~T€N Tx = O. By the same reasoning, we have ~T€G~ Tx = 0 for i = 0, 1, ... , q - 1 and ~T€G Tx = o. Hence, we have qx = 0 for each x E Rn+1, which is obviously a contradiction. QED. Recently, Wolf [1] classified the homogeneous Riemannian manifolds of constant curvature lja 2 • His result may be stated as follows.

3. Let M = snjG be a homogeneous Riemannian manifold of constant curvature lja 2 • (1) If n + 1 = 2m (but not divisible by 4), then THEOREM

Sn = {(z1, ... , zm)

E

em; Iz1 12 + ...

+ Izml2

= a2 },

and G is a finite group of matrices of the form )Jm' where A E e with IAI = 1 and 1m is the m X m identity matrix,(2) If n + 1 = 4m, then Sn = {(q1, ... , qm) E Qm; Iql 12 + ... + Iq ml2 = a2} (where Q is the field of quaternions), and G is afinite group of matrices of the form plm, where p E Q with Ipl = 1. Conversely, if G is a finite group of the type described in (1) or (2), then M = snjG is homogeneous. In view of Theorem 1, we do not have to consider the case where n is even. The reader interested in the classification problem of elliptic spaces, i.e., spaces of constant positive curvature, is referred to the following papers: Vincent [1], Wolf [5]; for n = 3, H. Hopf [1] and Seifert and Threlfall [1]. Milnor [1] partially generalized

NOTES

297

Theorem 2 to the case ·where G is a group of homeomorphisms acting freely on sn. Calabi and Markus [1] and Wolf [3, 4] studied Lorentz manifolds of constant positive curvature. See also Helgason [1]. For the study of spaces covered by a homogeneous RiemannIan manifold, see Wolf [2].

Note 5. Flat Riemannian manifolds Let M = RnjG be a compact flat Riemannian manifold, where G is a discrete subgroup of the group of Euclidean motions ofRn. Let N be the subgroup of G consisting of pure translations. Then (1) N is an abelian normal subgroup of G and is free on n generators; (2) N is a maximal abelian subgroup of G; (3) GjN is finite; (4) G has no finite subgroup. Indeed, (1) and (3) have been proved in (4) of Theorem 4.2 of Chapter V. To prove (2), let K be any abelian subgroup of G containing N. Since GjK is also finite by (2), RnjK is a compact flat Riemannian manifold. Since K is an abelian normal subgroup of K, K contains nothing but translations by Lemma 6 for Theorem 4.2 of Chapter V. Hence K = N. Finally, (4) follows from the fact that G acts freely on Rn. In fact, any finite group of Euclidean motions has a fixed point (cf. the proof of Theorem 7.1 of Chapter IV) and hence G has no finite subgroup. Auslander and Kuranishi [1] proved the converse: Let G be a group with a subgroup N satisfying the above conditions (1), (2), (3) and (4). Then G can be realized as a group of Euclidean motions tifRn such that RnjG is a compactjlat Riemannian manifold. Let RnjG and RnjG' be two compact flat Riemannian manifolds. We say that they are equivalent, if there exists an affine transformation cp such that cpGcp-l = G', that is, if G and G' are conjugate in the group of affine transformations of Rn. In addition to (4) of Theorem 4.2 of Chapter V, Bieberbach [1] obtained the following results: (a) If G and G' are isomorphic as abstract group, then RnjG and RnjG' are equivalent. (b) For each n, there are only a finite number of equivalence classes of compactjlat Riemannian manifolds RnjG.

298

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

We shall sketch here an outline of the proof. We denote by (A, p) an affine transformation of Rn with linear part A and translation part p. Let N be the subgroup of G consisting of pure translations and let (I, t 1), ••• , (I, tn) be a basis of N, where I is the identity matrix and t i ERn. Since (A,p)(I, ti )(A,p)-l = (I, Ati ) E N for any (A, p) E G, we can write Ati = ~j=1 a{tj} where each a{ is an integer. Let T be an (n X n)-matrix whose i-th column is given by t"" that is, T = (t 1 ... tn). Then (a{) = T-IA T is unimodular. (A matrix is called unimodular if it is nonsingular and integral together with its inverse.) To prove (a), let (A', p') E G' be the element corresponding to (A, p) E G by the isomorphism G' ~ G. Let N' be the subgroup of G' corresponding to N by the isomorphism G' ~ G. Then N' is normal and maximal abelian in G'. Hence N' is the subgroup of G' consisting of pure translations. Let (I, tn correspond to (I, ti ). Since (A',p') (I, tn(A',p')-1 = (I, A'tn, (I, A'tn corresponds to (I, Ati ). Hence we have A't; = ~;=1 a~t;. In other words, if we set T' (t{ ... t~), then T'-1A'T' = T-1 A T. Set G*

=

{( T-1AT,

T-lp - T'-lp');

(A, p)

E

G}.

Then G* is a group which contains no pure translations and hence is finite. Let U ERn be a point left fixed by G*. Then we have

(T, TU)-1(A,p)(T, Tu) - (T', O)-l(A',P')(T, 0) for all (A, p) E G. This completes the proof of (a). To prove (b), it suffices to show that there are only a finite number of mutually non-isomorphic groups G such that Rn/G are compact flat Riemannian manifolds. Each G determines a group extension 0.......;- N.......;- G.......;- K .......;-1, where the finite group K G/N acts linearly on N when N is considered as a subgroup of Rn. Given such a finite group K, the set ofgroup extensions 0 . . . .;- N . . . .;- G . . . .;- K . . . .;- 1 is given by H2(K, N). Since K is finite and N is finitely generated, H2(K, N) is finite. As we have seen in the proof of (a), if we identify N with the integral lattice points of Rn, then K = G/ N is given by unimodular matrices. Let K and K' be two finite groups ofunimodular matrices of degree n which are conjugate in the group GL(n; Z) of all

NOTES

299

unimodular matrices so that SKS-l = K' for some S E GL(n; Z). The mapping which sends tEN into St E N is an automorphism of N. Hence S induces an isomorphism H2(K, N) ~ H2(K', N), and if 0 ---+ N ---+ G' ---+ K' ---+ 1 is the element of H2(K', N) corresponding to an element 0 ---+ N ---+ G ---+ K ---+ 1 of H2 (K, N), then G and G' are isomorphic. Thus our problem is reduced to the following theorem of Jordan: There are only a finite number of conjugate classes offinite subgroups of

GL(n; Z). This theorem of Jordan follows from the theory of MinkowskiSiegel. Let H n be the space of all real symmetric positive definite matrices of degree n. Then GL(n; Z) acts properly discontinuously on H n as follows: X ---+ tsxs for X E H n and S E GL(n; Z). Let R be the subset of H n consisting of reduced matrices in the sense of Minkowski. Denote tSXS by SeX]. Then (i) U S[R] = H n ; SEGL(n; Z)

(ii) The set F defined by F = {S E GL(n; Z); S[R] n R = nonempty} is a finite set. The first property of R implies that any finite subgroup K of GL(n; Z) is conjugate to a subgroup of GL(n; Z) contained in F. Indeed, let X o E H n be a fixed point of K (for instance, set X o = ~.AEKtAA). Let SEGL(n;Z) be such that S[Xo]ER. Then S-lKS c F. Since F is finite, there are only a finite number of conjugate classes of finite subgroups of GL(n; Z). QED. As references we mention Minkowski [1], Bieberbach [2], Bieberbach and Schur [1] and Siegel [1]. Note that (a) implies that two compact flat Riemannian manifolds are equivalent if and only if they are homeomorphic to each other. Although (b) does not hold for non-compact flat Riemannian manifolds, there are only a finite number of homeomorphism classes of complete flat Riemannian manifolds for each dimension (Bieberbach [3]). For the classification of3-dimensional complete flat Riemannian manifolds, see Hantzche and Wendt [1] and Nowacki [1]. Most of the results for flat Riemannian manifolds cannot be generalized to flat affine connections, see, for example, Auslander [1].

300

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

Note 6. Parallel displacement of curvature Let M and M' be Riemannian manifolds and qJ: M ---+ M' a diffeomorphism which preserves the curvature tensor fields. In general, this does not imply the existence of an isometry of M onto M'. For instance, let M be a compact Riemannian manifold obtained by attaching a unit hemisphere to each end of the right circular cylinder 8 1 X [0, 1], where 8 1 is the unit circle, and then smoothing out the corners. Similarly, let M' be a compact Riemannian manifold obtained by attaching a unit hemisphere to each end of the right circular cylinder 8 1 X [0, 2] and then smoothing out the corners in the same way. Let qJ: M ---+ M' be a diffeomorphism which induces an isometry on the attached hemispheres and their neighborhoods. Since the cylinder parts of M and M' are flat, qJ preserves the curvature tensor fields. However, M and M' cannot be isometric with each other. Ambrose [1] obtained the following result, which generalizes Theorem 7.4 of Chapter VI in the Riemannian case. Let M and M' be complete, simply connected Riemannian manifolds, x an arbitrarily fixed point of M and x' an arbitrarily fixed point of M'. Letf: Tx(M) ---+ Tx,(M') be a fixed orthogonal transformation. Let 7" be a simply broken geodesic of M from x to a point y and 7"' the corresponding simply broken geodesic of M' from x' to a point y', the correspondence being given by f through parallel displacement. Let p (resp. p') be a plane in T x ( M) (resp. Tx,(M')) and q (resp. q') the plane in Ty(M) (resp. Ty,(M')) obtained from p (resp. p') by parallel displacement along 7" (resp. 7"'). Assume thatp' corresponds to p by j. If the sectional curvature K(q) is equal to the sectional curvature K'(q') for all simply broken geodesics 7" and all planes p in Tx(M), then there exists a unique isometry F: M ---+ M' whose differential at x coincides withj. Hicks [1] obtained a similar result in the case of affine connection; his result generalizes Theorem 7.4 of Chapter VI.

Note 7. Symmetric spaces Although the theory of symmetric spaces, in particular, Riemannian symmetric spaces, will be taken up in detail in Volume II, we shall give here its definition and basic properties.

NOTES

301

Let G be a connected Lie group with an involutive automorphism a (a 2 = 1, a =F 1). Let H be a closed subgroup which lies between the (closed) subgroup of all fixed points of a and its identity component. We shall then say that GjH is a symmetric homogeneous space (defined by a). Denoting by the same letter a the involutive automorphism of the Lie algebra 9 induced by a, we have 9 = m + 9 (direct sum), where 9 = {X E g; XO' = X} coincides with the subalgebra corresponding to .H and m = {X E g; XO' = -X}. We have obviously [9, m] c m and [m, m] c 9. The automorphism a of G also induces an involutive diffeomorphism ao of GjH such that ao(7Tx) = 7T(XO') for every x E G, where 7T is the canonical projection of G onto GjH. The origin o = 7T(e) of GjH is then an isolated fixed point of ao. We call ao the symmetry around o. By Theorem 11.1 of Chapter II, the bundle G(GjH, H) admits an invariant connection r determined by the subspace m. We call this connection the canonical connection in G(GjH, H).

For a symmetric space G jH, the canonical connection r in G(GjH, H) has the following properties: (1) r is invariant by the automorphism a of G (which is a bundle automorphism of G(G jH, H)) ~. (2) The curvature form is given by {leX, Y) = -(lj2) [X, y] E 9, where X and Yare arbitrary left invariant vector fields belonging to m' (3) For any X E m, let at = exp tX and let Xt = 7T(a t) = at(o). The parallel displacement of the fibre H along the curve xt coincides with the left translation h ~ ath, h E H. Proof. (1) follows easily from mO' = m. (2) is contained in Theorem 11.1 of Chapter II. (3) follows from the fact that ath for any fixed h E H is the horizontal lift through h of the curve X t • QED. The projection 7T gives a linear isomorphism of the horizontal subspace mat e of r onto the tangent space To(GjH) at the origin o. If h E H, then a~ (h) on m corresponds by this isomorphism to the linear isotropy h, i.e., the linear transformation of To (GjH) induced by the transformation h of GjH which fixes o. Now, denoting GjH by M, we define a mappingf of G into the bundle offrames L(M) over M as follows. Let U o be an arbitrarily THEOREM

~

1.

302

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

fixed frame Xl' ... ,Xn at 0, which can be identified with a certain basis of m. For any a E G, f(a) is the frame at a(o) consisting of the images of Xi by the differential of a. In particular, for h E H, f(h) = h . Uo = U o • rp(h), where rp(h) E GL(n; R) is the matrix which represents the linear transformation of To(M) induced by h with respect to the basis Uo• It is easy to see thatfis a bundle homomorphism of G into L(M) corresponding to the homomorphism rp of H into GL(n; R). If G is effective on GjH (or equivalently, if H contains no non-trivial invariant subgroup of G), then f and rp are isomorphisms. By Proposition 6.1, of Chapter II, the canonical connection r in G(M, H) induces a connection in L(M), which we shall call the canonical linear connection on GjH and denote still by r.

2.

The canonical linear connection on a symmetric space GjH has the following properties: (1) r is invariant by G as well as the symmetry (J0 around 0; (2) The restricted homogeneous holonomy group of r at 0 is contained in the linear isotropy group H; (3) For any X E m, let at = exp tX and X t = 'Tr(a t) = at(o). The parallel displacement of vectors along X t is the same as the transformation by at. In particular, X t is a geodesic; (4) The torsion tensor field is 0; (5) Every G-invariant tensor field on GjH is parallel with respect to r. In particular, the curvature tensor field R is parallel, i.e., V R = o. THEOREM

Proof. (1), (2) and (3) follow from the corresponding properties in Theorem 1. (4) follows from (1); since the torsion tensor field T is invariant by (Jo, we have T(X, Y) = (T(X'o, ya = - T( -X, - Y) = - T(X, Y) and hence T(X, Y) = 0 for any X and Y in To(M). Thus T = 0 at 0 and hence everywhere. (5) follows from (3). In fact, if K is a G-invariant tensor field, then VxoK = 0 for any X o E To(M), since there exists X E m such that X t in (3) has the initial tangent vector X o• QED.

o)to

Remark.

r

is the unique linear connection on GjH which has property (1). This justifies the name of canonical linear connection. Let GjH be a symmetric space with compact H. There exists a G-invariant Riemannian metric on GjH. For any such metric g, the Riemannian connection coincides with r. In fact, the metric

303

NOTES

tensor field g is parallel with respect to r by (5). Since r has zero torsion, it is the Riemannian connection by the uniqueness (Theorem 2.2, Chapter II).

Example. In G = SO(n + 1), let (J be the involutive automorphism A E SO(n + 1) ~ SAS-l E SO(n + 1) where S is the matrix of the form

-1 0) (o

with identity matrix In of degree n.

In The identity component HO of the subgroup H of fixed points of (f

consists of all matrices of the form

(~ ~), where B

€

SO(n). We

shall write SO(n) for HO with this understanding. The symmetric homogeneous space SO(n + l)jSO(n) is naturally diffeomorphic with the unit sphere Sn in Rn+l. In fact, let eo, e1, • • • , en be the standard orthonormal basis in Rn+l. The mapping A E SO(n + I) ~ Aeo E sn induces a diffeomorphism of SO (n + I) jSO(n) onto sn. The set of vectors Ae 1 , • • • , Aen can be considered as an orthonormal frame at the point Aeo of sn. This gives an isomorphism of the bundle SO(n + 1) over SO(n --r l)jSO(n) onto the bundle of orthonormal frames over Sn. The canonical linear connection on SO(n + l)jSO(n) coincides with the Riemannian connection of sn with respect to the Riemannian metric of sn as imbedded submanifold of Rn+l.

A linear connection r on a differentiable manifold M is said to be locally symmetric at x E M, if there exists an involutive affine transformation of an open neighborhood U of x which has x as an isolated fixed point. This local symmetry at x, if it exists, must be of the form (Xi) ~ (-Xi) with respect to any normal coordinate system with origin x, since it induces the linear transformation X ~ -X in Tx(M). We say that r is locally symmetric, ifit is locally symmetric at every point x of M.

3. A linear connection only if T = 0 and \1R = O. THEOREM

r

on M is locally symmetric

if and

Proof. If r is locally symmetric, then any tensor field of type (r, s) with odd r + s which is invariant by the local symmetry at x is 0 at x. Hence T = 0 and \1R = 0 on M. The converse follows from Theorem 7.4 of Chapter VI. QED.

304

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

4. Let r be a locally symmetric linear connection on M. If M is connected, simply connected and complete, then the group ~r(M) of all affine transformations is transitive on M. Let G = mO(M). Then M = G/H is a symmetric space for which r is the canonical linear connection. Proof. The first assertion follows from Corollary 7.9 of Chapter VI. Let ao be the local symmetry at a point 0 of lvI. By Corollary 6.2, ao can be extended to an affine transformation of M onto itself which is involutive. Define an involutive automorphism of G by a(J = ao a ao • Then H lies between the subgroup of all fixed QED. elements of a and its identity component. THEOREM

0

0

The Riemannian versions of Theorems 3 and 4 are obvious. The Riemannian symmetric spaces were introduced and studied extensively by Cartan [7]. For the canonical linear connection on symmetric G/H, see Nomizu [2] and Kobayashi [3]. Nomizu [4, 6] proved the converse of(2) of Theorem 2 that if the restricted linear holonomy group of a complete Riemannian manifold M is contained in the linear isotropy group at every point, then M is locally syn1metric. Simons [1] has a similar theorem. Nomizuand Ozeki [3] proved that,foranycompleteRiemannian manifold M, the condition VmR = 0 for some m > 1 implies VR = O. (This was known by Lichnerowicz [3, p. 4] when M is compact.) They remarked later that the assumption of completeness is not necessary.

Note 8. Linear connections with recurrent curvature Let M be an n-dimensional manifold with a linear connection r. A non-zero tensor field K of type (r, s) on M is said to be recurrent if there exists a I-form ex. such that V K = K ® ex.. The following result is due to Wong [1]. THEOREM 1. In the notation of §5 of Chapter III, let f: L(M) ---+ T~(Rn) be the mapping which corresponds to a given tensor field K of type (r, s). Then K is recurrent if and only if, for the holonomy bundle P(uo) through any U o E L(M), there exists a differentiable function q;(u) with no zero on P(u o) such that feu) = q;(u) ·f(uo) for u E P(uo). As a special case, K is parallel if and only if feu) is constant on P(u o)·

Using this result and the holonomy theorem (Theorem 8.1 of Chapter II), Wong obtained

2. Let r be a linear connection on M with recurrent curvature tensor R. Then the Lie algebra ofits linear holonomy group '¥(u o) is spanned by all elements of the form Quo (X, Y), where Q is the curvature form and X and Yare horizontal vectors at Uo' In particular, we have THEOREM

dim '¥(uo) < In(n - 1). As an application of Theorem 1, we shall sketch the proof of the following

For a Riemannian manifold M with recurrent curvature tensor whose restricted linear holonomy group is irreducible, the curvature tensor is necessarily parallel provided that dim M > 3. Proof. Let RJkl be the components of the T§(Rn)-valued function on O(M) which corresponds to the curvature tensor field R. We apply Theorem 1 to R. Since ~i.j,k,l (Rjkl)2 is constant on each fibre of OeM), cp2 is constant on each fibre of P(u o). Since cp never vanishes on P( uo), it is either always positive or always negative. Hence cp itself is constant on each fibre of P( uo)' Let A be the function on M defined by A(X) = l/cp(u), where x 7T(U) t M. Then AR is a parallel tensor field. If we denote by S the Ricci tensor field, then AS is also parallel. The irreducibility of M implies that AS = c . g, where c is a constant and g is the metric tensor (cf. Theorem 1 of Appendix 5). If dim M > 3 and if the Ricci tensor S is non-trivial, then A is a constant function by Theorem 1 of Note 3. Since AR is parallel and since A is a constant, R is parallel. Next we shall consider the case where the Ricci tensor S vanishes identically. Let V R = R ® a and let Rhl and am be the components of R and a with respect to a local coordinate system xl, ... , xn • By Bianchi's second identity (Theorem 5.3 of Chapter III; see also Note 3), we have THEOREM

3.

Multiply by gjm and sum with respect to j and m. Since the Ricci . h es 1'dentIca . 11y, we h ave ~j,mg "" j1nRijlm "" ~j,mg jmRijmk - 0. tensor vanIS Hence, jm H . "" /Vj = 0 where Hi -- ""mg ~ .Ri'klUl. ~ m J

J

'

UI.

UI.

306

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

This equation has the following geometric implication. Let x be an arbitrarily fixed point of M and let X and Y be any vectors at x. If we denote by V the vector at x with components cx.j(x), then the linear transformation R(X, Y): Tx(M) ~ TAM) maps V into the zero vector. On the other hand, by the Holonomy Theorem (Theorem 8.1 of Chapter II) and Theorem 1 of this Note (see also Wong [1]), the Lie algebra of the linear holonomy group 'Y(x) is spanned by the set of all endomorphisms of T x (M) given by R(X, Y) with X, Y E Tx(M). It follows that V is invariant by 'Y(x) and hence is zero by the irreducibility of 'Y(x). Consequently, V R vanishes at x. Since x is an arbitrary point of M, R is parallel. QED. On the other hand, every non-flat 2-dimensional Riemannian manifold is of recurrent curvature if the sectional curvature does not vanish anywhere.

If M

is a complete Riemannian manifold with recurrent curvature tensor, then the universal covering manifold M of M is either a symmetric space or a direct product of the Euclidean space Rn-2 and a 2dimensional Riemannian manifold. Proof. Use the decomposition theorem of de Rham (Theorem 6.2 of Chapter IV) and Theorem 3 above together with the following fact which can be verified easily. Let M and M' be manifolds with linear connections and let Rand R' be their curvature tensors, respectively. If the curvature tensor of M X M' is recurrent, then there are only three possibilities: (1) v R = 0 and v R' = 0; (2) R = 0 and v R' =1= 0; (3) v R =1= 0 and R' = o. (See also Walker [1].) QED. COROLLARY.

Note 9. The automorphism group of a geometric structure Given a differentiable manifold M, the group ofall differentiable transformations of M is a very large group. However, the group of differentiable transformations of M leaving a certain geometric structure is often a Lie group. The first result of this nature was given by H. Cartan [1] who proved that the group of all complex analytic transformations of a bounded domain in en is a Lie group. Myers and Steenrod [1] proved that the group of all isometries of a

307

NOTES

Riemannian manifold is a Lie group. Bochner and Montgomery [I, 2] proved that the group of all conlplex analytic transformations ofa compact conlplex manifold is a complex Lie group; they made use of a general theorem concerning a locally compact group of differentiable transformations which is now known to be valid in the form of Theorem 4.6, Chapter I. The theorem that the group of all affine transformations ofan affinely connected manifold is a Lie group was first proved by Nomizu [I] under the assumption of completeness; this assumption was later removed by Hano and Morimoto [I]. Kobayashi [I, 6] proved that the group of all automorphisms of an absolute parallelism is a Lie group by imbedding it into the manifold. This method can be applied to the absolute parallelism of the bundle of frames L(M) of an affinely connected manifold M (cf. Proposition 2.6 of Chapter III and Theorem 1.5 of Chapter VI). Automorphisms of a complex structure and a Kahlerian structure will be discussed in Volume II. A global theory of Lie transformation groups was studied in Palais [I]. We shall here state one theorem which has a direct bearing on us. Let G be a certain group of differentiable transformations acting on a differentiable manifold M. Let g' be the set of all vector fields X on M which generate a global I-parameter group of transformations which belong to the given group G. Let 9 be the Lie subalgebra of the Lie algebra X(M) generated by g'. THEOREM.

If

9 is finite-dimensional, then G admits a Lie group

structure (such that the mapping G X M fllf is differentiable) and 9 = g'. The Lie algebra of G is naturally isomorphic with g. ---,)0

We have the following applications of this result. If G is the group of all affine transformations (resp. isometries) of an affinely connected (resp. Riemannian) manifold M, then g' is the set of all infinitesimal affine transformations (resp. infinitesimal isometries) which are globally integrable (note that if M is complete, these infinitesimal transformations are always globally integrable by Theorem 2.4 of Chapter VI). By virtue of Theorem 2.3 (resp. Theorem 3.3) of Chapter VI, it follows that 9 is finite-dimensional. By the theorem above, G is a Lie group. The Lie algebra i(M) of all infinitesimal isometries,ofa Riemannian manifold M was studied in detail by Nomizu [8, 9]. At each

308

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

point x of M, a certain Lie algebra i(x) is constructed by using the curvature tensor field and its covariant differentials. If M is simply connected and analytic together with the metric, then i(M) is naturally isomorphic with i(x), where x is an arbitrary point.

.lVote 10. Groups of isometries and affine trans.formations with maximum dimensions In Theorem 3.3 of Chapter VI, we proved that the group ~(M) of isometries of a connected, n-dimensional Riemannian manifold M is of dimension at most in(n + 1) and that if dim .3(M) = in (n + 1), then M is a space of constant curvature. We shall oudine the proof of the following theorem. THEOREM

1. Let M be a connected, n-dimensional Riemannian dim ~(M) = in(n + 1), then M is isometric to one of

manifold. If the following spaces of constant curvature: (a) An n-dimensional Euclidean space R n ; (b) An n-dimensional sphere Sn ; (c) An n-dimensional real projective space Sn/ {± I} ; (d) An n-dimensional, simply connected hyperbolic space. Proof. From the proof of Theorem 3.3 of Chapter VI, we see that M is homogeneous and hence is complete. The universal covering space M of M is isometric to one of (a), (b) and (d) above (cf. Theorem 7.10 of Chapter YI). Every infinitesimal isometry X of M induces an infinitesimal isometry X of M. Hence, in(n + 1) = dim .3(M) ::;: dim .3(M) < in(n + 1), which implies that every infinitesimal isometry X of M is induced by an infinitesimal isometry X of M. If M is isometric to (a) or (d), then there exists an infinitesimal isometry X of M which vanishes only at a single point of M. Hence, M is simply connected in case the curvature is nonpositive. If M is isometric to a sphere Sn for any antipodal points x and x', there exists an infinitesimal isometry X of M = Sn which vanishes only at x and x'. This implies that M = Sn or M = snj{ ± f}. We see easily that if M is isometric to the projective space Sn j{ ± l}, then .3 (M) is isomorphic to o(n + 1) modulo its center and hence of dimension in(n + 1). QED. In Theorem 2.3 of Chapter VI, we proved that the group

NOTES

309

m(M) of affine transformations of a connected, n-dimensional manifold M with an affine connection is of dimension at most n2 + n and that if dim ~(M) = n2 + n, then the connection is flat. We prove THEOREM 2. ..if dim ~(M) = n2 + n, then M is an ordinary affine space with the natural flat affine connection. Proof. Every element of ~(M) induces a transformation of L(M) leaving the canonical form and the connection form invariant (cf. §l of Chapter VI). From the fact that ~(M) acts freely on L(M) and from the assumption that dim ~(M) = n2 + n = dim L(M), it follows that ~O(M) is transitive on each connected component of L(M). This implies that every standard horizontal vector field on L(M) is complete; the proof is similar to that of Theorem 2.4 of Chapter VI. In other words, the connection is complete. By Theorem 4.2 of Chapter V or by Theorem 7.8 of Chapter VI, the universal covering space M of M is an ordinary affine space. Finally, the fact that M = M can be proved in the same way as Theorem 1 above. QED. Theorems 2.3 and 3.3 are classical (see, for instance, Eisenhart [1]). Riemannian manifolds and affine connections admitting very large groups of automorphisms have been studied by Egorov, Wang, Yano and others. The reader will find references on the subject in the book of Yano [2].

Note 11. Conformal transformations of a Riemannian manifold Let M be a Riemannian manifold with metric tensor g. A transformation rp of M is said to be conformal if rp*g = pg, where p is a positive function on M. If p is a constant function, cp is a homothetic transformation. If p is identically equal to 1, rp is nothing but an isometry. An infinitesimal transformation X of M is said to be conformal if Lxg = ag, where a is a function on M. It is homothetic if a is a constant function, and it is isometric if a = O. The local I-parameter group of local transformations generated by an infinitesimal transformation X is conformal if and only if X is conformal.

310

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

1. The group of conformal transformations of a connected, n-dimensional Riemannian manifold M is a Lie transformation group of dimension at most ten + I)(n + 2), provided n > 3. This can be proved along the following line. The integrability conditions of L xg = ag imply that the Lie algebra of infinitesimal conformal transformation X is of dimension at most ten + I)(n + 2) (cf. for instance, Eisenhart [1, p. 285J). By the theorem of Palais cited in Note 9, the group of conformal transformations is a Lie transformation group. In §3 of Chapter VI we showed that, for almost all Riemannian manifolds M, the largest connected group 'l{O(M) of affine transformations of M coincides with the largest connected group 3 0 (M) of isometries of M. For the largest connected group G:0( M) of conformal transformations of M, we have the following several results in the same direction. THEOREM

THEOREM

2. Let M be a connected n-dimensional Riemannian

manifoldfor which G:0 (M) =1= ,30(M). Then, (1) If M is compact, there is no harmonic p-Jorm of constant length for 1

3 (Goldberg and Kobayashi [2J); (3) If M is a complete Riemannian manifold of dimension n > 3 with parallel Ricci tensor, then M is isometric to a sphere (Nagano [1 J) ; (4) M cannot be a compact Riemannian manifold with constant nonpositive scalar curvature (Vano [2; p. 279J and Lichnerowicz [3; p. I34J). (3) is an improvement of the result of Nagano and Yano [1 J to the effect that if M is a complete Einstein space of dimension > 3 for which G:°(M) =1= ~O(M), then M is isometric to a sphere. Nagano [1 J made use of a result of Tanaka [1 J. On the other hand, it is easy to construct Riemannian manifolds (other than spheres) for which G:°(M) =1= 3°(M). Indeed, let M be a Riemannian manifold with metric tensor g which admits a I-parameter group of isometries. Let p be a positive function on M which is not invariant by this I-parameter group of isometries. Then, with respect to the new metric pg, this group is a I-parameter group of non-isometric, conformal transformations. To show that dim G:°(M) = ten

+ 1) (n + 2) for a sphere M

of

311

NOTES

dimension n, we imbed M into the real projective space of dimension n + 1. Let xo, xl, ... , xn +l be a homogeneous coordinate system of the real projective space Pn + l of dimension n + 1. Let M be the n-dimensional sphere in Rn+l defined by (yl)2 + ... + (yn+I)2 = 1. We imbed Minto Pn+ l by means of the mapping defined by Xo

=

1

V 2 (1 +yn+ I) ,

Xl

= y\ ... ,xn = yn,

The image of M in P n+ I is given by

(XI )2

+ ... +

(X n )2 - 2xOxn +1 =

o.

Let h be the Riemannian metric on P n+ I given by

* _ (~~!J (xi) 2) (~~!J (dX i)2) - (~~!J Xi dxi) 2 p h- 2 (~~!J (Xi)2)2 ' where p is the natural projection from Rn+2 - {OJ onto Pn + l . Then the imbedding M ---+ P n + l is isometric. Let G be the group of linear transformations of Rn+2 leaving the quadratic form (X I)2 + ... + (X n )2 - 2xOxn +1 invariant. Then G maps the image of M in Pn + l onto itself. It is easy to verify that, considered as a transformation group acting on M, G is a group of conformal transformations of dimension l(n + 1) (n + 2). The case n = 2 is exceptional in most of the problems concerning conformal transformations for the following reason. Let M be a complex manifold of complex dimension 1 with a local coordinate system z = x + b. Let g be a Riemannian metric on M which is of the form f(dx 2 + dy2) = f dz dz, where f is a positive function on ltv!. Then every complex analytic transformation of M is conformal.

•

SUMMARY OF BASIC NOTATIONS We summarize only those basic notations which are used most frequently throughout the book. 1. l;i' l;i,j, ... ' etc., stand for the summation taken over i or i,j, ... , where the range of indices is generally clear from the context.

2. Rand

e denote the real and complex number fields, respec-

tively. , xn ) Rn: vector space of n-tuples of real numbers (xl, en: vector space of n-tuples of complex numbers (zl, , zn) (x,y): standard inner product l;i xyi in Rn (l;i xjJ~ in cn) GL(n; R): general linear group acting on Rn gI(n; R): Lie algebra of GL(n; R) GL(n; C) : general linear group acting on Cn gI(n; C): Lie algebra of GL(n; C) O(n): orthogonal group o(n): Lie algebra of O(n) U(n) : unitary group u(n): Lie algebra of U(n) T~( V): tensor space of type (r, s) over a vector space V T (V): tensor algebra over V An: space R n regarded as an affine space A(n; R) : group of affine transformations of An a(n; R): Lie algebra of A(n; R)

3. M denotes an n-dimensional differentiable manifold. Tx(M): tangent space of M at x ~(M): algebra of differentiable functions on M :{ (M): Lie algebra of vector fields on M ;:r(M): algebra of tensor fields on M !) (M) : algebra of differential forms on M T(M): tangent bundle of M L(M): bundle of linear frames of M 313

314

FOUNDATIONS OF DIFFERENTIAL GEOMETRY

o(M) : bundle of orthonormal frames of M

(with respect to

a given Riemannian metric) o = (Oi): canonical I-form on L(M) or O(M) A (M): bundle of affine frames of M T;(M): tensor bundle of type (r, s) of M f* : differential of a differentiable mappingf f*w: the transform of a differential form w by f xt : tangent vector of a curve x t , 0 < t ::;;: 1, at the point Lx: Lie differentiation with respect to a vector field X

Xt

4. For a Lie group G, GO denotes the identity component and 9 the Lie algebra of G. La: left translation by a E G Ra : right translation by a E G ad a: inner automorphism by a E G; also adjoint representation in 9 P(M, G) : principal fibre bundle over M with structure group G

A *: fundamental vector field corresponding to A E 9 w = (wj): connection form o = (OJ): curvature form E(M, F, G, P) : bundle associated to P(M, G) with fibre F 5. For an affine (linear) connection r on M, o = (0j): torsion form rJk: Christoffel's symbols 'Y(x): linear holonomy group at x E M (x) : affine holonomy group at x E M Vx: covariant differentiation with respect to a vector (field) X

R: curvature tensor field (with components R;kZ) T: torsion tensor field (with components T;k) S: Ricci tensor field (with components R ii ) 2l(M): group of all affine transformations a(M): Lie algebra of all infinitesImal affine transformations ~(M): group of all isometries i(M): Lie algebra of all infinitesimal isometries

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INDEX

Canonical decomposition (= de Rham decomposition), 185, 192 flat connection, 92 form on L(M), 118 invariant connection, 110, 301 invariant Riemannian metric, 155 linear connection, 302 metric, 155 l-form on a group, 41 parameter of a geodesic, 162 Chart, 2 Christoffel's symbols (r~k)' 141 Compact-open topology, 46 Complete, linear connection, 139 Riemannian manifold, 172 Riemannian metric, 172 vector field, 13 Components of a linear connection, 141 of a l-form, 6 of a tensor (field), 21, 26 of a vector (field), 5 Conformal transformation, 309 infinitesimal, 309 Connection, 63 affine, 129 canonical, 110, 301 canonical flat, 92 canonical linear, 302 flat, 92 flat affine, 209 form, 64 generalized affine, 127 induced, 82 invariant, 81, 103 by parallelism, 262 Levi-Civita, 158

Absolute parallelism, 122 Adjoint representation, 40 Affine connection, 129 generalized, 127 frame, 126 holonomy group, 130 mapping, 225 parallel displacement, 130 parameter, 138 space, 125 tangent, 125 transformation, 125, 226 infinitesimal, 230 Alternation, 28 Analytic continuation, 254 Arc-length, 157 Atlas, 2 complete, 2 Automorphism of a connection, 81 of a Lie algebra, 40 of a Lie group, 40 Bianchi's identities, 78, 121, 135 Bundle associated, 55 holonomy, 85 homomorphism of, 53 induced, 60 of affine frames, 126 of linear frames, 56 of orthonormal frames, 60 principal fibre, 50 reduced, 53 sub-, 53 tangent, 56 tensor, 56 trivial, 50 vector, 113 325

326 Connection linear, 119 metric, 117, 158 Riemannian, 158 universal, 290 Constant curvature, 202 space of, 202, 204 Contraction, 22 Contravariant tensor (space), 20 Convex neighborhood, 149, 166 Coordinate neighborhood, 3 Covariant derivative, 114, 115, 122 differential, 124 differentiation, 115, 116, 123 tensor (space), 20 Covector, 6 Covering space, 61 Cross section, 57 adapted to a normal coordinate system, 257 Cubic neighborhood, 3 Curvature, 132 constant, 202 form, 77 recurrent, 305 scalar, 294 sectional, 202 tensor (field), 132, 145 Riemannian, 201 transformation, 133 Cylinder, 223 Euclidean, 210 twisted, 223 Derivation of ~(M), 33 of 1: (M), 30 of the tensor algebra, 25 Development, 131 Diffeomorphism, 9 Differential covariant, 124 of a function, 8 of a mapping, 8 Discontinuous group, 44 properly, 43

INDEX Distance function, 157 Distribution, 10 involutive, 10 Divergence, 281 Effective action of a group, 42 Einstein manifold, 294 Elliptic, 209 Euclidean cylinder, 210 locally, 197, 209, 210 metric, 154 motion, 215 subspace, 218 tangent space, 193 torus, 210 Exponential mapping, 39, 140, 147 Exterior covariant derivative, 77 covariant differentiation, 77 derivative, 7, 36 differentiation, 7, 36 Fibre, 55 bundle, principal, 50 metric, 116 transitive, 106 Flat affine connection, 209 connection, 92 canonical, 92 linear connection, 210 Riemannian manifold, 209, 210 Form curvature, 77 l-form,6 r-form, 7 tensorial, 75 pseudo-,75 torsion, 120 Frame affine, 126 linear, 55 orthonormal, 60 Free action of a group, 42 Frobenius, theorem of, 10 Fundamental vector field, 51

INDEX

Geodesic, 138, 146 minimizing, 166 totally, 180 Green's theorem, 281 G-structure, 288 Holomorphic, 2 Holonomy bundle, 85 Holonomy group, 71, 72 affine, 130 homogeneous, 130 infinitesimal, 96, 151 linear, 130 local, 94, 151 restricted, 71, 72 Holonomy theorem, 89 Homogeneous Riemannian manifold, 155, 176 space (quotient space), 43 symmetric, 301 Homomorphism of fibre bundles, 53 Homothetic transformation, 242, 309 infinitesimal, 309 Homotopic, 284 Ck- , 284 Horizontal component, 63 curve, 68, 88 lift, 64, 68, 88 subspace, 63, 87 vector, 63 Hyperbolic, 209 H ypersurface, 9 Imbedding, 9, 53 isometric, 161 Immersion, 9 isometric, 161 Indefinite Riemannian metric, 155 Induced bundle, 60 connection, 82 Riemannian metric, 154 Inner product, 24 Integral curve, 12

Integral manifold, 10 Interior product, 35 Invariant by parallelism, 262 connection, 81, 103 Riemannian metric, 154 Involutive distribution, 10 Irreducible group of Euclidean motions, 218 Riemannian manifold, 179 Isometric, 161 imbedding, 161 immersion, 161 Isometry, 46, 161, 236 infinitesimal, 237 Isotropy group, linear, 154 subgroup, 49 Killing-Cartan form, 155 Killing vector field, 237 Klein bottle, 223 Lasso, 73, 184, 284 Leibniz's formula, 11 Levi-Civita connection, 158 Lie algebra, 38 derivative, 29 differentiation, 29 group, 38 subgroup, 39 transformation group, 41 Lift, 64, 68, 88 horizontal, 64, 68, 88 natural, 230 Linear connection, 119 frame, 55 holonomy group, 130 isotropy group, 154 Local basis of a distribution, 10 coordinate system, 3

327

328 Locally affine, 210 Euclidean, 197, 209, 210 symmetric, 303 Lorentz manifold (metric), 292, 297 Manifold, 2, 3 complex analytic, 3 differentiable, 2, 3 oriented, orientable, 3 real analytic, 2 sub-, 9 Maurer-Cartan, equations of, 41 Metric connection, 117, 158 Mobius band, 223 Natural lift of a vector field, 230 Non-prolongable, 178 Normal coordinate system, 148, 162 I-parameter group of transformations, 12 subgroup, 39 Orbit, 12 Orientation, 3 Orthonormal frame, 60 Paracompact, 58 Parallel cross section, 88 displacement, 70, 87, 88 affine, 130 tensor field, 124 Partition of unity, 272 Point field, 131 Projection, covering, 50 Properly discontinuous, 43 Pseudogroup of transformations, 1, 2 Pseudotensorial form, 75 Quotient space, 43, 44 Rank of a mapping, 8 Real projective space, 52 Recurrent curvature, 305

INDEX

Recurrent tensor, 304 Reduced bundle, 53 Reducible connection, 81, 83 Riemannian manifold, 179 structure group, 53 Reduction of connection, 81, 83 of structure group, 53 Reduction theorem, 83 de Rham decomposition, 185, 192 Ricci tensor (field), 248, 292 Riemannian connection, 158 curvature tensor, 201 homogeneous space, 155 manifold, 60, 154 metric, 27, 154, 155 canonical invariant, 155 indefinite, 155 induced, 154 invariant, 154 Scalar curvature, 294 Schur, theorem of, 202 Sectional curvature, 202 Segment, 168 Simple covering, 168 Skew-derivation, 33 Space form, 209 Standard horizontal vector field, 119 Structure constants, 41 equations, 77, 78,118,120,129 group, 50 Subbundle, 53 SUbmanifold, 9 Symmetric homogeneous space, 301 locally, 303 Riemannian, 302 Symmetrization, 28 Symmetry, 301 Tangent affine space, 125

INDEX

Tangent bundle, 56 space, 5 vector, 4 Tensor algebra, 22, 24 bundle, 56 contravariant, 20 covariant, 20 field, 26 product, 17 space, 20, 21 Tensorial form, 75 pseudo-,75 Torsion form, 120 of two tensor fields of type (1,1), 38 tensor (field), 132, 145 translation, 132 Torus, 62 Euclidean, 210 twisted, 223

329

Total differential, 6 Totally geodesic, 180 Transformation, 9 Transition functions, 51 Trivial fibre bundle, 51 Twisted cylinder, 223 torus, 223 Type ad G, 77 of tensor, 21 Universal factorization property, 17 Vector, 4 bundle, 113 field, 5 Vertical component, 63 subspace, 63, 87 vector, 63 Volume element, 281