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: 6319209 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 selfcontained 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, MontgomeryZippin, 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 selfcontained. 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 socalled 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 1110. Proposition 6.2 requires Section 1II4. Corollary 2.4 requires Proposition 7.4 in Chapter III. Theorem 4.1, (4) requires Section 1II4 and Proposition 6.2 in Chapter III. V: Proposition 2.4 requires Section 1117. VI: Theorem 3.3 requires Section V2. 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 V2 .
.
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, thenfl 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 nonempty, then the homeomorphism f' f of fl(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 ofntuples 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 orientationpreserving transformations of class Cr of Rn. Let Cn be the space of ntuples 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 nonempty, 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
fIJl: fIJ( U n Ui)
+
flJt (U nUt)
is an element of r whenever U n U i is nonempty. 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 ndimensional 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. SimilarIy, 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 ntuple 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)Yg(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 IJorm (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 Iform 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 Iform 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 Iforms.
I.
7
DIFFERENTIABLE MANIFOLDS
A Iform 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 rform 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 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:
.. ~r1 ( CK)~1' J1 ••• J8 1

~
k
Ki1 ··· k ... i r h .. • k ••. j 8
'
where the superscript k appears at the ith position and the subscript k appears at the jth 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 rlinear mappings of V X . . . X V into F. PROPOSITION
2.9.
Tr is isomorphic, in a natural way, to the vector space of all rlinear 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 rlinear 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 rlinear 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 rlinear 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 rlinear mappings of V X . . . X V into V. QED. With this interpretation, any tensor K of type (1, r) is an rlinear 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 rlinear 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 typepreserving, 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 typepreserving, 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 (Ymodule 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 rlinear 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 rlinear 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 rlinear 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 rlinear 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 rlinear 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 skewsymmetric: 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 skewsymmetric for any K and that K is skewsymmetric 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 Tcpl(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( cpI(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(Kcpl(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 Iparameter 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 Iparameter 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 typepreserving: 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 Iparameter 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{Jtt = f{Jt is a local1parameter 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 typepreserving, 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 Iform 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 Iparameter group of local transformations generated by a vector field X. For any tensor field K, we have PROPOSITION
Proof.
By definition,
LxK =
Iim~ lO
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=fPs1oL}{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) , fPy = 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., skewsymmetric 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. skewderivation) 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 skewderivation 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 skewderivation of degree 1. As a general result on derivations and skewderivations 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 skewderivation of degree k', then [D, D'] is a skewderivation of degree k + k'. PROPOSITION
34
FOUNDATIONS OF DIFFERENTIAL GEOMETRY
..
(c) If D and D' are skewderivations of degree k and k' respectively, then DD' + D'D is a derivation of degree k + k'. (d) A derivation or a skewderivation 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 = (cpl) *w and, hence, rp commutes with d. Let CPt be a local Iparameter 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 Iform 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 skewderivation (, 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 skewderivation 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 rform 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 skewderivation 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 skewderivation 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 Iform 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 rform,
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 Iform, then A
(dw) (X, Y)  !{X(w(Y))  Y(w(X))
w([X, Y])}, X, Y € X (.A1) .
If w
is a 2form, 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 rform 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 vectorspace 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) Iparameter group of transformations ofG. Indeed, if 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 MontgomeryZippin [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 compactopen topology. Proof. We recall that the compactopen 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 gJl 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))
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 nonempty, 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 . . noo 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 koo 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) koo cP:;kI ( cP (x) ), x) = koo = koo 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, 1pI 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
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 cpl(X) for every x E M. Proof. For any x E M, lety = cpl(X). Then LEMMA
d(cp;;I(X), cpl(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 compactopen 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 compactopen topology), then Lemma 6 implies that cp;;I(X) + cpl(X) for every x E M, and the convergence is uniform on every compact subset by Lemma 5. Thus cp;;1 + cpl in G. This means that the mapping G + G which maps cP into cpl 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 compactopen 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 (aI)) A E g. Proof. Since A* is induced by the Iparameter group of transformations Rat where at = exp tA, the vector field (R a ) *A * is induced by the Iparameter group of transformations RaRatRal = Ralata by Proposition 1.7. Our assertion follows from the fact that alata is the Iparameter group generated by (ad (aI))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 1TI ( 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 peq uivalence class of (rx, x, a) into the peq 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 pequivalence 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 7T1 ( 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 7T1 ( Uct ). This is possible since every point of P is contained in 7T1 ( 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 € 7T1 ( U) is the pequivalence 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 7T1 ( 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 aLie 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 ) ~ 7T1 ( 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, al~) 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 7T1 ( U) is isomorphic to U X G. Identifying 7T1 ( 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,
bl~)
for (x, a,
~)
E
U
X
G
X
F
and
bEG.
It follows that the isomorphism 7T1 ( 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 € 1T1 (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 u1 where u is an arbitrarily fixed point of 1T1 (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. 2833]. 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 nonsingular 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 nonsingular 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 Pl(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 fll( 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 fll( a(x)) is nonempty. 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 nonempty; 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 ndimensional 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) = (uIX, 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 fIP. 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 pl( 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. 6771] or Hu [1, pp. 8997]. 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 ndimensional 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 ndimensional 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 ndimensional 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 Iform 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 (aI)w, that is, w((Ra)*X) = ad (ai) . 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 gvalued 1form 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 (aI) . 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 (aI)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 7T1 ( 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 7T1 ( 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 gvalued) canonical Iform on G defined in §4 of Chapter I (p. 41). For each nonempty Ua. n Uf3 , define a gvalued Iform ()a.f3 on Ua. n Uf3 by ()a.f3 = "p:f3(). For each
C't,
define a gvalued Iform Wa. on Ua. by
PROPOSITION
1.4.
The forms ()a.f3 and Wa. are subject to the conditions:
Conversely, for every family of gvalued lJorms {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 nonempty, 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 (aI) 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 gvalued Iform 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 Iform on U determines uniquely a connection form on U X G. Thus Lemma 2 is reduced to the extension problem for gvalued Iforms on U. If {A i} is a basis for g, then w = ~ wi Ai where each wi is a usual Iform. Thus it is sufficient to consider the extension problem of usual Iforms on U. Let xl, ... , x n be a local coordinate system in U. Then every Iform 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 7T1 ( 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(at1)w(v t ) + at1d t , where at1d 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 Iparameter subgroup of G generated by A. Our differential equation dta t 1 = Y t is hence a generalization of the differential equation for Iparameter 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 Iparameter 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  Sil < 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 = btslaSl fors l < t ::;;: S2; ... ; f{Ju(e, Skl) = (c u, Skl + u) is defined for 0 ::;;: u < Sk  Skl' where cuC;;1 = Yu + skl , and we define at = Ct skl askl , 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 cpl 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.
72
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 1T1(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 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 CIcurve 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 CIcurve. 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 Clhorizontal 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
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.
II.
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 1Jorm 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 ojlex, 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 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.
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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 mth covariant differential vm K is defined inductively to be v(vmlK). We use the notation (vmK) (;X l ; . .. ; X m l ; X m) for (V x m (vml 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 ntuples 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,
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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 Iform 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 Iform on L(M) and cp is an Rn_ valued Iform 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 Iform 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 Rnvalued. 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 Iform 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 (aI) (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 Rncomponents 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 Rnvalued Iform 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 (Y~, Z~).
=
X~
f
corresponds to V x T. Hence, (X:f)(r;,~)
u1((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*)) = uI((VxT)(Y, Z)). It is therefore sufficient to prove that
28([X*, Y*]w Z:) = uI(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*))
=
tuI((VxR)(Y, Z)).
As in the proof of the first identity, we have
o.([X*, Y*]u, Z:) = iU1(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')) = u1 ((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)) = ut1X 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 Iparameter 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 'TTl(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*) = Ul(~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 Iform 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 Iform 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 Rnvalued 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 Iforms ()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 Iforms and functions. Since the torsion form 0 and the curvature form n are tensorial forms, they can be expressed in terms of n Iforms ()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 * TiJOle = T JiOle ,
i i a *Rjkl = 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 Iparameter 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 neighborhooElY 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
11,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 lpl(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 kI is a linear combination of tensor fields of type A j , 0 <j < k  1. We have (VkR)(X, Y; VI; ... ; V k )
=
VVk((VkIR) (X, Y; VI;··· ; Vk I ))
 (VkIR) (VV kX , Y; VI;  (VfcIR) (X, Vv kY ; VI;

~r:t
(VkIR) (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 ndimensional 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 ) = ~1j (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(a1X, a1Y), 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 KillingCartan 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 KillingCartan 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 ) = 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 ndimensional 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 LeviCivita 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 skewsymmetric 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 skewsymmetric. 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 skewsymmetric 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 (a1) (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 Iforms 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) = utl(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 skewsymmetric. 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 nonempty}. 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
c sin
ilX
for small s.
On the other hand, h is a Euclidean metric and hence
;s
0 such that exp: N(x; 2r) ~ U(x; 2r) is a diffeomorphism. Let {xt, x~, ... } be the set pl(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) pl( 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* € pl( 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 thatpl(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 pl(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 nonprolongeable 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 nonprolongeab1e. 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 nontrivial 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 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 ndimensional 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 such that expx is a diffeomorphism of N(x; r) onto U(x; r). Since the differential (expx)* is nonsingular 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 ndimensional 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 ndimensional 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), UlX 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 skewsymmetric 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.
202
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
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
0 (resp. 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. 120122]). This shows that if X, Y € S, then X + Y € S. There exists a curve W t such that W t2 = xtYtXtYt1 and we have Wo = [xo,Yo] (cf. Chevalley [1, pp. 120122] 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 nonsingular. 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 ndimensional 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 nonsingular and Akl = AI. 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 skewsymmetric, 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 skewsymmetric 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 skewsymmetric 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 Idimensional. Let A =F 0 and BE 3. Since A, B are skewsymmetric 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 skewsymmetric linear transformation wi.th J2 = In. Now J is representable by a matrix which is a block form, each
01) (
block being 1
0 ' with respect to a certain orthonormal basis
ofRn. The Iparameter 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 skewsymmetric 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 ndimensional differentiable manifold. An nform 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. 161167]). 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 Iparameter 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 (qJt1*w)] = Lxw. Hence t t=O
o =[ddf qJt1*w] t M t=O = 
r Lxw =
JM
=J [dd (qJt1*w)] M t t=O
1
div X dv.
M
QED.
Remark 1. The above formula is valid for a noncompact 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 nform 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 Ckhomotopic, 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 Ckcurves x(t) andy(t) are homotopic to each
other, then they are piecewise Ckhomotopic. 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 Ckhomotopy 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 Ulasso 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 Ulassos. (b) If the curve is moreover piecewise Ck, then each Ulasso in the product can be chosen to be of the form fl 1 • a . fl, where fl is a piecewise Ckcurve and a is a Ckcurve. 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 UIassos
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 3dimensional 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 LeviCivita and Ricci [1] in 1901. Covariant differentiation.which was formally introduced in this tensor calculus was given a geometric interpretation by LeviCivita [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], AmbroseSinger [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 AmbroseSinger [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( 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 ndimensional manifold with a linear connection r. A nonzero tensor field K of type (r, s) on M is said to be recurrent if there exists a Iform 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 nontrivial, 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 nonflat 2dimensional 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 Rn2 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 Iparameter 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 finitedimensional, 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 finitedimensional. 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, ndimensional 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, ndimensional Riemannian dim ~(M) = in(n + 1), then M is isometric to one of
manifold. If the following spaces of constant curvature: (a) An ndimensional Euclidean space R n ; (b) An ndimensional sphere Sn ; (c) An ndimensional real projective space Sn/ {± I} ; (d) An ndimensional, 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, ndimensional 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 Iparameter 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, ndimensional 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 ndimensional Riemannian
manifoldfor which G:0 (M) =1= ,30(M). Then, (1) If M is compact, there is no harmonic pJorm 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 Iparameter group of isometries. Let p be a positive function on M which is not invariant by this Iparameter group of isometries. Then, with respect to the new metric pg, this group is a Iparameter group of nonisometric, 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 ndimensional 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 ntuples of real numbers (xl, en: vector space of ntuples 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 ndimensional 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 Iform 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 lform on a group, 41 parameter of a geodesic, 162 Chart, 2 Christoffel's symbols (r~k)' 141 Compactopen topology, 46 Complete, linear connection, 139 Riemannian manifold, 172 Riemannian metric, 172 vector field, 13 Components of a linear connection, 141 of a lform, 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 LeviCivita, 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 Arclength, 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 lform,6 rform, 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 Gstructure, 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 KillingCartan form, 155 Killing vector field, 237 Klein bottle, 223 Lasso, 73, 184, 284 Leibniz's formula, 11 LeviCivita 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 MaurerCartan, equations of, 41 Metric connection, 117, 158 Mobius band, 223 Natural lift of a vector field, 230 Nonprolongable, 178 Normal coordinate system, 148, 162 Iparameter 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 Skewderivation, 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