Differential Geometry and Symmetric Spaces

D i Re rent i a1 Geometry and

Symmetric

Spaces

PURE A N D APPLIED MATHEMATICS A Series of Monographs and Textbooks

Edited by

PAULA. SMITHand SAMUEL EILENBERG Columbia University, New York I: ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) 11: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 111: HERBERT BUSEMANN AND PAULJ. KELLY.Projective Geometry and Projective Metrics. 1953 BERGMAN AND M. SCHIFFER. Kernel Functions and Elliptic IV: STEFAN Differential Equations in Mathematical Physics. 1953 V: RALPHPHILIPBOAS,JR. Entire Functions. 1954 VI: HERBERT BUSEMANN. T h e Geometry of Geodesics. 1955 VII: CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 Hu. Homotopy Theory. 1959 VIII: SZE-TSEN IX: A. OSTROWSICI. Solution of Equations and Systems of Equations. 1960 ~. of Modern Analysis. 1960 X: J. D I E U D O N NFoundations Curvature and Homology. 1962 XI: S. I. GOLDBERG. HELGASON.Differential Geometry and Symmetric XII: SIGURDUR Spaces. 1962 In preparation XIII: T . H. HILDEBRANDT. Introduction to the Theory of Integration.

Differential Geometry Symmetric Spaces 5ig u rdu r

Helgason

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

1962

ACADEMIC PRESS NEW YORK AND LONDON

COPYRIGHT 0 1962, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED

N O PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM BY PHOTOSTAT, MICROFILM OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTHAVENUE NEW YORK 3, N. Y.

United Kingdom Edition Published by

ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London,

IN.

1

Library of Congress Catalog Card Nztniber 62-13107

PRINTED I N THE UNITED STATES OF AMERICA

To Artie

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PREFACE According to its original definition, a symmetric space is a Riemannian manifold whose curvature tensor is invariant under all parallel translations. T h e theory of symmetric spaces was initiated by E. Cartan in 1926 and was vigorously developed by him in the late 1920’s. By their definition, symmetric spaces form a special topic in Riemannian geometry ; their theory, however, has merged with the theory of semisimple Lie groups. This circumstance is the source of very detailed and extensive information about these spaces. They can therefore often serve as examples for the testing of general conjectures. On the other hand, symmetric spaces are numerous enough and their special nature among Riemannian manifolds so clear that a properly formulated extrapolation to general Riemannian manifolds often leads to good questions and conjectures. T h e definition above does not immediately suggest the special nature of symmetric spaces (especially if one recalls that all Riemannian manifolds and all Kahler manifolds possess tensor fields invariant under the parallelism). However, the theory leads to the remarkable fact that symmetric spaces are locally just the Riemannian manifolds of the form Rn x GIK where Rn is a Euclidean n-space, G is a semisimple Lie group which has an involutive automorphism whose fixed point set is the (essentially) compact group K , and G / K is provided with a G-invariant Riemannian structure. E. Cartan’s classification of all real simple Lie algebras now led him quickly to an explicit classification of symmetric spaces in terms of the classical and exceptional simple Lie groups. On the other hand, the semisimple Lie group G (or rather the local isomorphism class of G) above is completely arbitrary; in this way valuable geometric tools become available to the theory of semisimple Lie groups. I n addition, the theory of symmetric spaces helps to unify and explain in a general way various phenomena in classical geometries. Thus the isomorphisms which occur among the classical groups of low dimensions are geometrically interpreted by means of isometries ; the analogy between the spherical geometries and the hyperbolic geometries is a special case of a general duality for symmetric spaces. On a symmetric space with its well-developed geometry, global function vii

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PREFACE

theory becomes particularly interesting. Integration theory, Fourier analysis, and partial differential operators arise here in a canonical fashion by the requirement of geometric invariance. Although these subjects and their relationship are very well developed in Euclidean space (Lebesgue integral, Fourier integral, differential operators with constant coefficients) the extension to general symmetric spaces leads immediately to interesting unsolved problems. T h e two types of nonEuclidean symmetric spaces, the compact type and the .noncompact type, offer different sorts of function-theoretic problems. T h e symmetric spaces of the noncompact type present no topological difficulties (the spaces being homeomorphic to Euclidean spaces) and their function theory ties up with the theory of infinite-dimensional representations of arbitrary semisimple Lie groups, which has made great progress in recent years. For the symmetric spaces of the compact type, on the other hand, the classical theory of finite-dimensional representations of compact Lie groups provides a natural framework, but the geometry of the spaces enters now in a less trivial fashion into their function theory. T h e objective of the present book is to provide a self-contained introduction to Cartan’s theory, as well as to more recent developments in the theory of functions on symmetric spaces. Chapter I deals with the differential-geometric prerequisites, and the basic geometric properties of symmetric spaces are developed in Chapter IV. From then on the subject is primarily Lie group theory, and in Chapter I X Cartan’s classification of symmetric spaces is presented. Although this classification may be considered as the culmination of Cartan’s theory, we have confined Chapter I X to proofs of general theorems involved in the classification and to a description of Cartan’s list. T h e justification of this notable omission is first that the usefulness of the classification for experimentation is based on its existence rather than on the proof that it exhausts the class of symmetric spaces; secondly this omission enabled us to include Chapter X (on functions on symmetric spaces) where it is felt that more open questions present themselves. At some places we indicate connections with topics in classical analysis, such as Fourier analysis, theory of special functions (Bessel, Legendre), and integral theorems for invariant differential equations. However, no account is given of the role of symmetric spaces in the theory of automorphic functions and analytic number theory, nor have we found it possible to include more recent topological investigations of symmetric spaces. Each chapter begins with a short summary and ends with an identification of sources as well as some comments on the historical development.

SUGGESTIONS TO READER

ix

The purpose of these historical notes is primarily to orient the reader in the vast literature and secondly they are an attempt to give credit where it is due, but here we must apologize in advance for incornpleteness as well as possible inaccuracies. This book grew out of lectures given at the University of Chicago 1958 and at Columbia University 1959-1960. At Columbia I had the privilege of many long and informative discussions with Professor HarishChandra; large parts of Chapters VIII and X are devoted to results of his. I am happy to express here my deep gratitude to him. I am also indebted to Professors A. Korhnyi, K. deleeuw, E. Luft and H. Mirkil who read large portions of the manuscript and suggested many improvements. Finally I want to thank my wife who patiently helped with the preparation of the manuscript and did all the typing.

Suggestions t o the Reader

Since this book is intended for readers with varied backgrounds we give here some suggestions for its use. Chapter I, Chapter IV, 5 1, and Chapter VIII, 5 1 - 5 3 can be read independently of the rest of the book. These parts would give the reader an incomplete but short and elementary introduction to modern differenctial geometry, with only advanced calculus and some point-set topology as prerequisites. Chapter I, 5 1 - 5 6, Chapter 11, and Chapter I11 can be read independently of the rest of the book as an introduction to semisimple Lie groups. However, Chapters I1 and I11 assume some familiarity with the elements of the theory of topological groups. Chapters I - I X require no further prerequisites. Chapter X, however, makes use of a few facts from Hilbert space theory and assumes some knowledge of measure theory. Each chapter ends with a few exercises. With a few possible exceptions (indicated with a star) the exercises can be worked out with methods developed in the text. T h e starred exercises are theorems which might have been included in the text, but were not found necessary for the subsequent chapters. S. Helgason

t “En hvatki es rnissagt es i fraeaurn pessurn, reynisk” ; [Ari Frbai: fslendingabbk (1 124)].

es skylt at hafa bat heldr, es sannara

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CONTENTS PREFACE . . SUGGESTIONS TO

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vii ix

CHAPTER I

Elementary Differential Geometry . . . . . . . . . . . . . . . . . . . . . .

1. Manifolds . 2. Tensor Fields 1. Vector Fields and 1-Forms . . . 2. The Tensor Algebra . . . . . 3. The Grassmann Algebra . . . . 4 . Exterior Differentiation . . . . 3 . Mappings . . . . . . . . I . The Interpretation of the Jacobian . . 2. Transformation of Vector Fields . . 3. Effect on Differential Forms . . . 4 . Affine Connections . . . . . . 5 . Parallelism . . . . . . . . 6 . T h e Exponential Mapping . . . . 7 . Covariant Differentiation . . . . . 8. The Structural Equations . . . . . 9 . The Riemannian Connection . . . . 10. Complete Riemannian Manifolds . . . 11. Isometries . . . . . . . . 12. Sectional Curvature . . . . . . 13. Riemannian Manifolds of Negative Curvature . . . 14. Totally Geodesic Submanifolds Exercises . . . . . . . . Notes . . . . . . . . .

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2 8 8 13 17 19 22 22 24 25 26 28 32 40 43 47 55 60 64 10 18 82 85

CHAPTER I1

Lie Groups and Lie Algebras I . T h e Exponential Mapping . . 1. The Lie Algebra of a Lie Group 2 . The Universal Enveloping Algebra 3. Left Invariant Affine Connections 4 . Taylor’s Formula and Applications

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2. Lie Subgroups and Subalgebras . . 3 . Lie Transformation Groups . . . 4 . Coset Spaces and Homogeneous Spaces .

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88 88 90 92 94 102 110 113

xii

CONTENTS

5. The Adjoint Group . 6 . Semisimple Lie Groups . . . Exercises . . . Notes .

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CHAPTER 111

Structure of Semisimple Lie Algebras 1. 2. 3. 4.

Preliminaries . . . . Theorems of Lie and Engel . Cartan Subalgebras . . . Root Space Decomposition . 5 . Significance of the Root Pattern . . . . 6. Real Forms 7. Cartan Decompositions . . Exercises . . . . . Notes . . . . . .

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161

CHAPTER IV

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

Affine Locally Symmetric Spaces . . . . Groups of Isometries . . . . . . Riemannian Globally Symmetric Spaces . . T h e Exponential Mapping and the Curvature . Locally and Globally Symmetric Spaces . . Compact Lie Groups . . . . . . Totally Geodesic Submanifolds . Lie Triple Systems . . . . . . . . . Exercises Notes . . . . . . . . . .

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189

CHAPTER V

Decomposition of Symmetric Spaces 1. 2. 3. 4. 5. 6.

Orthogonal Symmetric Lie Algebras . T h e Duality . . . . . . Sectional Curvature of Symmetric Spaces Symmetric Spaces with Semisimple Groups Notational Conventions . . . . Rank of Symmetric Spaces . . . Exercises . . . . . . . Notes . . . . . . . .

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of Isometries

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CHAPTER VI

Symmetric Spaces of the Noncompact Type 1. 2. 3. 4.

Decomposition of a Semisimple Lie Group . . Maximal Compact Subgroups and Their Conjugacy The Iwasawa Decomposition . . . . . Nilpotent Lie Groups . . . . . .

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...

CONTENTS

5. Global Decompositions 6. The Complex Case . Exercises . . . Notes . . . .

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234

CHAPTER VII

Symmetric Spaces of the Compact Type 1. The Contrast between the Compact Type and the Noncompact Type . . . . . 2. The Weyl Group . . . . . . . . 3. Conjugate Points . Singular Points . The Diagram . . . . . . . . . . . 4 . Applications to Compact Groups . . . 5. Control over the Singular Set . . . . . . . . . . 6. The Fundamental Group and the Center . . . . . . . 7. Application to the Symmetric Space U / K . . . . . . . 8. Classification of Locally Isometric Spaces . . . . . . . 9. Appendix . Results from Dimension Theory . . . . . . . Exercises . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . .

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24 1

. 243

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250 254 260 264 271 273 275 278 280

Almost Complex Manifolds . . . . . . . . . . . Complex Tensor Fields . T h e Ricci Curvature . . . . . . . Bounded Domains . The Kernel Function . . . . . . . . Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type . . . . . . 5 . Irreducible Orthogonal Symmetric Lie Algebras . . . . . . . . . 6. Irreducible Hermitian Symmetric Spaces 7. Bounded Symmetric Domains . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . .

281 285 293 301 306 310 311 322 325

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CHAPTER Vlll

Hermitian Symmetric Spaces 1. 2 3. 4.

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CHAPTER IX

O n the Classification of Symmetric Spaces 1. Reduction of the Problem . . . . . . . . . . . 2. Automorphisms . . . . . . . . . . . . . . 3. Involutive Automorphisms . . . . . . . . . . . . 4 . 8 . Cartan’s List of Irreducible Riemannian Globally Symmetric Spaces . 1 . Some Matrix Groups and Their Lie Algebras . . . . . . . 2 . The Simple Lie Algebras over C and Their Compact Real Forms . The Irreducible Riemannian Globally Symmetric Spaces of Type 11 and Type I V 3 . The Involutive Automorphisms of Simple Compact Lie Algebras . The Irreducible Globally Symmetric Spaces of Type I and Type 111 . . . 4 . Irreducible Hermitian Symmetric Spaces . . . . . . . 5 . Two-Point Homogeneous Spaces . Symmetric Spaces of Rank One . Closed . . . . . . . . . . . . . . Geodesics . . Exercises . . . . . . . Notes . . . . . .

326 331 334 339 339 346 347 354 355 358 359

xiv

CONTENTS

CHAPTER X

Functions on Symmetric Spaces I . Integral Formulas . . . . . . . . . 1. Generalities . . . . . . . . . 2 . Invariant Measures on Coset Spaces . . . . 3. Some Integral Formulas for Semisimple Lie Groups . 4 . Integral Formulasfor the Cartan Decomposition . . 5 . The Compact Case . . . . . . . . . . . . . . 2 . Invariant Differential Operators 1 . Generalities. The Laplace-Beltrami Operator . . . 2 . Invariant Differential Operators on Reductive Coset Spaces 3. The Case of a Symmetric Space . . . . . . . . 3. Spherical Functions . Definition and Examples . . . 4. Elementary Properties of Spherical Functions 5 . Some Algebraic Tools . . . . . . . . 6 . T h e Formula for the Spherical Function . . . . 1. The Euclidean Type . . . . . . . . 2 . The Compact Type . . . . . . . . 3. The A'oncompact Type . . . . . . . 7 . Mean Value Theorems . . . . . . . . 1 . The Mean Value Operators . . . . . . 2. Approximations by Analytic Functions . . . . 3. The Darboux Equation in a Symmetric Space . . 4. Poisso?i's Equation in a Two-Point Homogeneous Space . Exercises Notes .

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BIBLIOGRAPHY . . . . . LISTOF NOTATIONAL CONVENTIONS SYMBOLS FREQUENTLY USED . . AUTHOR INDEX . . . . . SUBJECT INDEX . . . . .

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CHAPTER I

ELEMENTARY DIFFERENTIAL GEOMETRY This introductory chapter divides in a natural way into three parts: 91-93 which deal with tensor fields on manifolds, 94-98 which treat general properties of affine connections, and $9- $14 which give an introduction to Riemannian geometry with some emphasis on topics needed for the later treatment of symmetric spaces. 51-93. When a Euclidean space is stripped of its vector space structure and only its differentiable structure retained, there are many ways of piecing together domains of it in a smooth manner, thereby obtaining a so-called differentiable manifold. Local concepts like a differentiable function and a tangent vector can still be given a meaning whereby the manifold can be viewed “tangentially,” that is, through its family of tangent spaces as a curve in the plane is, roughly speaking, determined by its family of tangents. This viewpoint leads to the study of tensor fields, which are important tools in local and global differential geometry. They form an algebra D(M), the mixed tensor algebra over the manifold M . The alternate covariant tensor fields (the differential forms) form a submodule X ( M ) of D(M) which inherits a multiplication from B ( M ) , the exterior multiplication. The resultifig algebra is called the Grassmann algebra of M . Through the work of 8. Cartan the Grassmann algebra with the exterior differentiation d has become an indispensable tool for dealing with submanifolds, these being analytically described by the zeros of differential forms. Moreover, the pair ( % ( M )d, ) determines the cohomology of IIZ via de Rham’s theorem, which however will not be dealt with here. 94-58. The concept of an affine connection was first defined by Levi-Civita for Riemannian manifolds, generalizing significantly the notion of parallelism for Euclidean spaces. On a manifold with a countable basis an affine connection always exists (see the exercises following this chapter). Given an affine connection on a manifold M there is to each curve y ( t ) in M associated an isomorphism between Thus, an affine connection makes it any two tangent spaces and possible to relate tangent spaces a t distant points of the manifold. If the tangent vectors of the curve y ( t ) all correspond under these isomorphisms we have the analog of a straight line, the so-called geodesic. The theory of affine connections M under which straight mainly amounts to a study of the mappings Exp, : M , lines (or segments of them) through the origin in the tangent space M , correspond to geodesics through p in M . Each mapping Exp, is a diffeomorphism of a neighborhood of 0 in M , into M , giving the so-called normal coordinates at p . Some other local properties of Exp, are given in 96, the existence of convex neighborhoods and a formula for the differential of Exp,. An affine connection gives rise to two important tensor fields, the curvature tensor field an$ the torsion tensor field which in turn describe the affine connection through E. Cartan’s structural equations [ ( 6 ) and (7), $8)].

my(t2).

--f

1

,2

[Ch. I

ELEMENTARY DIFFERENTIAL GEOMETRY

$9-914. A particularly interesting tensor field on a manifold is the so-called Riemannian structure. This gives rise to a metric on the manifold in a canonical fashion. It also determines an affine connection on the manifold, the Riemannian connection; this affine connection has the property that the geodesic forms the shortest curve between any two (not too distant) points. The relation between the metric and geodesics is further developed in $9-$10. The treatment is mainly based on the structural equations of 8.Cartan and is independent of the Calculus of Variations. The higher-dimensional analog of the Gaussian curvature of a surface was discovered by Riemann. Riemann introduced a tensor field which for any pair of tangent vectors at a point measures the corresponding sectional curvature, that is, the Gaussian curvature of the surface generated by the geodesics tangent to the plane spanned by the two vectors. Of particular interest are Riemannian manifolds for which the sectional curvature always has the same sign. The irreducible symmetric spaces are of this type. Riemannian manifolds of negative curvature are considered in $13 owing to their importance in the theory of symmetric spaces. Much progress has been made recently in the study of Riemannian manifolds whose sectional curvature is bounded from below by a constant > 0. However, no discussion of these is given since it is not needed in later chapters. The last section deals with totally geodesic submanifolds which are characterized by the condition that a geodesic tangent to the submanifold at a point lies entirely in it. In contrast to the situation for general Riemannian manifolds, totally geodesic submanifolds are a common occurrence for symmetric spaces.

$1. Manifolds

Let Rm and Rn denote two Euclidean spaces of m and n dimensions, respectively. Let 0 and 0‘ be open subsets, 0 C Rm, 0‘ C Rn and suppose p is a mapping of 0 into 0’. T h e mapping p is called dzflerentiable if the coordinates y i ( p ( p ) ) of p ( p ) are differentiable (that is, indefinitely differentiable) functions of the coordinates x , ( ~ ) p, E 0. T h e mapping p is called analytic if for each point p E 0 there exists a neighj n) in m variables such borhood U of p and n power series P j ( I that Y ~ ( T ( Q )= ) P i ( ~ l ( q) .,(PI, ..., xm(q) - .,(PI) (1 G j n ) for q E U. A differentiable mapping p : 0 -+ 0’ is called a dtfleomorphism of 0 onto 0’ if p ( 0 ) = 0‘, p is one-to-one, and the inverse mapping p-l is differentiable. I n the case when n = 1 it is customary to replace the term “mapping” by the term “function.” An analytic function on Rm which vanishes on an open set is identically 0. For differentiable functions the situation is completely different. In fact, if A and B are disjoint subsets of Rm, A compact and B closed, then there exists a differentiable function p which is identically 1 on A and identically 0 on B. T h e standard procedure for constructing such a function q is as follows:

<
0 for eachp E M . T h e functions y , = #a/# have the desired properties (i) and (ii). T h e system {ya}asAis called a partition of unity subordinate to the covering { Ua}aeA.

v,

>

§ 2. Tensor Fields

1 . Vector Fields and I-Forms

Let A be an algebra over a field K . A derivation of A is a mapping D : A --t A such that for a, /3 E K , f , g E A ; Pg> = aDf PDg (i> D(.f [ii) W f g ) = f(%> (Df)g for f,g E A.

+

+

+

§ 21

2. Tensor Fields

9

Definition. A vector field X on a C" manifold is a derivation of the algebra Cm(M). Let W (or W ( M ) )denote the set of all vector fields on M . Iff E Cm(M) and X , Y E,)'&(I% then f X and X Y denote the vector fields

+

fX : R - f ( X g ) ,

x+

Y:g+Xg+

Yf,

g g

E E

Crn(M), C"(M).

This turns V ( M ) into a module over the ring 5 = Cm(M). If X , Y E W ( M ) ,then X Y - Y X is also a derivation of Cm(M)and is denoted by the bracket [ X , Y ] .As is customary we shall often write B(X) Y = [ X , Y ] .T h e operator 6 ( X ) is called the Lie derivative with respect to X . The bracket satisfies the Jucobi identity [ X , [ Y ,Z ] ] [ Y ,[Z, X I ] [Z, [ X , Y ] ]= 0 or, otherwise written B ( X ) ( [ Y ,Z])= [O(X)Y , Z ] [ Y , B(X)21. I t is immediate from (ii) that iff is constant and X E W,then Xf = 0. Suppose now that a function g E C"(M) vanishes on an open subset V C M . Let p be an arbitrary point in V . According to Lemma 1.2 there exists a function f E C m ( M ) such that f ( p ) = 0, and ,f = 1 outside V . Then g = f g so

+

+ +

which shows that Xg vanishes at p . Since p was arbitrary, Xg = 0 on V. We can now define X f on V for every function f E C"(V). If p E V , select f E C m ( M )such that f and f coincide in a neighborhood of p and put ( X j )( p ) = (Xf) (p). T h e consideration above shows that this is a valid definition, that is, independent of the choice o ff. This shows that a vector field on a manifold induces a vector field on any open submanifold. On the other hand, let Z be a vector field on an open submanifold V C M and p a point in V . Then there exists a vector field 2 on M and an open neighborhood N , p E N C V such that 2 and Z induce the same vector field on N . In fact, let C be any compact neighborhood of p contained in V and let N be the interior of C. Choose v', E Cm(M) of compact support contained in V such that $ = 1 on C. For any g E C m(M ),let g v denote its restriction to V and define z g by

Then g -+ z g is the desired vector field on M . Now, let ( U , y ) be a local chart on M , X a vector field on U , and let p be an arbitrary point in U. We put y(q) = (x,(q), ..., x,,(q)) ( q E U ) ,

I0

[Ch. I

ELEMENTARY DIFFERENTIAL GEOMETRY

and f * = f o y-* for f E C m ( M ) .Let I/' be an open subset of c' such that ?(I/') is an open ball in Rm with center p;(p) = (ul, ...,a,). If ( x l , ..., x,,,) E p;( V ) , we have

f*(%*.., =f * ( a l ,

..., a,)

= f*(al,

..., a,)

+ s: if*(., + f ( x , 4,..., + a,)) dt + 2 a,) rf:(al + t(xl - 4, ..., a , + -

a,

t(x,,

-

m

t(xm - am))dt.

(xj -

j=1

(Here fj* denotes the partial derivative o f f * with respect to the j t h argument.) Transferring this relation back to M we obtain

where gi

E

Cm(V), (1

< i < m], and

It follows that

T h e mapping f --t (af*/ax,) o y ( f E Cm(U ) )is a vector field on U and is dknoted a/axi. We write af/ax, instead of a/axi(f). Now, by ( 2 )

a x =2 (XXt)-i=l axi m

on U .

(3)

<
;(p) are dual to each other under the nondegenerate bilinear form ( , ) on zi(p)x D X p ) defined by the formula

where ei, e; are members of a basis of M,, fi,fl are members of a dual basis of &I;. I t is then obvious that the formula holds if ei, ei are arbitrary fl are arbitrary elements of Mg. I n particular, elements of MI) and fi, the form ( ) is independent of the choice of basis used in the definition. Each T E Di(M) induces an @linear mapping of D:(M) into 8 given by the formula )

for S E %:(M). ( T ( S ) )(P) = (TV, S , I > If T ( S ) = 0 for all S E D:(M), then T, = 0 for all p E M , so T = 0. Consequently, Di(M) can be regarded as a subset of ( Q ( M ) ) * .We have now the following generalization of Lemma 2.3.

>

The module is the dual of D:(M) (Y, s 0). Except for a change in notation the proof is the same as that of Lemma 2.3. T o emphasize the duality we sometimes write ( T , S ) instead of T ( S ) ,( T E Di,S E D:). Let (or D(M)) denote the direct sum of the 8-modules Di(M), Lemma 2.3'.

m

D

=

2 3;. 1'. s=o

Similarly, if

p

E

M we consider the direct sum m

16

[Ch. I

ELEMENTARY DIFFERENTIAL GEOMETRY

T h e vector space a(p) can be turned into an associative algebra over R as follows: Let a = el @ ... @ e , @ f l @ ... @ fs, b = e; 6 ... 6 e i @ f; ... @ f:, where ei, el are members of a basis for M,, fj, are members of a dual basis for Mg. Then a @ b is defined by the formula

fi

a

0b

=

el @

... 0er @ e l 0... @ eb Of, 0... Of, Of; @ ... Of:.

We put a 6 1 = a, 1 @ b = b and extend the operation (a, b) -+a @ b to a bilinear mapping of D(p) x D ( p ) into D(p). Then D(p) is an associative algebra over R. T h e formula for a @ b now holds for arbitrary elements ei, ei E M p and fj, E M;. Consequently, the multiplication in %(p) is independent of the choice of basis. T h e tensor product 6 in D is now defined as the 3-bilinear mapping (S, T ) -+ S @ T of 3 x il, into D such that

fi

( S 0T ) ,

=

s E q,T E a",p E M .

0T , ,

% I

This turns the @module

into a ring satisfying

f(S@ T ) = fS @ 1' = S @fl' for f E 8, S , T E 3.In other words, D is an associative algebra over the ring 8. T h e algebras 2, and D ( p ) are called the mixed tensor algebras over A4 and M p , respectively. T h e submodules n

00

are subalgebras of 3 (also denoted %*(Ad) and

a*(&'))

and the subspaces

s=o

T=O

are subalgebras of D ( p ) . Now let Y, s be two integers 3 1, and let i, j be integers such that 1 i Y, 1 < j s. Consider the R-linear mapping Cii: D i ( p ) -+ Di:;(p) defined by

<