An Introduction to Differentiable Manifolds and …

1 An Introduction to Differentiable Manifolds and Riemannian Geometry An Introduction to Differentiable Manifolds and Riemannian Geometry Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Ellenberg and Hyman Earns Columbia University, New York RECENT TITLES GLEN E. BREDON. Introduction to Compact Transformation Groups WERNER GREUB, STEPHEN HALPERIN, AND RAY VANSTONE. Connections, Curvature, and Cohomology : Volume I, De Rham Cohomology of Manifolds and Vector Bundles. Vol- ume 11, Lie Groups, Principal Bundles, and Characteristic Classes. In preporolioti: Volume 111, Cohomology of Principal Bundles and Homogeneous Spaces XIA DAO-XING. Measure and Integration Theory of Infinite-Dimensional Spaces : Abstract Harmonic Analysis RONALD G. DOUGLAS. Banach Algebra Techniques in Operator Theory WILLARD MILLER, JR. Symmetry Groups and Their Applications ARTHUR A.

2 SAGLE AND RALPH E. WALDE. Introduction to Lie Groups and Lie Algebras T. BENNY RUSHING. Topological Embeddings JAMES W. VICK. Homology Theory : An Introduction to Algebraic Topology E. R. KOLCHIN. Differential Algebra and Algebraic Groups GERALD J. JANUSZ. Algebraic Number Fields A. S. B. HOLLAND. Introduction to the Theory of Entire Functions WAYNE ROBERTS AND DALE VARBERG. Convex Functions A. M. OSTROWSKI. Solution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations H. M. EDWARDS. Riemann s Zeta Function SAMUEL EILENBERC. Automata, Languages, and Machines : Volume A. In preparotioti: Volume B MORRIS HIRSCH AND STEPHEN SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS. Noneuclidean Tesselations and Their Groups J. DIEUDONN~. Treatise on Analysis, Volume IV FRANCOIS TREVES.

3 Basic Linear Partial Differential Equations WILLIAM M. BOOTHBY. An Introduction to Differentiable Manifolds and Riemannian Geometry BRAYTON GRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERT A. ADAMS. Sobolev Spaces 1,s PreParafion D. V. WIDDER. The Heat Equation IRVING E. SECAL. Mathematical Cosmology and Extragalactic Astronomy J. DIEUDOXN~. Treatise on Analysis, Volume 11, enlarged and corrected printing An Introduction to Differentiable Manifolds and Riemannian Geometry William M. Boothby DEPAHTMliNT OF MAI'HEMATIC'S WASHINGTON 1 JNlVEKSlTY ST. MISSOURI ACADEMIC PRESS New York San Francisco London 1975 A Siihsidiary of Harcourt Brace Jovnnovich. Publishers COPYRIGHT (Q 1975, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLlCATlON MAY BE REPRODUCED OR TRANSMITED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAC, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITUOUT PERMISSION IN WRITING FROM THE PUBLISHER.)

4 ACADEMIC PRESS, INC. 111 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1 Library of Congress Cataloging in Publication Data Boothby, William Munger, Date Riemannian geometry. An Introduction to Differentiable Manifolds and (Pure and applied mathematics, a series of monographs Bibliography: p. Includes index. 1. Differentiable Manifolds . 2. Riemannian mani- and textbooks ; no. folds. I. Title. 11. Series. [ ] 5 73-18967 ISBN 0-12-116050-5 AMS(M0S) 1970 Subject Classifications: 2241,5341,5741,5841 PRINTED IN THE UNITED STATES OF AMERICA This book is dedicated with love and afection to my wife, Ruth, and to my sons, Daniel, Thomas, and Mark. This Page Intentionally Left BlankContents 1. Introduction to Manifolds I. Preliminary Comments on R" 1 2. R" and Euclidean Space 4 3. Topological Manifolds 6 4.)

5 Further Examples of Manifolds . Cutting and Pasting I I 5. Abstract Manifolds . Some Examples 14 Notes 18 11. Functions of Several Variables and Mappings 1. Differentiability for Functions of Several Variables 20 2. Differentiability of Mappings and Jacobians 25 3. The Space of Tangent Vectors at a Point of R" 4. Another Definition of T,(R") 32 5. Vector Fields on Open Subsets of R" 37 6. The Inverse Function Theorem 41 7. The Rank of a Mapping 46 29 Notes 50 111. Differentiable Manifolds and Submanifolds I. The Delinition of ii Differentiable Manifold 52 2. Further Examples 60 3. Ditferentiable Functions and Mappings 65 4. Rank of a Mapping. Immersions 69 5. Submanifolds 75 6. Lie Groups 81 7. The Action of a Lie Group on a Manifold. Transformation Groups 8. The Action of a Discrete Group on a Manifold 9. Covering Manifolds 100 Notes 104 89 95 vii viii CONTENTS IV.

6 Vector Fields on a Manifold I. The Tangent Space at a Point of a Manifold 2. Vector Fields I I5 3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 4. The Existence Theorem for Ordinary Differential Equations 130 5. Some Examples of One-Parameter Groups Acting on a Manifold 13X 6. One-Parameter Subgroups of Lie Groups 145 7. The Lie Algebra of Vector Fields on a Manifold X. Frobenius' Theorem 156 9. Homogeneous Spaces 164 Appendix Partial Proof of Theorem 172 106 I22 149 Notes 171 V. Tensors and Tensor Fields on Manifolds 1. 7 -. 3. 4. 5. 6. 7. 8. Tangent Covectors 175 Covectors on Manifolds 176 Covector Fields and Mappings 17X Bilinear Forms. The Riemannian Metric Riemannian Manifolds as Metric Spaces Partitions of Unity 191 Tensor Fields 197 IXI 1x5 Some Applications of the Partition of Unity Tensors on a Vector Space Tensor Fields 199 Mappings and Covariant Tensors 200 The Symmetrizing and Alternating Transformations Multiplication of Tensors on a Vector Space 205 Multiplication of Tensor Fields 206 Exterior Multiplication of Alternating Tensors The Exterior Algebra on Manifolds Orientation of Manifolds and the Volumc Element Exterior Differentiation 217 221 Notes 225 193 197 201 Multiplication of Tensors 204 207 213 21 I An Application to Frobenius' Theorem Vl.

7 Integration on Manifolds I. Integration in R'. Domains of Integration 227 2. A Generalization to Manifolds 233 3. Integration on Lie Groups 241 4. Manifolds with Boundary 248 5. Stokes's Theorem for Manifolds with Boundary 256 6. Homotopy or Mappings. The Fundamental Group 263 7. Basic Properties of the Riemann Integral Integration on Riemannian Manifolds 237 228 Homotopy of Paths and Loops. The Fundamental Group The Homotopy Operator 274 265 Some Applications of Differential Forms. The de Rham Groups 271 CONTENTS ix X. Some Further Applications ofde Rham Groups 278 9. Covering Spaces and the Fundamental Group 2x6 The de Rham Groups of Lie Groups 282 Notes 292 VII. Differentiation on Riemannian Manifolds I. Dilferentiation of Vector Fields along Curves in R" 294 The Geometry of Space Curves Curvature of Plane Curves 301 Formulas for Covariant Derivatives 30X Vx,, Y and Differentiation of Vector Fields 310 Constant Vector Fields and Parallel Displacement The Curvature Tensor 321 The Riemannian Connection and Exterior Differential Forms 297 2.

8 Differentiation of Vector Fields on Submanifolds of R" 303 3. Differentiation on Riemannian Manifolds 313 4. Addenda to the Theory of DiRerentiation on a Manifold 319 321 324 5. Geodesic Curves on Riemannian Manifolds 326 6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 331 7. Some Further Properties of Geodesics 338 X. Symmetric Riemannian Manifolds 347 9. Some Examples 353 Notes 360 VIII. Curvature I. The Geometry of Surfaces in EJ 362 2. The Gaussian and Mean Curvatures of a Surface 370 3. Basic Properties of the Riemann Curvaturc Tensor 37X 4. The Curvature Forms and the Equations of Structure 385 5. Differentiation of Covariant Tensor Fields 391 6. Manifolds of Constant Curvature 399 Spaces of Positive Curvature 402 Spaces of Zero Curvature 404 Spaces of Constant Negative Curvature The Principal Curvatures at a Point of a Surface The Theorema Egregium of Gauss 366 373 405 Notes 410 INDEX 417 This Page Intentionally Left BlankPreface Apart from its own intrinsic interest, a knowledge of Differentiable Manifolds has become useful-ven mandatory-in an ever-increasing number of areas of mathematics and of its applications.

9 This is not too surprising, since Differentiable Manifolds are the underlying, if unacknow- ledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on Manifolds may be to those who have made it, it is not at all easy for most students. It usually involves many weeks of concentrated work with very general concepts (whose importance is not clear until later) during which the relation to the already familiar ideas in calculus and linear algebra become lost-not irretrievably, but for all too long. Simple but nontrivial examples that illustrate the necessity for the high level of abstraction are not easy to present at this stage, and a realization of the power and utility of the methods must often be postponed for a dismayingly long time.

10 This book was planned and written as a text for a two-semester course designed, it is hoped, to overcome, or at least to minimize, some of these difficulties. It has, in fact, been used successfully several times in preliminary form as class notes for a two-semester course intended to lead the student from a reasonable mastery of advanced (multivariable) calculus and a rudimentary knowledge of general topology and linear algebra to a solid fundamental knowledge of Differentiable Manifolds , including some facility in working with the basic tools of manifold theory: tensors, differential forms, Lie and covariant derivatives, multiple integrals, and so on. Although in overall content this book necessarily overlaps the several available excellent books on manifold theory, there are differences in presentation and emphasis which, it is hoped, will make it particularly suitable as an introduc- tory text.