Lectures on Differential Geometry

368 Pages·2000·11.63 MB·English

by  Shiing-Shen Chern

Lectures on
Differential
Geometry
SERIES ON UNIVERSITY MATHEMATICS
Editors:
W Y Hsiang : Department of Mathematics, University of California, Berkeley,
CA 94720, USA
T T Moh : Department of Mathematics, Purdue University, W. Lafayette
IN 47907, USA
e-mail: ttmQmath.purdue.edu Fax: [phone]
S S Ding : Department of Mathematics, Peking University, Beijing, China
M C Kang : Department of Mathematics, National Taiwan University,
Taiwan, China
M Miyanishi : Department of Mathematics, Osaka University, Toyonaka,
Osaka 560, Japan
Published
Vol. 1: Lectures on Differential Geometry
S S Chern, W H Chen and K S Lam
Vol. 5: Algebra
T T Moh
Vol. 6: A Concise Introduction to Calculus
W Y Hsiang
Vol. 7: Number Theory with Applications
W C Winnie Li
Vol. 8: Analytical Geometry
I Vaisman
Forthcoming
Vol. 2: Lectures on Lie Group
W Y Hsiang
Vol. 3: Classical Geometries
W Y Hsiang
Vol. 4: Foundation of Linear Algebra
W Y Hsiang & T T Moh
Lectures on
Differential
Geome trv
J
S. S. Chern
University of California, USA
W. H. Chen
Peking University, China
K. S. Lam
California State Polytechnic University, Pomona, USA
World Scientific
Singapore New Jersey. London Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P 0 Box 128, Farrer Road, Singapore 912805
LISA ofice: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Chern, Shiing-Shen, 191 1-
[Wei Fen chi ho chiang 1. English]
Lectures on differential geometry / S.S. Chern, W.H. Chen, K.S.
Lam.
p. cm.
ISBN 9810234945 ISBN 9810241828 (pbk)
1. Geometry, Differential. I. Chen, Wei-hum 11. Lam, K. S.
(Kai Shue), 1949- . 111. Title.
QA641.C4913 1998
516 .3'6--dc21 98-2203 I
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
First published 1999
Reprinted 2000
Copyright 0 2000 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means,
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photocopy is not required from the publisher.
This book is printed on acid and chlorine free paper.
Printed in Singapore by World Scientific Printers
V
Preface
The present book is a translation and an expansion of an introductory text
based on a lecture series delivered in Peking University (the People’s Republic
of China) in 1980 by a renowned leader in differential geometry, S.S. Chern.
The original Chinese text resulted from the efforts of several colleaguesa, and,
in its final formb, was compiled by Wei-Huan Chen of Peking University. The
final rendition of this English translation is carried out by the undersigned,
under the guidance of S.S. Chern. It has been revised from a preliminary draft
by Hung-Chieh Chang, which w a p repared under the supervision of Professor
T.T. Moh of Purdue University.
This translation aims at preserving, as far as possible, both the contents
and style of Professor Chern’s lectures, the hallmarks of which are simplicity,
directness, and economy of approach together with in-depth treatments of
fundamental topics. It should be suitable as a text or a work of reference for
a wide audience, including (but not limited to) advanced undergraduate and
beginning graduate students in mathematics, as well as physicists interested in
the diverse applications of differential geometry to physics. Our hope is that
the material in this text will provide a solid and comprehensive background
for more advanced and specialized studies.
It is Professor Chern’s opinion that the time is ripe for the subject of Finsler
geometry to occupy a more prominent position within university curricula in
basic differential geometry. It was already alluded to by Riemann in his famous
Habilitation speech of 1854; and its relevance to the calculus of variations was
stressed by Hilbert in his 1900 Paris Lecture. Since Finsler’s thesis work on
it in 1918, the subject has seen many important developments, but has lacked
the kind of coherence that characterizes Riemannian geometry. Some remark-
able recent work has shown, however, that the more natural starting point
of Riemannian geometry is the more general Finsler setting, and that many
of the beautiful and deep results in the former have Finslerian counterparts.
Professor Chern himself, beginning with his early work in the 1940’s and in
recent collaborations with David Bao, has initiated crucial steps and paved the
way in this research.
In view of these developments, a new and rather lengthy chapter prepared
by Kai S. Lam and S.S. Chern on Finsler geometry (Chapter Eight) has been
added. The last section in Chapter Five of the original Chinese text on com-
pleteness in Riemannian geometry has also been revised and reincorporated as
section 7 of the new chapter, which treats completeness in Finsler geometry.
To bring the entire subject of differential geometry into perspective, Pro-
fessor Chern has written a valuable piece, “Historical Notes”, specially for the
vi
present English edition. This appears as the new Appendix A. Appendix B,
entitled “Differential Geometry and Theoretical Physics”, was originally au-
thored by Professor Chern in Chinese and included in the Chinese text. The
English translation, by the undersigned, appears for the first time in this book.
It is hoped that this essay will stimulate a degree of fruitful discussion between
mathematicians and physicists.
Professor Chern is well known for his masterful synthesis of deep geomet-
rical insights and skillful calculations. The present text will bear witness to
this immensely fruitful mathematical style. A central theme of the text is that
gbbal and local problems of differential geometry are equally interesting and
important. Even though local objects such as coordinates in a manifold are
devoid of intrinsic meaning, local tools, such as Cartan’s exterior differential
calculus and Ricci’s tensor analysis, are extremely useful in the study of man-
ifolds. Hence these tools have been developed and used extensively; but at
the same time, the importance of intrinsic objects with invariant properties
under a change of coordinates, such as tangent vector fields, differential forms,
etc., is also stressed. Throughout the text, the relationship between local and
global properties of a manifold, as exemplified, for example, by the Gauss-
Bonnet Theorem (Chapter Five) and the Chern classes (Chapter Seven), is
emphasized.
As a physicist with relatively little formal training in mathematics, I take
great pleasure in expressing my sincere gratitude to Professor Chern. Not only
has he graciously put up with the plodding attempts of a novice, but has also,
over the course of many months, provided me with generous support and guid-
ance. In addition, he took great pains to introduce me to the beautiful and
fascinating developments in Finsler geometry. I have deeply benefitted from
this unique opportunity to collaborate with and learn from a great mathemati-
cian. The new chapter on Finsler geometry has relied heavily on the joint work
of Bao and Chern [Houston Journal of Mathematics, Vol. 19, No. 1, 135-180
(1993)], and preliminary drafts of an upcoming comprehensive treatise, “An
Introduction to Riemann-Finsler Geometry,” by Bao, Chern and Shen, to be
published by Springer-Verlag. I am deeply grateful to Professor David Bao of
the University of Houston, to Professor Zhongmin Shen of Indiana University-
Purdue University at Indianapolis, and again to Professor Chern, for allowing
me to draw from these materials. A special note of heartfelt thanks is owed
to David Bao, who has rendered inestimable help in the preparation of the
new chapter, by carefully going over the drafts, offering freely expert advice,
and generously providing much needed reference material. In many ways, our
chapter on Finsler geometry may be viewed as providing an introduction to the
Bao-Chern-Shen treatise mentioned above, and the serious reader who wishes
to explore the subject at greater depth is well-advised to pursue that definitive
work.
vii
I would also like to thank my colleagues in both the Physics and Math-
ematics Departments of my home institution for their various kind acts of
assistance, support, and encouragement, especially physicists John Fang and
Soumya Chakravarti, mathematical physicist Martin Nakashima, and math-
ematicians Bernard Banks and Charles Amelin. To physicist Dr. Barbara
Hoeling I owe a special word of gratitude for helping me translate an early pa-
per by L. Berwald from German to English. In addition, I am grateful to the
Faculty Sabbatical Program of the California State University for providing
the necessary freedom and time for this project. Last but not least, I wish to
thank my wife, Dr. Bonnie J. Buratti, and our three boys, Nathan, Reuben,
and Aaron, for their simply being part of a wonderful and supportive family.
The authors are indebted to Hung-Chieh Chang and T.T. Moh for their
efforts in the initial translation; and Hu Sen, Chen Wei, A.N. Kobayashi,
S.H. Gan and Jitan Lu of World Scientific for their marvelous expediency and
professionalism in bringing this book into print. Finally, a big word of thanks
to Messrs. Geoff Simms and Andres Cardenas, who endured with good humor
the incessant alterations in the draft, and who, with their insuperable skills in
UT$, rendered the manuscript into its present form.
Kai S. Lam
California State Polytechnic University, Pomona
aThe lecture series was made possible by the kind concern and support of the following
(&%a),
colleagues in Peking University: Professors Duan Xue-Fu Jiang Ze-Han (%?1),I
Wu JiangLei (%%anad )W u Da-Ren (Ak45)T.h e latter two, as well as Tien Chow (He),
also read the Chinese manuscript and contributed valuable suggestions. In addition, the
following individuals all expended considerable efforts in note-taking and teaching assistance
(%%a),
during the lectures: Jiang Xue-Cheng ($!?%), You Cheng-Ye Liu Wang-Jin ($1 If€&),
Han Nien-Guo (%%H),Z hou Zuo-Ling (MI$%), Liu YingM;Og (%JEflfSl)u, n Zhen-Zu (a%&),
(egw).
and Li An-Min
bLectures on Differential Geometry (?%%n@by i;S#.S). ’Cch)e rn and Wei-Huan Chen,
Peking University Press, 1983. The present translation is based on this edition. A second
Chinese edition, under the same title and with no textual changes from the first, was
published in 1990 by the Lien Ching Ch‘u Pan Shih Yeh Go., Taipei, Republic of China.

Contents
1 Differentiable Manifolds 1
$1-1 Definition of Differentiable Manifolds . . . . . . . . . . . . . . . 1
$1-2 Tangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
fj-3 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
$1-4 Frobenius’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Multilinear Algebra 39
§2-1 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 39
$2-2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
$2-3 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Exterior Differential Calculus 65
$3-1 Tensor Bundles and Vector Bundles . . . . . . . . . . . . . . . 65
$3-2 Exterior Differentiation . . . . . . . . . . . . . . . . . . . . . . 74
$3-3 Integrals of Differential Forms . . . . . . . . . . . . . . . . . . . 85
53-4 Stokes’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Connections 101
§4-1 Connections on Vector Bundles . . . . . . . . . . . . . . . . . . 101
$4-2 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . 113
$4-3 Connections on Frame Bundles . . . . . . . . . . . . . . . . . . 121
5 Riemannian Geometry 133
$5-1 The Fundamental Theorem of Riemannian Geometry . . . . . . 133
$5-2 Geodesic Normal Coordinates . . . . . . . . . . . . . . . . . . . 143
85-3 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . 155
$5-4 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . 162
6 Lie Groups and Moving Frames 173
$6-1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
$6-2 Lie Transformation Groups . . . . . . . . . . . . . . . . . . . . 186
56-3 The Method of Moving Frames . . . . . . . . . . . . . . . . . . 198
ix