Riemann-finsler Geometry

204 Pages·6.695 MB·English

by  Shiing-shen Chern, Zhongmin Shen

RIEMANN-FINSLER
GEOMETRY
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NANKAI TRACTS IN MATHEMATICS
Series Editors: Yiming Long and Weiping Zhang
Nankai Institute of Mathematics
Published
Vol. 1 Scissors Congruences, Group Homology and
Characteristic Classes
by J. L. Dupont
Vol. 2 The Index Theorem and the Heat Equation Method
by Y. L Yu
Vol. 3 Least Action Principle of Crystal Formation of Dense Packing Type
and Kepler's Conjecture
by W. Y. Hsiang
Vol. 4 Lectures on Chern-Weil Theory and Witten Deformations
by W. P. Zhang
Vol. 5 Contemporary Trends in Algebraic Geometry and Algebraic
Topology
edited by Shiing-Shen Chern, Lei Fu & Richard Hain
Vol. 6 Riemann-FinslerGeometry
by Shiing-Shen Chern & Zhongmin Shen
Vol. 7 Iterated Integrals and Cycles on Algebraic Manifolds
by Bruno Harris
Vol. 8 ' Minimal Submanifolds and Related Topics
by Yuanlong Xin
Nankai Tracts in Mathematics - Vol. 6
RIEMANN-FINSLER
GEOMETRY
Shiing-Shen Chern
Nankai Institute of Mathematics
P. Ft. China
Zhongmin Shen
Indiana University Purdue University Indianapolis
USA
Y|? World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI
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Library of Congress Cataloging-in-Publication Data
Chern, Shiing-Shen 1911-2004
Riemann-Finsler geometry / S.S. Chern, Zhongmin Shen.
p. cm. -- (Nankai tracts in mathematics ; v. 6)
Includes bibliographical references and index.
ISBN 981-238-357-3 (alk. paper) - ISBN 981-238-358-1 (pbk. : alk. paper)
1. Finsler spaces. 2. Geometry, Riemannian. I. Shen, Zhongmin, 1963- II. Title. III.
Series.
QA689.C48 2005
516.3'75-dc22
2005040818
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Preface
Two years ago David Bao, Zhongmin Shen and I published a book on
Riemann-Finsler geometry through Springer Verlag. Riemann-Finsler ge-
ometry is not a generalization of Riemannian geometry. Riemann knew
and began with the general case. He saw the main features of Riemannian
geometry and remarked that the general case does not involve new ideas.
In this assessment he was only partially correct. It certainly cannot
include global problems. In local problems these are also subtleties, which
need manipulation.
The aim of this book is to provide an elementary account of Finsler
geometry to show that Finsler geometry is essentially not more difficult.
Such an account is desirable, as Finsler metrics have come up in many
applications.
I am ashamed to say that Shen actually wrote the whole book, although
the idea of the book originated from me. The manuscript was written at
least five times and I am impressed by its clarity and simplicity. I went
through it in a seminar but do not wish to evade any responsibility if
mistakes are found.
S.S. Chern
January 2003
V
vi Preface
At the end of 2001, I visited S.S. Chern at Nankai Institute of Math-
ematics in Tianjin, P.R. China. During my visit, Chern told me that he
wished to have a concise book written for graduate students and young
geometers who are interested in Riemann-Finsler geometry. The primary
goal is to introduce some basic concepts, examples, theorems and to bring
the readers to the most current research areas in Finsler geometry. We had
a thorough discussion on topics and I started to collect some materials for
the manuscript. Soon I realized that it is very difficult to write such a book.
There is no simple proof for some important examples and theorems. Of-
ten times, the computation has to be carried out using a computer program
such as MAPLE.
Since the winter of 2001, I have been meeting Chern twice a year at
Nankai Institute of Mathematics to work on our book project. The first
draft of the manuscript was completed in the summer of 2002. However, I
continued to make changes based on my discussion with Chern and com-
ments from our colleagues. Chern wanted to check all the details by himself.
Thus the submission of the manuscript was postponed for several times.
On December 3,1 was stricken with the saddest news from my colleagues
that S.S. Chern was no longer with us. Personally, I lost a great advisor.
Without Chern's support and encouragement throughout the last decade,
I would not have done any work in Finsler geometry.
I would like to take this opportunity to thank X. Chen, L. Kozma, X.
Mo, C. Robles, H. Shimada, and G. C. Yildirim for their valuable comments.
Zhongmin Shen
December 2004
Contents
Chapter 1 Finsler Metrics 1
1.1 Minkowski Norms 2
1.2 Finsler Metrics 9
1.3 Length Structure and Volume Form 15
1.4 Navigation Problem 20
1.5 Cartan Torsion 25
Chapter 2 Structure Equations 31
2.1 Chern Connection 31
2.2 Structure Equations 40
2.3 Finsler Metrics of Constant Flag Curvature 43
2.4 Bianchi Identities 47
Chapter 3 Geodesies 51
3.1 Sprays 51
3.2 Shortest Paths 56
3.3 Projectively Equivalent Finsler Metrics 60
3.4 Projectively Flat Metrics 63
Chapter 4 Parallel Translations 71
4.1 Parallel Vector Fields 71
4.2 Parallel Translations 76
4.3 Berwald Metrics 79
4.4 Landsberg Metrics 81
vii
viii Contents
Chapter 5 S-Curvature 87
5.1 Distortion and S-Curvature 87
5.2 Randers Metrics of Isotropic S-Curvature 92
5.3 An Equation on the S-Curvature 104
Chapter 6 Riemann Curvature 107
6.1 Riemann Curvature 107
6.2 Second Variation of a Geodesic 115
6.3 Nonpositive Flag Curvature 120
Chapter 7 Finsler Metrics of Scalar Flag Curvature 127
7.1 Some Basic Properties 127
7.2 Global Rigidity Theorems 131
7.3 Randers Metrics of Scalar Flag Curvature 139
Chapter 8 Projectively Flat Finsler Metrics 149
8.1 Projectively Flat Randers Metrics 149
8.2 Projectively Flat Metrics with Constant Flag Curvature . . .. 155
8.3 Projectively Flat Metrics with Almost Isotropic S-Curvature . 166
Appendix A Maple Programs 171
A.I Spray Coefficients of Two-dimensional Finsler Metrics 171
A.2 Gauss Curvature 176
A.3 Spray Coefficients of (a, /3)-Metrics 178
Bibliography 185
Index 191
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The Simple 3 step formula
RIEMANN-FINSLER
GEOMETRY