Transformation Groups in Differential Geometry

192 Pages·1972·14.177 MB·English

by  Shoshichi Kobayashi

Classics in Mathematics
Shoshichi Kobayashi Transformation Groups in Differential Geometry
Shoshichi Kobayashi
Transformation Groups
in Differential Geometry
Reprint of the 1972 Edition
Springer
Shoshichi Kobayashi
Department of Mathematics, University of California
Berkeley, CA 94720-3840
USA
Originally published as Vol. 70 of the
Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd sequence
Mathematics Subject Classification {1991}:
Primary 53C20, 53ClO, 53C55, 32M05, 32J25, 57S15
Secondary 53C15, 53AlO, 53A20, 53A30, 32H20, 58DOS
ISBN-13: 978-3-540-58659-3 e-ISBN-I3: 978-3-642-61981-6
001: 10.1007/978-3-642-61981-6
Photograph by kind permission of George Bergman
CIP data applied for
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@ Springer. Verlag Berlin Heidelberg 1995
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SPIN 10485278 4113140 -543210 -Printed on acid-free paper
Shoshichi Kobayashi
Transformation Groups
in Differential Geometry
Springer-Verlag Berlin Heidelberg New York
1972
Shoshichi Kobayashi
University of California, Berkeley, California
AMS Subject Classifications (1970):
Primary 53 C 20,53 C 10,53 C 55,32 M 05,32 J 25,57 E 15
Secondary 53 C 15, 53 A 10, 53 A 20, 53 A 30, 32 H20, 58 D 05
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro_
duction by photocopying machine or similar means, and storage in data banks. Under § S4 of the German
Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the
amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidel
berg 1972. Library of Congress Catalog Card Number 72-80361. Printed in Germany. Printing and bin
ding: Universitiitsdruckerei H. StUrtz AG, WUrzburg
Preface
Given a mathematical structure, one of the basic associated mathematical
objects is its automorphism group. The object of this book is to give a
biased account of automorphism groups of differential geometric struc
tures. All geometric structures are not created equal; some are creations
of ~ods while others are products of lesser human minds. Amongst the
former, Riemannian and complex structures stand out for their beauty
and wealth. A major portion of this book is therefore devoted to these
two structures.
Chapter I describes a general theory of automorphisms of geometric
structures with emphasis on the question of when the automorphism
group can be given a Lie group structure. Basic theorems in this regard
are presented in §§ 3, 4 and 5. The concept of G-structure or that of
pseudo-group structure enables us to treat most of the interesting geo
metric structures in a unified manner. In § 8, we sketch the relationship
between the two concepts. Chapter I is so arranged that the reader who
is primarily interested in Riemannian, complex, conformal and projective
structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec
tures I gave in Tokyo and Berkeley in 1965.
Contents of Chapters II and III should be fairly clear from the
section headings. It should be pointed out that the results in §§ 3 and 4
of Chapter II will not be used elsewhere in this book and those of §§ 5 and
6 of Chapter II will be needed only in §§ 10 and 12 of Chapter III. I
lectured on Chapter II in Berkeley in 1968; Chapter II is a faithful version
of the actual lectures.
Chapter IV is concerned with automorphisms of affine, projective
and conformal connections. We treat both the projective and the con
formal cases in a unified manner.
Throughout the book, we use Foundations of Differential Geometry
as our standard reference. Some of the referential results which cannot be
found there are assembled in Appendices for the convenience of the
reader.
As its title indicates, this book is concerned with the differential
geometric aspect rather than the differential topological or homological
VI Preface
aspect of the theory of transformation groups. We have confined our
selves to presenting only basic results, avoiding difficult theorems. To
compensate for the omission of many interesting but difficult results,
we have supplied the reader with an extensive list of references.
We have not touched upon homogeneous spaces, partly because they
form an independent discipline of their own. While we are interested in
automorphisms of given geometric structures, the differential geometry of
homogeneous spaces is primarily concerned with geometric objects
which are invariant under given transitive transformation groups. For
the convenience of the reader, the Bibliography includes papers on the
geometry of homogeneous spaces which are related to the topics discussed
here.
In concluding this preface, I would like to express my appreciation
to a number of mathematicians: Professors Yano and Lichnerowicz,
who interested me in this subject through their lectures, books and papers;
Professor Ehresmann, who taught me jets, prolongations and infinite
pseudo-groups; K. Nomizu, T. Nagano and T. Ochiai, my friends and
collaborators in many papers; Professor Matsushima, whose recent
monograph on hoi om orphic vector fields influenced greatly the presen
tation of Chapter III; Professor Howard, who kindly made his manu
script on hoi om orphic vector fields available to me. I would like to thank
Professor Remmert and Dr. Peters for inviting me to write this book and
for their patience.
I am grateful also to the National Science Foundation for its un
failing support given to me during the preparation of this book.
January, 1972 S. Kobayashi
Contents
I. Automorphisms of G-Structures .
1. G-Structures . . . . . . . . 1
2. Examples of G-Structures . . 5
3. Two Theorems on Differentiable Transformation Groups. 13
4. Automorphisms of Compact Elliptic Structures 16
5. Prolongations of G-Structures . . . . . . 19
6. Volume Elements and Symplectic Structures . 23
7. Contact Structures . . . . . . . . . . . . 28
8. Pseudogroup Structures, G-Structures and Filtered Lie Alge-
bras. . . . . . . . . . . . . . . . . . . . . . . . . 33
II. Isometries of Riemannian Manifolds. 39
1. The Group of Isometries of a Riemannian Manifold. . . . 39
2. Infinitesimal Isometries and Infinitesimal Affine Trans-
formations . . . . . . . . . . . . . . . . . . . . 42
3. Riemannian Manifolds with Large Group of Isometries 46
4. Riemannian Manifolds with Little Isometries. . . . 55
5. Fixed Points of Isometries . . . . . . . . . . . . 59
6. Infinitesimal Isometries and Characteristic Numbers. 67
III. Automorphisms of Complex Manifolds. . . . . . . . 77
1. The Group of Automorphisms of a Complex Manifold. 77
2. Compact Complex Manifolds with Finite Automorphism
Groups . . . . . . . . . . . . . . . . . . . . . 82
3. Holomorphic Vector Fields and Holomorphic I-Forms 90
4. Holomorphic Vector Fields on Kahler Manifolds 92
5. Compact Einstein-Kahler Manifolds . . . . . . . . 95
6. Compact Kahler Manifolds with Constant Scalar Curvature 97
7. Conformal Changes of the Laplacian .......... 100
8. Compact Kahler Manifolds with Nonpositive First Chern
Class . . : . . . . . . . . . . . . . . . . . . . . . 103
VIII Contents
9. Projectively Induced Holomorphic Transformations. 106
10. Zeros of Infinitesimal Isometries . . . . . . . . . 11 2
11. Zeros of Holomorphic Vector Fields . . . . . . . 11 5
12. Holomorphic Vector Fields and Characteristic Numbers. 119
IV. Affine, Conformal and Projective Transformations ..... 122
1. The Group of Affine Transformations of an Affinely Con-
nected Manifold . . . . . . . . . . . . . . . 122
2. Affine Transformations of Riemannian Manifolds. 125
3. Cartan Connections. . . . . . . . . 127
4. Projective and Conformal Connections 131
5. Frames of Second Order . . . . . . . 139
6. Projective and Conformal Structures. . 141
7. Projective and Conformal Equivalences 145
Appendices. . . . . • . • • • • • • . . • 150
1. Reductions of I-Forms and Closed 2-Forms 150
2. Some Integral Formulas . . . . 154
3. Laplacians in Local Coordinates 157
4. A Remark on d'd"-Cohomology 159
Bibliography . 160
Index . . . . 181