Transformation Groups in Differential Geometry

By: Shoshichi Kobayashi

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Paperback | 15 February 1995 | Edition Number 2

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194 Pages

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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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Geometry Differential & Riemannian Geometry Algebra

Automorphisms of G-Structures	p. 1
G-Structures	p. 1
Examples of G-Structures	p. 5
Two Theorems on Differentiable Transformation Groups	p. 13
Automorphisms of Compact Elliptic Structures	p. 16
Prolongations of G-Structures	p. 19
Volume Elements and Symplectic Structures	p. 23
Contact Structures	p. 28
Pseudogroup Structures, G-Structures and Filtered Lie Algebras	p. 33
Isometries of Riemannian Manifolds	p. 39
The Group of Isometries of a Riemannian Manifold	p. 39
Infinitesimal Isometries and Infinitesimal Affine Transformations	p. 42
Riemannian Manifolds with Large Group of Isometries	p. 46
Riemannian Manifolds with Little Isometries	p. 55
Fixed Points of Isometries	p. 59
Infinitesimal Isometries and Characteristic Numbers	p. 67
Automorphisms of Complex Manifolds	p. 77
The Group of Automorphisms of a Complex Manifold	p. 77
Compact Complex Manifolds with Finite Automorphism Groups	p. 82
Holomorphic Vector Fields and Holomorphic 1-Forms	p. 90
Holomorphic Vector Fields on Kahler Manifolds	p. 92
Compact Einstein-Kahler Manifolds	p. 95
Compact Kahler Manifolds with Constant Scalar Curvature	p. 97
Conformal Changes of the Laplacian	p. 100
Compact Kahler Manifolds with Nonpositive First Chern Class	p. 103
Projectively Induced Holomorphic Transformations	p. 106
Zeros of Infinitesimal Isometries	p. 112
Zeros of Holomorphic Vector Fields	p. 115
Holomorphic Vector Fields and Characteristic Numbers	p. 119
Affine, Conformal and Projective Transformations	p. 122
The Group of Affine Transformations of an Affinely Connected Manifold	p. 122
Affine Transformations of Riemannian Manifolds	p. 125
Cartan Connections	p. 127
Projective and Conformal Connections	p. 131
Frames of Second Order	p. 139
Projective and Conformal Structures	p. 141
Projective and Conformal Equivalences	p. 145
App. 1. Reductions of 1-Forms and Closed 2-Forms	p. 150
App. 2. Some Integral Formulas	p. 154
App. 3. Laplacians in Local Coordinates	p. 157
App. 4. A Remark on d'd''-Cohomology	p. 159
Bibliography	p. 160
Index	p. 181
Table of Contents provided by Blackwell. All Rights Reserved.

Transformation Groups in Differential Geometry

At a Glance

Paperback

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How to return your order

Defective items

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