Transformation groups in differential geometry
著者
書誌事項
Transformation groups in differential geometry
（Classics in mathematics）
Springer, c1995
大学図書館所蔵 全66件
注記
Reprint of the 1972 edition
Includes bibliographical references and index
内容説明・目次
内容説明
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.
目次
I. Automorphisms of G-Structures.- 1. G -Structures.- 2. Examples of G-Structures.- 3. Two Theorems on Differentiable Transformation Groups.- 4. Automorphisms of Compact Elliptic Structures.- 5. Prolongations of G-Structures.- 6. Volume Elements and Symplectic Structures.- 7. Contact Structures.- 8. Pseudogroup Structures, G-Structures and Filtered Lie Algebras.- II. Isometries of Riemannian Manifolds.- 1. The Group of Isometries of a Riemannian Manifold.- 2. Infinitesimal Isometries and Infinitesimal Affine Transformations.- 3. Riemannian Manifolds with Large Group of Isometries.- 4. Riemannian Manifolds with Little Isometries.- 5. Fixed Points of Isometries.- 6. Infinitesimal Isometries and Characteristic Numbers.- III. Automorphisms of Complex Manifolds.- 1. The Group of Automorphisms of a Complex Manifold.- 2. Compact Complex Manifolds with Finite Automorphism Groups.- 3. Holomorphic Vector Fields and Holomorphic 1-Forms.- 4. Holomorphic Vector Fields on Kahler Manifolds.- 5. Compact Einstein-Kahler Manifolds.- 6. Compact Kahler Manifolds with Constant Scalar Curvature.- 7. Conformal Changes of the Laplacian.- 8. Compact Kahler Manifolds with Nonpositive First Chern Class.- 9. Projectively Induced Holomorphic Transformations.- 10. Zeros of Infinitesimal Isometries.- 11. Zeros of Holomorphic Vector Fields.- 12. Holomorphic Vector Fields and Characteristic Numbers.- IV. Affine, Conformal and Projective Transformations.- 1. The Group of Affine Transformations of an Affinely Connected Manifold.- 2. Affine Transformations of Riemannian Manifolds.- 3. Cartan Connections.- 4. Projective and Conformal Connections.- 5. Frames of Second Order.- 6. Projective and Conformal Structures.- 7. Projective and Conformal Equivalences.- Appendices.- 1. Reductions of 1-Forms and Closed 2-Forms.- 2. Some Integral Formulas.- 3. Laplacians in Local Coordinates.
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