Transformation Groups in Differential Geometry

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.

Autorentext

Biography of Shoshichi Kobayashi

Shoshichi Kobayashi was born January 4, 1932 in Kofu, Japan. After obtaining his mathematics degree from the University of Tokyo and his Ph.D. from the University of Washington, Seattle, he held positions at the Institute for Advanced Study, Princeton, at MIT and at the University of British Columbia between 1956 and 1962, and then moved to the University of California, Berkeley, where he is now Professor in the Graduate School.

Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with N. Nomizu, Hyperbolic Complex Manifolds and Holomorphic mappings and Differential Geometry of Complex Vector Bundles.

Inhalt

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-Kähler Manifolds.- 6. Compact Kähler Manifolds with Constant Scalar Curvature.- 7. Conformal Changes of the Laplacian.- 8. Compact Kähler 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.