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.
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.