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. 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.