## Transformation Groups in Differential GeometryGiven 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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### Contents

I Automorphisms of GStructures | 1 |

2 Examples of GStructures | 5 |

3 Two Theorems on Differentiable Transformation Groups | 13 |

4 Automorphisms of Compact Elliptic Structures | 16 |

5 Prolongations of GStructures | 19 |

6 Volume Elements and Symplectic Structures | 23 |

2 Infinitesimal Isometries and Infinitesimal Affine Transformations | 42 |

3 Riemannian Manifolds with Large Group of Isometries | 46 |

9 Projectively Induced Holomorphic Transformations | 106 |

10 Zeros of Infinitesimal Isometries | 112 |

11 Zeros of Holomorphic Vector Fields | 115 |

12 Holomorphic Vector Fields and Characteristic Numbers | 119 |

IV Affine Conformal and Projective Transformations | 122 |

2 Affine Transformations of Riemannian Manifolds | 125 |

3 Cartan Connections | 127 |

4 Projective and Conformal Connections | 131 |

4 Riemannian Manifolds with Little Isometries | 55 |

5 Fixed Points of Isometries | 59 |

6 Infinitesimal Isometries and Characteristic Numbers | 67 |

III Automorphisms of Complex Manifolds | 77 |

2 Compact Complex Manifolds with Finite Automorphism Groups | 82 |

3 Holomorphic Vector Fields and Holomorphic 1Forms | 90 |

4 Holomorphic Vector Fields on Kiihler Manifolds | 92 |

5 Compact EinstetnKa hler Manifolds | 95 |

6 Compact Kahler Manifolds with Constant Scalar Curvature | 97 |

7 Conformal Changes of the Laplacian | 100 |

8 Compact Kahler Manifolds with Nonpositive First Chern Class | 103 |

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### Common terms and phrases

acting affine connection affine transformations Amer assume automorphism basis belongs called Cartan Chapter closed compact complex manifold components conformal connection consider consisting constant contains coordinate system corresponding curvature defined definition denote derivation diffeomorphism Differential Geometry dimension element equations equivalent Example exists fact fibre finite fixed follows formula frames function G-structure geodesic Geometry given GL(n hand Hence holomorphic vector field homogeneous identity implies induced infinitesimal isometry integrable invariant isomorphism Kahler manifold Kobayashi leaves Lemma Lie algebra Lie group mapping Math matrix natural negative neighborhood nonzero normal obtain orientable origin parallel positive projective proof Proof of Lemma proof of Theorem Proposition prove rank resp respect restriction result Riemannian manifold space structure subgroup submanifold symmetric symplectic tangent tensor Theorem Theorem 2.1 trans transformation group transitive unique vanishes write zero