Transformation Groups in Differential GeometrySpringer Science & Business Media, 1995/02/15 - 182 ページ 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 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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多く使われている語句
1-parameter group affine connection affine transformations Amer automorphism group Cartan connection Chapter Chern class compact complex manifold compact Kähler manifold complex manifold conformal structure conformal transformations constant coordinate system x¹ Corollary corresponding curvature defined denote diffeomorphism Differential Geometry dimension Example fibre finite formula function G-structure g₁ geodesic given GL(m GL(n graded Lie algebra h₁ Hence holomorphic 1-form holomorphic transformations holomorphic vector field homogeneous imbedding induced infinitesimal automorphism infinitesimal isometry integrable invariant isomorphism Kähler manifold Kobayashi Kobayashi-Nomizu Lie group Lie transformation group line bundle linear frames linear transformations mapping matrix n-dimensional Nagoya Math neighborhood nonzero obtain principal bundle projective structure Proof of Lemma proof of Theorem Proposition prove pseudogroup resp Ricci tensor Riemannian manifold Riemannian metric simply connected subbundle submanifold subspace symmetric tangent space Theorem 2.1 trans transformation group V₁ vanishes variétés X₁ Yano zero set
