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AuthorKobayashi, Shoshichi. author
TitleTransformation Groups in Differential Geometry [electronic resource] / by Shoshichi Kobayashi
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1995
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Descript VIII, 182 p. online resource


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 strucยญ tures. All geometric structures are not created equal; some are creations of õds 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 ยงยง 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geoยญ metric structures in a unified manner. In ยง 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 ยงยง 5, 6, 7 and 8. This chapter is partly based on lecยญ 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-Kรคhler Manifolds -- 6. Compact Kรคhler Manifolds with Constant Scalar Curvature -- 7. Conformal Changes of the Laplacian -- 8. Compact Kรคhler 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

Mathematics Group theory Differential geometry Mathematics Differential Geometry Group Theory and Generalizations


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