Transformation groups in differential geometry
Transformation groups in differential geometry
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DOI：
10.1007/978-3-642-61981-6
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发表时间：
1972
期刊：
影响因子：
--
通讯作者：
Shôshichi Kobayashi
中科院分区：
文献类型：
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作者：
Shôshichi Kobayashi
给定一个数学结构，一个基本的相关数学对象是它的自同构群。这本书的目的是给一个有偏见的帐户自同构群的微分几何结构。并非所有的几何结构都是平等的;有些是自然的创造，而另一些则是人类思想的产物。在前者中，黎曼和复杂的结构因其美丽和财富而脱颖而出。因此，本书的一个主要部分专门讨论这两种结构。第一章介绍了一般理论的自同构的几何结构的问题，重点是自同构群时，可以给予李群结构。这方面的基本定理在§§ 3、4和5中给出。G-结构或伪群结构的概念使我们能够以统一的方式处理大多数有趣的几何结构。在第8节中，我们概述了这两个概念之间的关系。第一章是这样安排的，读者如果主要对黎曼结构、复结构、共形结构和射影结构感兴趣，可以跳过§§ 5、6、7和8。这一章的部分内容是根据我1965年在东京和伯克利所作的演讲编写的。
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.