Geometric properties of generic differentiable manifolds
Geometric properties of generic differentiable manifolds
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泛型可微流形的几何性质
DOI：
10.1007/bfb0085382
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发表时间：
1977
期刊：
影响因子：
--
通讯作者：
C. Wall
中科院分区：
文献类型：
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作者：
C. Wall
这是1976年7月在里约热内卢IMPA举办的三场讲座的放大版，当时正值ELAN4 I！我的目标是调查奇点理论在微分几何中的应用。然而，在准备这篇综述的过程中，我注意到大多数这样的应用都关键地使用了横截性和类属性质的概念，所以我决定将我自己限制在这些方面。这为下面研究的其他相当不同的结构提供了一定的统一性，这本身就使这个主题更容易被同化。会谈的准备工作相当仓促(所讨论的一些问题已经在会议上提出)，其中的一些错误已更正如下。这篇报道也是在压力下写的，因为我在从事博览会的过程中不断发现新的结果和新的来源，所以读者会发现一些零碎的东西需要整理。事实上，这项调查提出了许多有趣的悬而未决的问题。这篇论文分为三章(在三堂课之后)，每一章都有进一步的细分。在第一章中，我描述了它们在微分几何中的应用并给出了例子。我煞费苦心地举出了大量的例子，因为我觉得这些例子比有些抽象的一般定理更有启发性(有时也更有用)。解决这些问题涉及到大量的常规代数，我已经取消了其中的大部分。我试图证明，对于曲线和曲面，这种哲学产生了有趣的推广到更高维度的结果。这一章分为A-F节，C节的定理3(例如)在别处称为定理C3。
This is an amplified version of a course of three lectures given at IMPA in Rio in July 1976, on the happy occasion of ELAn4 I! I. The objective was to survey applications of singularity theory to differential geometry.~ owever, in preparing the survey I noticed that the majority of such applications made crucial use of transversality and the notion of generic property, so I decided to restrict myself to these. This lends a certain unity to the otherwise rather different constructions studied below, which itself makes the topic easier to assimilate. The talks were rather hurriedly prepared (some of the questions discussed having been raised at the conference itself), and some errors in them are corrected below. This account too is written under pressure, as I kept discovering new results and new sources while engaged on the exposition, so the reader will find a number of loose ends to tidy up. Indeed, this survey raises a large number of interesting unsolved problems.This paper is divided into three chapters (following the three lectures), each with further subdivisions. In the first, I describe the applications to differential geometry and give examples. I have been at pains to give a large number of examples, as i feel these are more instructive (and sometimes more useful) than the somewhat abstract general theorems. Working these out involves a lot of routine algebra, much of which I have suppressed. I have tried to show that this philosophy yields interesting generalisations to higher dimensions of results f~ miliar for curves and surfaces. Tnis chapter is divided into sections A-F, and Theorem 3 of section C (for example) is elsewhere referred to as Theorem C3.