Introduction to the Geometry of Foliations, Part A
Introduction to the Geometry of Foliations, Part A
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DOI：
10.1007/978-3-322-85619-7
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
G. Hector;U. Hirsch
中科院分区：
文献类型：
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作者：
G. Hector;U. Hirsch
部分B我们介绍了几何的Foliations是一个直接延续的部分A（第1-III章）已出版的数学方面于1981年。在第一章中，对表面的叶理进行了研究。B部分的目的是将其推广到任意维流形上的codimanas 1的叶理。事实证明，我们在曲面上观察到的许多现象只依赖于余维，因此在高维流形上的余维1叶理中也有类似的现象。此外，用于研究叶理表面的方法，例如胶合或湍流叶理，直接推广到高维情况。然而，它们不足以提供给定流形上所有余维为1的叶理的拓扑分类，而在紧曲面上的叶理的情况下是可能的（见I; § 4）。因此，我们必须满足于描述叶理的几何，因为它是由它们的极小集、饱和开集的结构或消失圈或完整不变测度的存在所反映的。我们所做的大多数工作都要求底层流形是紧的（通常没有边界）。让我们借此机会回顾一下，流形被假定（有可数基）是连通的，除非相反的情况是明显的，或者有明确的相反陈述。
Part B of our Introduction to the Geometry of Foliations is a direct continuation of Part A (chapters 1-III) which has been published in the Aspects of Mathematics in 1981. In chapter I the study of foliations was carried out for surfaces. The object of Part B is to extend this to foliations of codimansion one on manifolds of arbitrary dimension. It will turn out that many of the phenomena we have observed on surfaces depend only on the codimension and thus have an analogue in codimension-one foliations on manifolds of higher dimension. Also the methods used to investigate foliated surfaces, for example gluing or turbulizing foliations, generalize directly to the higher dimensional case. They do not, however, suffice to provide a topological classification of all codimension-one foliations on a given manifold as is possible in the case of foliations on compact surfaces (see I; § 4). We must therefore be content to describe the geometry of foliations as it is reflected, say, by their minimal sets, by the structure of saturated open sets or by the existence of vanishing cycles or holonomy invariant measures.Most of what we do requires the underlying manifold to be compact (often without boundary). Let us take this opportunity to recall that manifolds are assumed (to have a countable basis and) to be connected unless the contrary is obvious or there is an explicit statement to the contrary.