Riemannian Foliations
Riemannian Foliations
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DOI：
10.1007/978-1-4684-8670-4
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
P. Molino;G. Cairns
中科院分区：
文献类型：
--
作者：
P. Molino;G. Cairns
叶理理论起源于常微分方程解的整体分析：在一个n维流形M上，一个[自治]微分方程由一个向量场X定义;如果这个向量场没有奇点，那么它的轨迹将M分成曲线，即余维n-1的叶理。更一般地说，M上余维为q的叶理F对应于M划分为维数为，- p= n-q。第一个浮现在脑海中的全局图像是一堆“斑块”。1-;--从侧面看[横向1- 1-sally]，这样的1- 1-的叶子。堆叠是1--1--的点。di L的商流形W '_ mension q.-----~）WM实际上，这个图像对应于一种基本类型的叶子，人们说它是“简单的”。对于任意的叶理，只有l-uL ally [在”简单”开集U上]，叶理才表现为斑块的堆叠，并允许局部商流形。在全局范围内，一片叶子L可以返回并将一个简单的开集U切割成几个斑块，有时甚至是无限多个斑块。
Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X; if this vector field has no singularities, then its trajectories form a par tition of M into curves, ie a foliation of codimension n-1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension,--------,--.---p= n-q. The first global image that comes to mind is 1--------;------that of a stack of" plaques". 1---------;------Viewed laterally [transver 1--------1-----sally], the leaves of such a 1--------1-----. stacking are the points of a 1--------1-------. quotient manifold W of di L.....-'_ mension q.-----~) WM Actually, this image corresponds to an elementary type of folia tion, that one says is" simple". For an arbitrary foliation, it is only l-u L ally [on a" simpIe" open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may--return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.