Hausdorff向量场

Hausdorff Vector Fields

讨论了紧流形上的光滑无奇点向量场。建立了光滑无奇点向量场与流形的Ｓ－结构之间的一一对应。把轨道空间是Ｈａｕｓｄｏｒｆ空间的向量场称为Ｈａｕｓｄｏｒｆ向量场。Ｈａｕｓｄｏｒｆ向量场的轨道空间是一个余一维的光滑流形。本文的主要结果是：ｎ－维流形Ｍ上存在Ｈａｕｓｄｏｒｆ向量场的充要条件是Ｍ是某（ｎ－１）－维流形上的光滑Ｓ１－纤维丛。

In this paper, we investagate smooth vector fields without critical point on compact manifold. There is a 1-1 correspondence between smooth vector fields without critical point and the S structures on the manifold. A vector field without critical point is called a Hausdorff vector field when its trajectory space is a Hausdorff space. The main result is the following theoerm: there exists a Hausdorff vector fields on an n dimensional manifold M iff M is a smooth S 1 fibre boundle over an (n-1) dimensional manifold.