Foliations and Geometric Structures

叶状结构和几何结构

基本信息

批准号：
02640015
负责人：
INABA Takashi
金额：
$ 0.96万
依托单位：
Chiba Universtiy
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1990
资助国家：
日本
起止时间：
1990 至 1991
项目状态：
已结题

项目摘要

There are two natural ways to introduce geometric structures into foliations. One is to introduce them in the direction transverse to the leaves, and the other is to do so in the direction tangent to the leaves. The former way has already been studied very much by many people, but it seems that works about the latter way are not so many in the literature.In this research, we investigated these both types of geometric structures on foliations, mainly from the viewpoint of differential topology. Firstly, as for the transverse geometric structures, we studied co-dimension one foliations with transverse projective structure. We clarified the relation between the global holonomy group and the topological properties of the foliations. Furthermore, we showed that a transversely projective foliation cannot have exceptional leaves if the ambient manifold has amenable fundamental group. Secondly, as for the tangential geometric structures, we studied(1)tangentially affine foliations and(2)tangentially holomorphic foliations : (1)Note that tangentially affine foliations appear as Lagrangian foliations on symplectic manifolds. We determined the space of all leafwise affine, functions on a tangentially affine foliation on the torus. We also proved that the three dimensional sphere does not admit any co-dimension one tangentially affine foliation. (2)A Levi-flat real hypersurface in a complex surface has a tangentially holomorphic foliation, which is usually called the Levi foliation. We obtained some topological properties of compact Levi-flat hypersurfaces by investigating the holonomy of total leaves in their Levi foliations. In particular, we showed that the three dimensional sphere cannot be embedded as a Levi-flat hypersurface. This result is applied to two dimensional complex dynamical systems.

有两种自然的方法可以将几何结构引入叶子中。一种是将它们引入横向叶子的方向，另一个是朝着与叶子切线的方向进行。前者已经对许多人进行了很多研究，但是在文献中似乎并不多。首先，至于横向几何结构，我们研究了具有横向射击结构的共构叶。我们阐明了全球全体人类群体与叶子的拓扑特性之间的关系。此外，我们表明，如果环境歧管具有正面的基本组，则横向投射叶片不能具有出色的叶子。其次，至于切向几何结构，我们研究了（1）切向叶子叶子和（2）切向骨膜叶片：（1）请注意，切向植物在符号流形上显示为拉格朗日叶子。我们确定了所有叶轮仿射的空间，在圆环上的切向植物上起作用。我们还证明，三维球不承认任何一个切入的叶子叶子。（2）在复杂表面中的Levi-Flat真实超表面具有切向圆形的叶面，通常称为Levi叶叶。我们通过研究Levi叶子中的总叶子的全能，获得了紧凑型Levi-Flat Hyperfaces的一些拓扑特性。特别是，我们表明三维球不能嵌入为LEVI-FLAT超表面。该结果应用于二维复杂的动态系统。