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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.

有两种自然的方式将几何结构引入叶理。一种是在叶片的横向引入它们，另一种是在叶片的切线方向上这样做。前一种方法已经被很多人研究过了，但关于后一种方法的研究工作似乎并不多，本文主要从微分拓扑的角度研究了叶理上的这两种几何结构。首先，在横截几何结构方面，我们研究了具有横截射影结构的余维1叶理。阐明了整体完整群与叶理拓扑性质之间的关系。此外，我们还证明了如果周围流形具有顺从基本群，则横投射叶理不可能有例外叶。其次，对于切几何结构，我们研究了（1）切仿射叶理和（2）切全纯叶理：（1）注意，切仿射叶理表现为辛流形上的Lagrange叶理。我们确定了环面上切向仿射叶理上的所有叶向仿射函数的空间。我们还证明了三维球面不允许任何余维1切仿射叶理。(2)A复曲面中的Levi平坦真实的超曲面具有切向全纯叶理，通常称为Levi叶理。通过研究紧致Levi-平坦超曲面的Levi叶状结构中全叶的完整性，得到了紧致Levi-平坦超曲面的一些拓扑性质。特别地，我们证明了三维球面不能嵌入为列维平坦超曲面。这一结果适用于二维复杂动力系统。