Foliations and Geometric Structures

有两种自然的方式将几何结构引入叶状结构。一种是在横向于叶片的方向上引入它们，另一种是在与叶片相切的方向上这样做。前一种方法已经被很多人研究过，但关于后一种方法的研究在文献中似乎还不是很多。在本研究中，我们主要从微分拓扑学的角度来研究这两种类型的叶层几何结构。首先，对于横向几何结构，我们研究了具有横向射影结构的余维一叶状结构。阐明了整体完整群与叶层的拓扑性质之间的关系。此外，我们还证明了，如果环境流形有服从的基本群，则横向射影叶层不可能有例外叶。其次，对于切向几何结构，我们研究了(1)切向仿射叶层和(2)切向全纯叶层：(1)注意切向仿射叶层在辛流形上表现为拉格朗日叶层。我们确定了所有叶仿射的空间，即环面上切向仿射叶层上的函数。我们还证明了三维球面不允许有任何余维的切线仿射面片。(2)复曲面上的Levi平坦实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叶全分类，得到了紧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作为Levi平坦超曲面嵌入。这一结果被应用于二维复杂动力系统。

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.