Foliations and Geometric Structures

有两种自然的方法将几何结构引入叶状。一种是在叶片的横向方向上引入它们，另一种是在与叶片相切的方向上引入它们。前一种方式已经被很多人研究过，但关于后一种方式的作品在文献中似乎并不多。在本研究中，我们主要从微分拓扑的角度研究了这两种类型的叶上几何结构。首先，在横向几何结构方面，研究了具有横向投影结构的协维一叶理。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可服从的基群，那么横向投影叶理就不可能有例外叶。其次，对于切向几何结构，我们研究了(1)切向仿射叶和(2)切向全纯叶：(1)注意到切向仿射叶在辛流形上表现为拉格朗日叶。我们确定了环面上切向仿射叶上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协维切向仿射叶理。(2)复曲面上的列维平面实超曲面具有切向全纯叶理，通常称为列维叶理。通过研究列维叶中全叶的完整性，得到了紧致列维平面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列维平面超曲面嵌入。该结果适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.