Topology of nonintegrable plane fields

本研究的目的是从拓扑学的角度研究不可积平面场，并阐明它们的全局行为。首先，考虑了与恩格尔平面场相切的环的刚性。给定一条初始点固定的特征曲线，我们完全确定了曲线在切向曲线空间中对平面场的微小扰动如何改变曲线的端点。特别地，我们得到了以下结论：当且仅当曲线对正则射影结构的展开像与整个射影线重合时，在扰动作用下端点的迹成为开集。作为这一结果的应用，我们得到：任何非仿射特征环都是非刚性的。我们还证明了圆的每一个一维射影结构都可以被实现为某恩格尔流形中某特征环的正则射影结构。接下来，我们对高维刚度进行了研究。证明了高阶接触流形上符号面场的极大积分子流形总是局部刚性的。我们还给出了具有不可积平面场的流形中的刚性环面的一个例子。第三，我们尝试将叶的几何熵推广到不可积情况。为了定义熵，我们需要使用积分曲线。我们认识到，如果我们只使用有界几何的积分曲线，我们就能够定义不可积平面场的熵的概念。这些结果的一部分已经发表在2005年国际叶理会议的论文集上，题目是：关于与不可积叶理相切的子流形的刚性。

The purpose of this research was to study nonintegrable plane fields from the topological viewpoint and clarify their global behavior.First, we considered the rigidity of loops tangent to Engel plane fields. Given a characteristic curve with the initial point being fixed, we completely determined how the terminal point of the curve can vary by small perturbations of the curve in the space of tangential curves to the plane field. Especially, we obtained the following: The trace of terminal points under perturbations becomes an open set if and only if the developing image of the curve with respect to the canonical projective structure coincides with the whole projective line. As an application of this result we got the following: Any non-affine characteristic loop is non-rigid. We also showed that every 1-dimensional projective structure of the circle can be realized as the canonical projective structure of some characteristic loop in some Engel manifold.Next, we studied the rigidity in higher dimensions. We showed that maximal integral submanifolds of the symbol plane fields on contact manifolds of higher orders are always locally rigid. We also produced an example of a rigid torus in some manifold endowed with a nonintegrable plane field.Thirdly, we tried to generalize the Ghys-Langevin-Walczak geometric entropy of foliations to the nonintegrable cases. To define an entropy, we need to use integral curves. We recognized that if we exclusively use integral curves with bounded geometry we are able to define a notion of entropy for nonintegrable plane fields.Parts of these results have been published in the proceedings of the international conference FOLIATIONS 2005, under the title : On rigidity of submanifold a tangent to nonintegrable foliations.