Topology of nonintegrable plane fields

不可积平面场的拓扑

基本信息

批准号：
16540053
负责人：
INABA Takashi
金额：
$ 2.18万
依托单位：
Chiba University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540053/
关键词：
nonintegrable plane fields Engel structure rigidity characteristic curve projective structure contact manifolds of higher order geometric entropy backet generating アノソフ流エントロピー局所剛的特性葉剛的部分多様体

项目摘要

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.

这项研究的目的是从拓扑角度研究不可整合的平面场，并阐明其全球行为。首先，我们考虑了与恩格尔平面场相切的环的刚性。给定具有固定初始点的特征曲线，我们完全确定了曲线的端子点如何因切向曲线与平面场的曲线的小扰动而变化。尤其是，我们获得以下内容：当且仅当相对于规范的投影结构的开发图像与整个投影线一致时，扰动下的终端点的痕迹才成为一个开放集。作为此结果的应用，我们得到了以下内容：任何非携带特性循环都是非刚性的。我们还表明，圆的每个一维投影结构都可以实现为某些特征环的某些特征环的规范投射结构。我们研究了更高维度的刚度。我们表明，在较高阶的接触歧管上，符号平面场的最大积分子构架始终是局部刚性的。我们还制作了一个具有不可整合平面场的多种歧管中的刚性圆环的示例。三分之一，我们试图将叶子的Ghys-Langevin-Walczak几何熵推广到不可整合的情况下。要定义熵，我们需要使用积分曲线。我们认识到，如果我们专门使用有界几何形状的积分曲线，我们能够为非整合平面字段定义熵概念。这些结果的部分已在2005年国际会议叶子的会议记录中发表，标题为标题：Submanifold的刚性刚度是较小的，与不可整合的叶子相互依赖。