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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.

本研究的目的是从拓扑学的观点来研究不可积平面场，并阐明它们的整体行为。首先，我们考虑了与恩格尔平面场相切的环的刚性。给定一条初始点固定的特征曲线，我们完全确定了曲线的端点如何在平面场的切向曲线空间中通过曲线的小扰动而变化。特别地，我们得到了以下结果：端点的迹在扰动下成为开集当且仅当曲线关于标准射影结构的发展像与整个射影线重合。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1维射影结构都可以实现为Engel流形中某个特征环的正则射影结构。证明了高阶切触流形上符号平面场的极大积分子流形总是局部刚性的。第三，我们尝试将叶理的Gestival-Langevin-Walczak几何熵推广到不可积的情形。为了定义熵，我们需要使用积分曲线。我们认识到，如果我们专门使用积分曲线与有界几何，我们能够定义一个概念的熵为nonintegrable plane fields.Parts这些结果已发表在会议论文集的国际会议叶2005年，标题下：对刚性的子流形切线nonintegrable叶。