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Topological study of Engel structures and its characteristic foliations

恩格尔结构及其特征叶状结构的拓扑研究

基本信息

批准号：
14540064
负责人：
INABA Takashi
金额：
$ 2.5万
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：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

An Engel structure is a maximally non-integrable 2-dimensional plane field on a 4-dimensional manifold. In this research we investigated global properties of Engel structures from the viewpoint of topology. In particular, we paid attention to the characteristic 1-dimensional foliations canonically associated to Engel structures. First, we considered the following problem posed by Gershkovich : Does there exist an Engel structure on the 4-dimensional Euclidean space whose characteristic foliation admits a compact leaf ? We affirmatively solved this problem by constructing a concrete example. We also applied the construction to obtaining Engel structures on other open 4-manifolds. This result was published in Bulletin of the Australian Mathematical Society. Next, P.Walczak, the foreign joint investigator of this research, constructed an Engel structure using an Anosov flow. The head investigator remarked that this Engel structure is essentially new. Namely, it is not isotopic to any other known examples. Thirdly, we sought 1-dimensional transversely parallelizable foliations on 4-dimensional manifolds which cannot be topologically conjugate to the characteristic foliation of any Engel structure. It is known that a characteristic foliation admits a tangential projective structure and transverse contact structure. We observed the following fact : If a compact leaf of a characteristic foliation has finite holonomy, then the projective. structure of the leaf is not affine. Using this fact we found a l-dimensional transversely parallelizable foliation which is not topologically conjugate to the characteristic foliation of any Engel structure. Finally, we initiated the study of generalizing the rigidity property of characteristic foliations of Engel structures to the cases of higher dimensional plane fields.

恩格尔结构是四维流形上的最大不可积二维平面场。本文从拓扑学的角度研究了Engel结构的全局性质。特别是，我们注意到典型的1维叶理与恩格尔结构。首先，我们考虑了Gershkovich提出的以下问题：是否存在恩格尔结构的4维欧氏空间的特征叶承认一个紧凑的叶？我们通过构造一个具体的例子，肯定地解决了这个问题。我们也将这种构造应用于其它开4-流形上的Engel结构。这一结果发表在《澳大利亚数学学会公报》上。接下来，本研究的外国联合研究员P. Walczak使用Anosov流构建了恩格尔结构。首席研究员说，这种恩格尔结构基本上是新的。也就是说，它与任何其他已知的例子都不是同位素。第三，我们在4维流形上寻找1维横向可平行的叶理，这些叶理不能与任何恩格尔结构的特征叶理拓扑共轭。已知特征叶理具有切向投影构造和横向接触构造。我们观察到以下事实：如果一个紧凑的叶的特征叶理有有限的holonomy，那么投射。叶的结构不是仿射的。利用这一事实，我们发现了一个l维横向平行叶理，这是不是拓扑共轭的特征叶理的任何恩格尔结构。最后，我们开始了将恩格尔结构的特征叶理的刚性性质推广到高维平面场的研究。