Topological study of Engel structures and its characteristic foliations

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恩格尔结构的全局性质。特别地，我们关注了通常与恩格尔结构相关的一维叶状特征。首先，我们考虑了Gershkovich提出的以下问题：在四维欧几里得空间上是否存在一种恩格尔结构，其特征叶理允许紧叶？通过构建一个具体的例子，我们肯定地解决了这个问题。我们还将该构造应用于其它开放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，该项研究的外国联合研究者P.Walczak利用阿诺索夫流构造了一个恩格尔结构。首席调查员说，这个恩格尔结构本质上是新的。也就是说，它与任何其他已知的例子都不是同位素。第三，我们在四维流形上寻找不能与任何恩格尔结构的特征叶形拓扑共轭的一维横向可并行叶形。我们知道，一个典型的叶理具有切向投影结构和横向接触结构。我们观察到以下事实：如果一个特征叶的紧致叶具有有限完整性，则其投影。叶片的结构不是仿射的。利用这一事实，我们发现了一个与任何恩格尔结构的特征叶理拓扑不共轭的l维横向平行叶理。最后，我们开始了将Engel结构特征叶理的刚性性质推广到高维平面场的研究。

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.