T. Vogel
We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, there is an Engel structure on M_1#M_2#(S^2xS^2) which is closely related to the original Engel structures.
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We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete h-principle for overtwisted Engel structures with fixed overtwisted disc.
In [CPPP] it was shown that Engel structures satisfy an existence $h$-principle, and the question of whether a full $h$-principle holds was left open. In this note we address the classification problem, up to Engel deformation, of Cartan and Lorentz prolongations. We show that it reduces to their formal data as soon as the turning number is large enough. Somewhat separately, we study the homotopy type of the space of Cartan prolongations, describing completely its connected components in the overtwisted case.
This article introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete h-principle when some auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.
We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classification of Engel structures on circle bundles with characteristic line field tangent to the fibers.
Engel structures on M x S^1 and M x I are studied in this paper, where M is a 3-dimensional manifold. We suppose that these structures have characteristic line fields parallel to the fibres, S^1 or I. It is proved that they are characterized by contact structures on the cross section M, the twisting numbers, and Legendrian foliations on both ends M x dI in the case of M x I.
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg--Thurston and by Mitsumatsu.
4-manifolds and non-existence of open book
本文证明存在无穷多个可平行化的闭定向4-流形不具有开卷分解，从而说明Engel流形没有Giroux对应的类似物，且并非每个Engel流形都有支撑开卷。
We apply spectral sequences to derive both an obstruction to the existence of $n$-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on $M\times\mathbb{S}^1$ with characteristic line field tangent to the fibers is determined by the contact structure induced on a cross section and the twisting of the Engel structure along the fibers. Our results show that this statement needs some modification: to classify the diffeomorphism type of the Engel structure we additionally have to fix a class in the fi…
Transverse tori in Engel manifolds
研究Engel流形中环面的横结构，揭示其与接触3流形中结的类比关系，分类形式不变量并探讨其在同伦中的唯一性作用。
Engel CR submanifolds of $\mathbb{C}^3$
本文给出了3维流形上$\mathbb{S}^1$丛浸入或嵌入$\mathbb{C}^3$且复切定义Engel结构的充分条件，构造了首批复切为Engel的紧子流形实例，回答了Y. Eliashberg的问题。
Existence and classification of maximal growth distributions
该文研究光滑流形上极大增长分布的存在性与分类，证明秩>2的极大增长分布满足完全h原理，建立微分关系丰度检验新准则，正面解答了相关长期公开问题，同时说明秩2情形的非丰度性。
We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.
We establish a parametric extension $h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the $3$-dimensional result from \cite{Eli89}. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.
We exhibit a 3-manifold which admits no tight contact structure.
We exhibit homology spheres which never yield lens spaces by any integral Dehn surgery by using Ozsvath Szabo's contact invariant.
We prove $h$-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on $h$ principle of contact foliations in terms of the regular Jacobi structures.
Discrete conformal structures on surfaces with boundary (II) -- Rigidity and Existence
本文研究带边界曲面上离散共形结构的刚性与存在性，将Guo-Luo及Guo的相关结果推广到公理化框架下的一般情形。
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of Schwartzman, whereas the methods are adapted from work of Sullivan.
This is a continuation of a previous paper of same title. The degeneration, i.e. curvature blow-up, of sequences of metrics appoaching the Sigma constant, assumed non-positive, is analysed. The degeneration is related to the sphere decomposition of the 3-manifold M, in case M is sigma-tame.
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