Jiro Adachi
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.
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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.
An Engel manifold is a 4-manifold with a completely non-integrable 2-distribution called Engel structure. I research the functorial relation between Engel manifolds and Contact 3-orbifolds. And I construct an Engel manifold that the automorphism group is trivial.
In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if the characteristic foliation is "nice". In this paper, we introduce the Cartan prolongation of a contact 3-orbifold and the development map associated to a more general Engel manifold, and we give necessary and sufficient condition for the Cartan prolongation to be a manifold. Moreover, we explain the Cartan prolongation of a 3-dimensional contact étale Lie groupoid and the development map associated to any Engel…
We relate open book decompositions of a 4-manifold M with its Engel structures. Our main result is, given an open book decomposition of M whose binding is a collection of 2-tori and whose monodromy preserves a framing of a page, the construction of an En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a contactomorphism. As a consequence, on a parallelizable closed 4-manifold, every open book with toric binding carries in the previous sense an Engel structure. More…
We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges. We also study associated Engel defining forms and define a unique splitting of $TM$ associated with $\mathcal{D}$ $J$-Engel.
A completely nonintegrable $2$-dimensional distribution on a $4$-manifold is called an Engel structure. A $4$-manifold with an Engel structure is called an Engel manifold. The developing map for an Engel manifold is very important tool to determine the Engel structure. Montgomery used it to prove that an Engel automorphism is determined by the values on a global slice. Moreover, Montgomery constructed Engel manifolds whose automorphism group is small. In this paper, we prove that the automorphism group of an Engel manifold is embedded into the automorphism group of the Cartan prolongation of a…
Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles for which these characteristic classes are non-trivial. This shows that the foliated cohomology classes of Kotschick and Morita carry information that is not merely topological.
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。
We classify the irreducible components of the space of foliations on Fano 3-folds with rank one Picard group. As a corollary we obtain a classification of holomorphic Poisson structures on the same class of 3-folds.
On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability
该研究计算了一类紧时空的类光测地线切触流形，推广至Zoll流形乘积时空，并用Engel几何刻画了三维切触流形对应时空类光测地线空间的条件及时空恢复方法。
Vanishing cycles of symplectic foliations
本文提出强辛叶状结构作为紧叶状结构的高维推广，引入 Lagrange 消失闭链并证明其构成强性障碍，同时构造各余维数例子并梳理相关全纯曲线理论。
In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal Hamiltonian vector fields on the plane. We prove that this cohomology class can be decomposed as a product of a certain leaf cohomology class of degree 5 and the transverse symplectic class. This is similar to the well known factorization of the Godbillon-Vey class for codimension n foliations. We also interpret the characteristic classes of transversely symplectic foliations introduced by Kontsevich in terms of the known classes and prove …
Numerically flat foliations and holomorphic Poisson geometry
本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。
Contact line bundles, foliations, and integrability
本研究利用接触线丛截面空间的Jacobi结构构建接触系统非交换可积性，可统一处理两类接触流形的接触系统，还可用于研究耗散哈密顿系统。
Leafwise de Rham cohomology of generic Reeb foliations
本文证明了对于通用Reeb foliation，其叶wise de Rham cohomology的H^0为实数域，而H^1在三维情况下为无限维。
Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed
本文将测地圆叶状结构的结果推广到接触流形上的 Reeb 流与哈密顿流，证明连通接触流形上所有轨道闭合时周期完全相同，并给出谐振子、Lotka-Volterra 系统等例子及其在量子力学谱问题中的应用。
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