On the classification of tight contact structures I

On the coarse classification of tight contact structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξis tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dime…

测地向量场、诱导切触结构与三维中的紧致性

Geodesic vector fields, induced contact structures and tightness in dimension three

本文简化了Gluck-Harrison定理的证明，证明了在非负混合曲率条件下测地向量场诱导切触结构，并分类了具有等距Reeb流的闭切触3-流形，推广了紧致性结果。

Homotopy classification of contact foliations on an open contact manifold

We give a complete classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient manifold. The results are analogues of Haefliger's classification of foliations on open manifold.

Homologie de contact des varietes toroidales

We show that contact homology distinguishes infinitely many tight contact structures on any orientable, toroidal, irreducible 3-manifold. As a consequence of the contact homology computations, on a very large class of toroidal manifolds, all known examples of universally tight contact structures with nonvanishing torsion satisfy the Weinstein conjecture. ----- On montre que l'homologie de contact distingue une infinite de structures de contact tendues sur toute variete toroidale irreductible et orientable de dimension trois. En consequence des calculs d'homologie de contact, sur une tres large…

三维切触流形的定量紧性：一种次黎曼方法

Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach

本文采用次黎曼方法研究三维切触流形的定量紧性，引入切触Jacobi曲线给出紧性半径下界，验证经典模型下估计的紧性，还得到K-切触情形下Cartan-Hadamard定理的切触类比。

Tight Beltrami fields with symmetry

Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $α$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $α$, we provide a bound for the volume $\text{Vol}(M)$ and the curvature, which implies the universal tightness of the contact structure $ξ=\kerα$.

大圆纤维化与奇数维球面上的切触结构

Great Circle Fibrations and Contact Structures on Odd-Dimensional Spheres

本文证明从5-球面开始，所有奇数维球面都存在与纤维正交的超平面分布非切触结构的光滑大圆纤维化，打破了3-球面中的已知结论。

On the classification of contact metric $(k,μ)$-spaces via tangent hyperquadric bundles

We classify locally the contact metric (k,mu)-spaces whose Boeckx invariant is $\le -1$ as tangent hyperquadric bundles of Lorentzian space forms.

Tight holomorphic maps, a classification

We classify all tight holomorphic maps between Hermitian symmetric spaces of non-compact type.

S²上主S¹丛上不变紧接触形式的尖锐收缩不等式

Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

本文研究S²上非平凡主S¹丛中S¹不变紧接触形式的收缩问题，证明了依赖Euler类的尖锐不等式，并将其应用于旋转对称Finsler度量、可缩Reeb轨道及Viterbo猜想特例。

Almost contact B-metric structures and the Bianchi classification of the three-dimensional Lie algebras

The object of investigation are the almost contact manifolds with B-metric in the lowest dimension three, constructed on Lie algebras. It is considered a relation between the classes in the Bianchi classification of three-dimensional real Lie algebras and the classes of a classification of the considered manifolds. There are studied some geometrical characteristics in some special classes.

A classification of $ N(κ)$-contact metric manifolds with $ \mathcal{T} $-curvature tensor

In this paper, we present a classification of $ N(κ)$-contact metric manifolds with using some special flatness conditions on $ \mathcal{T} $-curvature tensor. We examine $\mathcal{T}$-flat, quasi-$\mathcal{T}$-flat, $ ξ$-$\mathcal{T}$-flat and $ φ$-$\mathcal{T}$-flat $ N(κ)$-contact metric manifolds .Also,we consider the conditions $ \mathcal{T}(ξ,X).R=0 $ and $ \mathcal{T}(ξ,X).S=0 $ for the Riemannian curvature tensor $ R $ and Ricci curvature tensor $ S $. Thus, we obtain a classification of $ N(κ)$-contact metric manifolds.

On the classification of almost contact metric manifolds

On connected manifolds of dimension higher than three, the non-existence of $132$ Chinea and González-Dávila types of almost contact metric structures is proved. This is a consequence of some interrelations among components of the intrinsic torsion of an almost contact metric structure. Such interrelations allow to describe the exterior derivatives of some relevant forms in the context of almost contact metric geometry.

Structures de contact sur les varietes fibrees en cercles au-dessus d'une surface

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isot…

A classification of the torsion tensors on almost contact manifolds with B-metric

The space of the torsion (0,3)-tensors of the linear connections on almost contact manifolds with B-metric is decomposed in 15 orthogonal and invariant subspaces with respect to the action of the structure group. Three known connections, preserving the structure, are characterized regarding this classification.

环面与球面上的接触结构和贝尔特拉米场

Contact structures and Beltrami fields on the torus and the sphere

本文在3-球面和3-环面上构造了新的显式紧和过度扭转接触结构，证明了弱相容度量下接触球面定理不成立，并建立了紧接触结构的几何刚性。

Classification results for three-dimensional (para)contact metric and almost (para)cosymplectic $(κ,μ)$-spaces

It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(κ,μ)$-manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic $(κ,μ)$-spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic $(κ,μ)$-spaces. There is obtained local classification of paracontact metric $(κ,μ)$-spaces, and almost paracosymplectic $(κ,μ)$-spaces, for every possible value of $κ$.

Classification of tight homomorphisms

A totally geodesic map $f:\mathcal X_1\to\mathcal X_2$ between Hermitian symmetric spaces is tight if its image contains geodesic triangles of maximal area. Tight maps were first introduced in [BIW09], and were classified in [Ham13, Ham14, HO14] in the case of irreducible domain. We complete the classification by analyzing also maps from reducible domains. This has applications to the study of maximal representations and of the Hitchin component for representations in the groups $Sp(2n,\mathbb R)$.

Dynamical Quantization of Contact Structures

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometri…

Homology computation for a class of contact structures on $T^3$ and equivariant reductions

We consider a family of tight contact structures on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show some algebraic equivariant homology reductions.

On the classification of the almost contact metric manifolds

The vector space of the tensors $\mathcal F$ of type (0,3) having the same symmetries as the covariant derivative of the fundamental form of an almost contact metric manifold is considered. A scheme of decomposition of $\mathcal F$ into orthogonal components which are invariant under the action of $U(n)\times 1$ is given. Using this decomposition there are found 12 natural basic classes of almost contact metric manifolds. The classes of cosymplectic, $α$-Sasakian, $α$-Kenmotsu, etc. manifolds fit nicely to these considerations. On the other hand, many new interesting classes of almost contact …

A classification result and contact structures in oriented cyclic orbifold

We prove every oriented compact cyclic $3$-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compact cyclic 3-orbifolds. We show every contact structure in an oriented compact cyclic $3$-orbifold contactified by our method is homotopic to an overtwisted structure with the overtwisted disc intersecting the singular locus of the orbifold. We pose Eliashberg's like characterizat…

Classification of almost contact metric structures on three-dimensional non-unimodular Lie groups

In this paper, left-invariant almost contact metric structures on three-dimensional non-unimodular Lie groups are investigated. It is proved that for every Riemannian Lie group, there is one of these structures. In addition, left-invariant normal almost contact metric structures on three dimensional non-unimodular Lie groups are classified.

K接触形式的分类及其子拉普拉斯算子的谱不变量

Classification of K-contact forms and spectral invariants of their sub-Laplacians

本文研究了K接触形式的分类及其子拉普拉斯算子的谱不变量，证明了紧致三维流形上存在此类形式的条件，并与Reeb轨道周期相关联。

The geometry of contact metric three manifolds

Determining the associated metrics we get a local classification of contact metric three manifolds.

Lie groups as 3-dimensional almost contact B-metric manifolds in the main vertical classes

Almost contact B-metric manifolds of dimension 3 are constructed by a two-parametric family of Lie groups. The class of these manifolds in a known classification of almost contact B-metric manifolds is determined as the direct sum of the main vertical classes. The type of the corresponding Lie algebras in the Bianchi classification is given. Some geometric characteristics and properties of the considered manifolds are obtained.

$b$-接触结构的存在性与分类

Existence and classification of $b$-contact structures

本文研究$b$-流形上的$b$-接触结构，证明三维过扭叶情形的存在性$h$-原理，并提出分类方法，完成临界曲面为二球面或未打结环面的$S^3$分类。

Classification of 1st order symplectic spinor operators over contact projective geometries

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one $1^{st}$…