3-dimensional methods in contact geometry

Local geometry of symplectic divisors with applications to contact torus bundles

In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.

Contact geometry

This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a detailed exposition of the original proof of the Lutz-Martinet theorem. The text ends with a guide to the literature.

The arc complex and contact geometry: non-destabilizable planar open book decompositions of the tight contact 3-sphere

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this result to show that there are planer open books of the standard contact structure on the 3-sphere with 5 (or any number larger than 5) boundary components that do not destabilize. We also show that any planar open book with 4 or fewer boundary components does destabilize.

Grid Diagrams, Braids, and Contact Geometry

We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.

Knots and Contact Geometry II: Connected Sums

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.

A property of the skein polynomial with an application to contact geometry

We prove a finiteness property of the values of the skein polynomial of homogeneous knots which allows to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of their Legendrian embeddings in the standard contact structure of R^3), and a give a short proof that there are only finitely many among these knots that have given genus and given braid index.

Topological methods in 3-dimensional contact geometry

These notes provide an introduction to Giroux's theory of convex surfaces in contact 3-manifolds and its simplest applications. They put a special emphasis on pictures and discussions of explicit examples. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surface is encoded by finitely many curves on the surface. Then we describe the bifurcations that happen in generic families of surfaces. As applications, we explain how Giroux used this technology to reprove Bennequin's theorem saying that the standard contact structure on S^3 is ti…

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

三维 Anosov 流的辛几何与双接触拓扑

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

本文从接触与辛几何角度刻画三维 Anosov 流，建立研究其动力学的几何拓扑框架，并给出双接触结构唯一性及基于 Reeb 流的 Anosov 性判据。

The contact geometry of the restricted 3-body problem

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dynamical system.

Quantitative Darboux theorems in contact geometry

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are c…

Rational linking and contact geometry

In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple. This version of the paper corrects the definition of rational self-linking number in the previous and published version of the paper. Wi…

Einstein Manifolds and Contact Geometry

We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.

Noncommutative integrability and action-angle variables in contact geometry

We introduce a notion of the noncommutative integrability within a framework of contact geometry.

利用双接触几何对 Anosov 流进行手术

Surgery on Anosov flows using bi-contact geometry

本文利用双接触几何定义了一类新的 Anosov 流 Dehn 手术，揭示了 Goodman 手术产生 Anosov 流的条件，并证明了沿简单闭测地线进行 Foulon-Hasselblatt Legendrian 手术是生成接触 Anosov 流的充要条件。

New steps in $C^0$ symplectic and contact geometry of smooth submanifolds

We provide a $C^0$ counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, we prove a quantitative $h$-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, we give another proof of the Dimitroglou Rizell and Sullivan theorem \cite{RS22} which states that Legendrian knots are preserved by contact homeomorphisms, provided their image is smooth. Moreover, our method gives related exa…

Christiaan Huygens and contact geometry

This article is based on the author's inaugural lecture at the University of Cologne on 24 January 2003.

Partially ordered groups and geometry of contact transformations

We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation number. Our approach is based on tools of Symplectic Topology.

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…

Cosphere bundle reduction in contact geometry

We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is taken by the cosphere bundle. We use Albert's method for reduction at zero and Willett's method for non-zero reduction. We provide examples for both cases.

Contact Geometry of the Pontryagin Maximum Principle

This paper gives a brief contact-geometric account of the Pontryagin maximum principle. We show that key notions in the Pontryagin maximum principle---such as the separating hyperplanes, costate, necessary condition, and normal/abnormal minimizers---have natural contact-geometric interpretations. We then exploit the contact-geometric formulation to give a simple derivation of the transversality condition for optimal control with terminal cost.

$\mathbb{S}^2$ 上限制性三体问题的接触几何

Contact Geometry of the Restricted Three-Body Problem on $\mathbb{S}^2$

研究S^2上对称限制性三体问题中能量超曲面的接触几何，证明低于第一临界值及略高时具有接触类型，通过Moser正则化紧化后与RP^3或其连通和接触同胚，并利用Taubes定理推断周期轨道存在。

无穷维中的接触几何

Contact geometry in infinite dimensions

本文将接触几何理论推广到无穷维空间，并研究了时间依赖和耗散哈密顿系统的模型示例。

Generating functions in symplectic and contact geometry

A translated point of a contactomorphism $ϕ$ on a contact manifold with contact form $α$ is a point $p$ where $α$ is preserved under $ϕ$ and whose image under $ϕ$ lies in the same Reeb trajectory. They were introduced as a contact analogon for fixed points of Hamiltonian diffeomorphisms by Sheila Sandon and can be understood as a special case of leafwise fixed points. She established a contact version of the non-degenerate Arnol'd conjecture on spheres using a generating function approach. It turns out that Sandon's proof only works under the assumption that there exists a generating function …

Contact spheres and hyperkähler geometry

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles. This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkahler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-S…

A Note on Toric Contact Geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant's theorem due to Lerman and Tolman [LT] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

Toric contact geometry in arbitrary codimension

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

The contact Banach-Mazur distance and large scale geometry of overtwisted contact forms

In the symplectic realm, a distance between open starshaped domains in Liouville manifolds was recently defined. This is the symplectic Banach-Mazur distance. It was proposed by Ostrover and Polterovich and developed by Ostrover, Polterovich, Usher, Gutt, Zhang and Stojisavljević. The natural question is, can an analogous distance in the contact realm be defined? One idea is to define the distance on contact hypersurfaces of Liouville manifolds and another one on contact forms supporting isomorphic contact structures. Rosen and Zhang recently defined such a distance working with manifolds that…

Anosov 3-流的强适配切触几何

Strongly adapted contact geometry of Anosov 3-flows

本文基于与Reeb动力学的相互作用提供了Anosov 3-流的3维切触几何刻画，证明了所得几何空间与Anosov 3-流空间同伦等价，并证明了关于适配范数渐近同步的技术定理。

Cauchy-Riemann geometry and contact topology in three dimensions

We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. This ``contact torsion'' is expected to be able to distinguish among ``contact lens'' spaces. We also propose the study of a certain kind of monopole equation associated with a contact structure.

3-流形中曲面的$C^0$-切触几何

$C^0$-Contact Geometry of Surfaces in 3-Manifolds

该研究探讨3-流形中曲面的$C^0$-切触几何，证明切触同胚保持特征叶状结构、正则余迷向曲面具有$C^0$-刚性，同时构造反例表明切触凸性具有$C^0$-柔性。

The contact geometry of the spatial circular restricted 3-body problem

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range.

广义接触几何中的Boothby-Wang构造

A Boothby-Wang construction in generalized contact geometry

本文提出了一种在广义接触几何中构造非Poon-Wade类型接触结构的方法，并展示了曲率诱导辛叶状结构的结论。

Differential graded contact geometry and Jacobi structures

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.

Superanalogs of symplectic and contact geometry and their applications to quantum field theory

The paper contains a short review of the theory of symplectic and contact manifolds and of the generalization of this theory to the case of supermanifolds. It is shown that this generalization can be used to obtain some important results in quantum field theory. In particular, regarding $N$-superconformal geometry as particular case of contact complex geometry, one can better understand $N=2$ superconformal field theory and its connection to topological conformal field theory. The odd symplectic geometry constitutes a mathematical basis of Batalin-Vilkovisky procedure of quantization of gauge …

The geometry of symplectic pairs

We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.

Contact topology and CR geometry in three dimensions

We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of $CR$ structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''sphe…

Remarks on Contact and Jacobi Geometry

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…

Liouville's Theorem and the canonical measure for nonconservative systems from contact geometry

Standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton's equations, Liouville's theorem and to find the canonical invariant measure. In this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry. In particular, we find out the generalized Hamilton's equations, Liouville's theorem and the microcanonical and canonical measures invariant under the contact flow. Remarkably, the latter measure has a power law density distribution with respect to the sta…

接触几何中余迷向子流形的刚性结果

A rigidity result for coisotropic submanifolds in contact geometry

本文研究接触流形中紧致正则余迷向子流形的余迷向形变，证明了在特征叶层微分同胚的邻近子流形中其具有刚性，并结合经典叶层刚性结果推广至所有邻近子流形。

Lorentzian distance functions in contact geometry

We define a Lorentzian distance function on the group of contactomorphisms of a closed contact manifold. This distance function is continuous with respect to the Hofer norm on the group of contactomorphisms defined by Shelukhin and finite if and only if the group of contactomorphisms is orderable. To prove this we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined by Rosen and Zhang is non-degenerate. In this case similar results hold for a…

A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain GL(1,R)-principal bundle P\to M which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an `odd-dimensional cousin' of symplectic geometry but rather a part of the latter, namely `…

Geometric decompositions of almost contact manifolds

These notes are intended to be an introduction to the use of approximately holomorphic techniques in almost contact and contact geometry. We develop the setup of the approximately holomorphic geometry. Once done, we sketch the existence of the two main geometric decompositions available for an almost contact or contact manifold: open books and Lefschetz pencils. The use of the two decompositions for the problem of existence of contact structures is mentioned.

Relative Invariants, Contact Geometry and Open String Invariants

In this paper we propose a theory of contact invariants and open string invariants, which are generalizations of the relative invariants. We introduce two moduli spaces $\bar{\mathcal{M}}_{A}(M^{+},C,g,m+ν,{\bf y},{\bf p},(\mathbf{k},\mathfrak{e}))$ and $\bar{\mathcal{M}}_{A}(M,L;g,m+ν,{\bf y},{\bf p},\overrightarrowμ)$, prove the compactness of the moduli spaces and the existence of the invariants.

Singular Contact Geometry and Beltrami Fields in Cholesteric Liquid Crystals

The description of point defects in chiral liquid crystals via topological methods requires the introduction of singular contact structures, a generalisation of regular contact structures where the plane field may have singularities at isolated points. We characterise the class of singularities that may arise in such structures, as well as the subclass of singularities that can occur in a Beltrami field. We discuss questions of global existence, and prove that all singular contact structures with nonremovable singularities are overtwisted. To connect the theory to experiment we also discuss no…

Contact geometry and thermodynamics of black holes in AdS spacetimes

In this paper we discuss a formulation of extended phase space thermodynamics of black holes in Anti de Sitter (AdS) spacetimes from the contact geometry point of view. Thermodynamics of black holes can be understood within the framework of contact geometry as flows of vector fields generated by Hamiltonian functions on equilibrium submanifolds in the extended phase space that naturally incorporates the structure of a contact manifold. Deformations induced by the contact vector fields are used to construct various maps among thermodynamic quantities. Thermodynamic variables and equations of st…

Geometry of contact transformations and domains: orderability versus squeezing

Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a cont…

Lectures on Holomorphic Curves in Symplectic and Contact Geometry

This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from readers, I've decided to make a presentable half-finished version available here. Further chapters will be added in future updates.

Symplectic, Poisson, and contact geometry on scattering manifolds

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introd…

空间限制性四体问题中 Hill 近似的接触几何

Contact geometry of Hill's approximation in a spatial restricted four-body problem

本文证明空间等边 CR4BP 的 Hill 近似系统在相应能量范围内具有接触性质，从而为研究特洛伊小行星附近小天体动力学引入全纯曲线与 Floer 理论工具。

Contact integral geometry and the Heisenberg algebra

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifo…

接触几何与Lagrange-Poincaré-Herglotz方程中的对称性约化与重构

Symmetry reduction and reconstruction in contact geometry and Lagrange-Poincaré-Herglotz equations

该研究针对对称群下不变的接触拉格朗日系统，基于Herglotz向量场、其投影场及准速度推导了Lagrange-Poincaré-Herglotz约化方程的显式坐标形式，并给出了相关示例，丰富了接触几何下的对称约化理论。

Twisted supersymmetric 5D Yang-Mills theory and contact geometry

We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills…

渐进接触几何与AKSZ形式主义

Graded Contact Geometry and the AKSZ Formalism

本文提出了一种基于微分graded接触流形的AKSZ形式主义，恢复了Jacobiσ模型并构建了与Courant-Jacobi李代数Bundle相关的三维拓扑场论。