Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory

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This weaker version of CR-structure: is it studied somewhere

When I study 5-dimensional $\mathcal{N} = 1$ supersymmetry, I came across such structure as follows. $(R, \kappa, \Phi, M)$ is an almost contact 5-manifold, such that \begin{equation} \kappa ...
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506 views

Is there a “unique” homogeneous contact structure on odd-dimensional spheres?

Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then $$ \mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in ...
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2answers
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The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...
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1answer
70 views

Legendrian knots on pages of a compatible open book

Suppose we have a Legendrian knot embedded on a page of an open book compatible with the given contact structure on the 3-manifold. Is it true that the page framing and Thurston-Bennequin framing of ...
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1answer
201 views

Meaning of “ Open Book cannot be Stabilized Further”?

I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...
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1answer
91 views

Factor of 2 In the Definition of Metric Contact Structure

In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation \begin{equation} d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right) ...
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1answer
124 views

3d-analog of “every 2d oriented manifold is complex”

Is there an analog of the statement of "every 2d oriented surface is a complex manifold"? I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR ...
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43 views

a section invariant under Reeb flow

Let $M$ be a compact contact manifold with $R$ the Reeb vector field. Let $E$ be some vector bundle of gauge group $G$, say, $G = SU(2)$ and $E$ is the adjoint vector bundle. So my question is: If ...
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1answer
170 views

Extending Reeb field from contact submanifold to ambient contact manifold

Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...
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1answer
101 views

contact metric structure on squashed spheres

My goal to write down an explicit (and simplest) contact metric structure on squashed $S_\omega^{2n + 1}$ defined as \begin{equation} S_\omega ^{2n + 1} = \left\{ {\left( {{z_i}} \right) \in ...
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142 views

Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures

Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...
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193 views

Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...
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39 views

Contact structures transverse to a given line bundle

Let M be a orientable three dimensional manifold with a 1 dimensional continuous distributon $F$ (i.e 1 dimensional subbundle of TM). It is known that every orientable three dimensional manifold ...
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51 views

Extendability of Contact Structures; Foliations of $S^2$

I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
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1answer
163 views

How to compute Conley-Zehnder indices on prequantization spaces?

My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits ...
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1answer
88 views

Determining Tightness/Overtwistedness of Contact Structure using Lift of Structure to the Universal Cover

All: I would appreciate any ideas, refs., etc. on the following: Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the contact structure, say $\eta$ on $M^3$ lifts to a ...
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2answers
281 views

Automorphisms of Surfaces, Open Books and Contact Structures

Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of MCG(S), the Mapping Class Group of S, i.e., the group of self-diffeomorphisms of S up to isotopy, but with $f$ ...
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1answer
106 views

Group of CR automorphisms

Let $(M, D, J)$ be a strictly pseudoconvex hypersurface type CR manifold with $J$ integrable. Let $D$ be the kernel of a $1$-form $\eta_0$. As known the automorphism group is defined to be $$ CR = \{ ...
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1answer
139 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$ embedded ...
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1answer
280 views

Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?

Context According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form ...
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1answer
334 views

pre-quantization of Jet bundle

We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
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2answers
466 views

Legendrian Tubular Neighborhood Theorem

Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian ...
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119 views

Classification of (almost) contact structures on $S^3$

Question: Is there a classification of almost contact or contact structures on $S^3$? What is it and references? The motivation of this question is as follows: (1) There is one paper showing that a ...
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2answers
178 views

question on Thurston-Bennequin number

I have three questions actually: 1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot? 2- If the above is true, suppose we ...
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1answer
296 views

Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic? For example, is this true for small ...
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1answer
67 views

Minimal Legendrian submanifolds and laplacian of particular functions

I'm reading the paper Lê, Hôngvân(D-MPI-NS); Wang, Guofang(PRC-ASBJ-MSY) A characterization of minimal Legendrian submanifolds in $S^{2n+1}$. Compositio Math. 129 (2001), no. 1, 87–93. Let $x: L^n ...
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107 views

Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
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1answer
160 views

Contact structures and adjunction inequality in 3-manifolds

It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the ...
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2answers
478 views

Tight vs. overtwisted contact structure

I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight ...
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1answer
383 views

the existence of (almost) contact (metric) structure

I am trying to understand some basic facts about (almost) contact (metric) structure, especially on 3-manifolds 5-manifolds. (1) I saw statement that "any compact oriented 3-manifold admit contact ...
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125 views

The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
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280 views

a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...
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1answer
113 views

Contact structures on pseudo-riemannian manifolds

I am looking for references on contact structures in the framework of pseudo-riemannian manifolds. For instance on the Lorentz-Minkowski 3-space (resp. (n+1)-space). Denoting by L (resp. H) the ...
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1answer
206 views

Normalized Hamiltonian holomorphic vector fields on Sasakian manifolds

Hello, I am reading the paper Futaki; Ono; Wang Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differential Geom. 83 (2009), no. 3, 585–635. For your ...
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From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
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1answer
589 views

What is knot contact homology?

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed ...
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1answer
795 views

'Contactization' and Symplectization

Given any manifold $M$, we can get a symplectic manifold by taking the cotangent bundle $T^\ast M$ with symplectic form $\omega=\sum dp_i\wedge dq_i$. Given any manifold $M$, we can get a contact ...
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4answers
582 views

When does a hypersurface have contact-type?

In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type if there is a contact form $\lambda$ such that $d\lambda=\omega|_Y$. Recall that a contact form is a 1-form ...
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1answer
194 views

Why is tb < 0 for boundary of a convex surface?

Why is $tb(K)$ (Thurston-Bennequin invariant) of a Legendrian knot $K$ which is the boundary of a convex surface $\Sigma$ is negative in a contact 3 manifold?
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1answer
358 views

Osculating spaces and distributions on (real) Grassmannian manifold

Hello! Recenlty, doing my research, I came across a quite natural construction, and I would like to know more about it. Unfortunately, being not expert neither in Grassmannians nor in Contact ...
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strong contactomorphism group inside contactomorphism group

Let $(M, \xi)$ be a closed contact manifold with co-oriented contact structure $\xi = \ker \alpha$. Let $\mathrm{Cont}(M, \alpha)$ be the group of diffeomorphisms that preserve the contact form ...
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1answer
149 views

Persistence of boundary punctured holomorphic curves

Given a Legendrian $\Lambda$ in a contact manifold $(Y,\alpha)$, suppose one has a rigid J-holomorphic curve into the symplectization $(M,\omega)=(Y\times\mathbb{R},\mathrm{d}(e^t\alpha))$. As ...
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2answers
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contactomorphism of $S^{2n+1}$ for n>1

Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1? For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the ...
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Thom polynomial for contact algebraic structures

Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$ and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume that contact structure has degree $p$ (see Polynomial contact ...
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1answer
295 views

“Rounding the corners” to get contact boundary

Suppose we have symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ with non-empty boundary of contact . Often we need to deal with the product $M_1 \times M_2$ with the product symplectic ...
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What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics. What are good examples to illustrate him of the usefulness of contact geometry in this context? On one hand the Hamiltonian ...
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contact manifolds dimension five

Hello, Which manifolds in dimension five admit contact structures? I am not too familiar with the contact realm so any references to look at would be much appreciated.
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823 views

Is there a table of (fibred knot) monodromies?

Background/motivation I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...
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6answers
1k views

Polynomial contact structures on $RP^3$

Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then ...
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Question about the dimension of a Contact (Symplectic) manifold

I am reading about contact geometry and I have a question: Why do we only consider contact structure of an odd-dimension manifold? and the same question for definition of symplectic geometry? I think ...