Contact Geometry studies what in some sense is a diametric opposite to integrable distributions. A contact structure is a maximally non-integrable hyperplane field on an odd-dimensional manifold.

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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“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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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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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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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 ...