**0**

votes

**1**answer

67 views

### Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...

**0**

votes

**1**answer

94 views

### Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds

We know every contact manifold admits a symplectic submanifold, e.g., by Giroux's bijection : if $\omega$ is a contact form for a 3-manifold $M^3$ , then $d \omega$ is symplectic on the fibers of the ...

**3**

votes

**1**answer

161 views

### Generalization of Giroux's Theorem for Higher Dimensions?

Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized:
In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any ...

**2**

votes

**1**answer

60 views

### Embedded Contact Homology and Manifold Decompositions

Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.
In the symplectic ...

**5**

votes

**0**answers

190 views

### Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...

**2**

votes

**1**answer

71 views

### non-isotopic but homotopic tight contact structure

By a theorem of Eliashberg, two overtwisted contact structures on a 3-manifold which belong to the same homotopy class (as plane fields), are also isotopic (through contact structures). Is there an ...

**1**

vote

**0**answers

83 views

### 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 ...

**11**

votes

**3**answers

543 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 ...

**8**

votes

**2**answers

249 views

### 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 ...

**1**

vote

**1**answer

86 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 ...

**2**

votes

**1**answer

221 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.) ...

**1**

vote

**1**answer

96 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)
...

**3**

votes

**1**answer

130 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 ...

**0**

votes

**0**answers

47 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 ...

**2**

votes

**1**answer

183 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 ...

**1**

vote

**1**answer

103 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 ...

**3**

votes

**1**answer

156 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 ...

**3**

votes

**0**answers

212 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, ...

**1**

vote

**0**answers

46 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 ...

**2**

votes

**0**answers

59 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 ...

**2**

votes

**1**answer

198 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 ...

**4**

votes

**1**answer

96 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 ...

**3**

votes

**2**answers

315 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 $f$ ...

**1**

vote

**1**answer

108 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 = \{ ...

**6**

votes

**1**answer

145 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 ...

**5**

votes

**1**answer

292 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 ...

**1**

vote

**1**answer

348 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 $J^kM×\mathbb{R}$?

**3**

votes

**2**answers

529 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 ...

**1**

vote

**0**answers

123 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 ...

**2**

votes

**2**answers

191 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 ...

**3**

votes

**1**answer

312 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 ...

**1**

vote

**1**answer

69 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 ...

**2**

votes

**0**answers

114 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.
...

**1**

vote

**1**answer

170 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 ...

**4**

votes

**2**answers

549 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 ...

**1**

vote

**1**answer

418 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 ...

**1**

vote

**0**answers

137 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 ...

**1**

vote

**0**answers

282 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}/ ...

**3**

votes

**1**answer

127 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 ...

**1**

vote

**1**answer

210 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 ...

**7**

votes

**0**answers

201 views

### 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 $\mathbb{R}^n$. ...

**5**

votes

**1**answer

633 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 ...

**6**

votes

**1**answer

850 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 ...

**5**

votes

**4**answers

603 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 ...

**3**

votes

**1**answer

199 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?

**2**

votes

**1**answer

361 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 ...

**9**

votes

**2**answers

527 views

### 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 ...

**2**

votes

**1**answer

150 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 ...

**4**

votes

**2**answers

330 views

### 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 ...

**5**

votes

**3**answers

528 views

### 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 ...