**1**

vote

**0**answers

48 views

### why is there such a 1-form on a planar open book?

Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform ...

**5**

votes

**1**answer

335 views

### Are there spaces in which there are no fibered knots?

I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...

**4**

votes

**1**answer

87 views

### Is there a known Legendrian simple link?

Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.
How about the same notion for link ...

**0**

votes

**0**answers

59 views

### Floer homology for manifolds with contact boundary

I am reading the paper " A survey of floer homology for manifolds with contact boundary" by A. Oancea. In theorem 2.1, he discussed the invariance of
Viterbo's theory of symplectic homology with ...

**0**

votes

**1**answer

70 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

104 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

222 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

69 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

195 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 ...

**3**

votes

**2**answers

91 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

84 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

550 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

252 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

88 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

223 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

97 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

48 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

186 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

158 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

216 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

207 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

98 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

324 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

293 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

351 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

548 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

124 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

193 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

316 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

116 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

173 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

561 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

428 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

139 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

283 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

133 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

211 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

202 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

649 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

860 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

610 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

202 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 ...