# Questions tagged [contact-geometry]

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

152
questions

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49 views

### Tightness/Overtwistedness of genus one open book decomposition

Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})...

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136 views

### An analogue of the Poisson bracket in contact geometry?

I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...

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**1**answer

75 views

### $2$-Form inducing a non-degenerate form on $\Gamma(T\mathbb{R}^{2n+1})$

Every $2$-form $\omega\in \Omega^2(\mathbb{R}^{2n+1})$ induces a skew-symmetric map
$$
\omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1})
$$
...

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54 views

### How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...

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126 views

### Overtwisted contact forms on open manifolds

I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here.
It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...

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244 views

### Why is the standard contact structure on $\mathbb R^{2n+1}$ called “standard”?

The standard contact structure on $\mathbb R^{2n+1}=(x_1,y_1,\dots,x_n,y_n,z)$ is given by $\ker\alpha$, where $\alpha=dz-\sum_{i=1}^ny_idx_i$. But is there a reason why this contact structure is ...

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196 views

### Maximal Thurston--Bennequin number of boundary knot classes in contact handlebodies

Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\...

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195 views

### Physical motivation for tight/overtwisted dichotomy

I'm learning about tight vs. overtwisted contact structures in contact geometry. I understand that we care about the existence/nonexistence of overtwisted disks in a contact structure in part because ...

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90 views

### Maximal dimension guaranteed for integral manifolds of hyperplane distributions

To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...

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115 views

### Sheaves with specified singular support at infinity coming from hyperplane arrangements

Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\...

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91 views

### Genericity of contact structures all of whose closed Reeb orbits are nondegenerate

First, a contact form $\alpha$ on $M$ with Reeb vector field $R$ is said to be non-degenerate if, for any point $p$ such that $\phi_T^R(p) = p$, we have $\det{(\textrm{id}_{T_pM} - (d \phi_T^R)_p)} \...

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43 views

### How does the knot contact homology augmentation polynomial change under a surgery on the base manifold?

So for every knot $K \in S^3$, there is a knot contact homology of $K$, and we can find the augmentation variety for this homology. The defining polynomial is known as the augmentation polynomial $A(K)...

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80 views

### Convex surfaces with transverse boundary (contact geometry)

Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...

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149 views

### $S^3$ as a Sasakian Manifold

Reading about Sasakian manifolds one come across two slogans:
A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."
B) "A Sasakian manifold sits between two Kahler manifolds -...

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82 views

### Transverse knots with knot types of strongly quasi-positive knots

In 2008, Etnyre and Van Horn-Morris proved that if $L$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $L$ in the standard ...

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91 views

### Superlevel sets of a parametrized quadratic forms

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...

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209 views

### Contactomorphisms have in general no fixed points

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...

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189 views

### boundary connect sum of Ganatra-Pardon-Shende

In Section 3.4 of https://arxiv.org/pdf/1809.03427.pdf, Ganatra-Pardon-Shende define the boundary connnect sum of two exact conical Lagrangians in a Liouville domain. In particular, in Figure 10, they ...

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50 views

### Global solutions for an analytic family of differential operators with initial condition

This is related to this other question question of mine.
Let $M$ be a $3$-dimensional closed (compact without boundary) strongly pseudoconvex manifold and let $\Delta_t$ be a collection of ...

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66 views

### Lifting of Contact isotopies on a symplectization

Let $\mathbb{R}^{2n}\times\mathbb{R}=\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}=\{(q,p,z)|{q},{p}\in\mathbb{R}^n,z\in\mathbb{R}\}$ be a contact manifold with the standard contact form $\alpha=pdq+...

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96 views

### Intuition for the Liouville one-form restricted to the unit cotangent-bundle

So, it seems to be a fairly classical result that the Liouville one-form restricts to the unit cotangent bundle of a Riemann surface equipped with a Riemannian metric. I know that the flow of Reeb ...

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124 views

### Are these two arguments incompatible?

I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...

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118 views

### Is every hyperplane distribution in $\mathbb{R}^n$ given by a nowhere vanishing one form?

From wikipedia: Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-...

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54 views

### «Euclidean» local systems

The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries ...

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360 views

### Symplectisation as a functor between appropriate categories

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its ...

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208 views

### What does it mean for two natural numbers to be *approximately equal*?

This is related to this other question of mine about a paper of Colin and Honda.
I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...

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**1**answer

506 views

### On a corollary of a paper by Colin and Honda

The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...

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91 views

### Reference for “holomorphic contact geometry”

Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact ...

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192 views

### Legendrian surgery and invertible elements in zeroth degree symplectic cohomology

Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...

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154 views

### Stabilizing an open book with Anosov piece

It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...

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89 views

### Contact 3-manifolds with hyperkahler Stein fillings?

Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is
1. Hyperkahler (s.t. Stein structure is the Kahler part of it)
2. not ...

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59 views

### Evolute hypersurfaces

Do you have any references on studies examining the evolute (or focal) hypersurface to a given hypersurface in dimension greater than 3 ?
The evolute can for instance be defined as the envelope of ...

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1k views

### Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?

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295 views

### Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...

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234 views

### First Chern Class of Contact Structure which is not Torsion

Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...

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254 views

### Is there any known relationship between sutured contact homology and Legendrian contact homology?

On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...

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117 views

### On different definitions of a prequantization space

Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).
Without talking ...

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**1**answer

104 views

### On the existence and classification of prequantization spaces over a closed symplectic manifold

Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...

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1k views

### The Lefschetz operator

Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...

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114 views

### Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy

Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...

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1k views

### Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

$\require{AMScd}$
Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.
Theorem 1: (Lagrangian Neighborhood Theorem) Let $...

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188 views

### Contact geometry: approximation of Legendrian mappings

Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$.
Question 1. Is it true that a $C^1$-contact ...

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380 views

### Physical intuition behind prequantization spaces

Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...

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201 views

### Where can I find good surveys on Symplectic and Contact geometry

Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ?
In the symplectic case : I am ...

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1k views

### Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux:
Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and ...

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397 views

### Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?

The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...

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301 views

### Non-trivial examples of overtwisted contact structures

Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\...

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96 views

### Is there a simply connected contact manifold, “non-exactly” fillable, cappable, such that the whole thing is symplectically aspherical?

Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...

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313 views

### Correct notion of chain homotopy for linearized homology of augmented DGAs?

$\require{AMScd}$
Preliminaries: Let $(A,\partial)$ be a differential graded $k$-algebra with an augmentation $\epsilon$. That is, $\epsilon$ is a DGA map $\epsilon:(A,\partial) \to k$ where $k$ is ...

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160 views

### Lutz twist and open book decompositions

Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...