# Questions tagged [contact-geometry]

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

177
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### Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$.
According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...

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### Are these contact structures on the open solid torus diffeomorphic?

Let $M=S^1\times \mathbb{R}^2$ and $\alpha_1, \alpha_2$ be a pair of contact one-forms on $M$ such that the restrictions $\alpha_1|_{S^1\times \{0\}}$, $\alpha_2|_{S^1\times \{0\}}$ coincide and ...

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### Expression of the Riemannian metric on the Siegel domain?

I'm looking for proof that, for the complex Siegel domain in $\mathbb C^{n}$ defined by:
$$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |...

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### Looking for examples of 3rd-order contact transformations

In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated:
As a final ...

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### Can differential forms be exact and positive on a distribution?

Let $M$ be a manifold of dimension $d$, and let $\mathscr D$ be a distribution of rank $d - 1$ on $M$ (I would also be interested in lower rank distributions, but mainly I am interested in codimension ...

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### Almost contact structures on spheres

I will write $(M,\xi)=\mathbf{OB}(P,\phi)$ to denote that $M$ admits an open book decomposition with page $P$ and monodromy $\phi$ supporting a contact structure $\xi$. I will focus on the case where $...

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### Characterization of contact vector fields

Let $H$ be a subbundle of the tangent bundle $TM$ of a smooth manifold $M$.
A vector field $K$ on $M$ is contact if its flow $\Phi_K^t$ preserves $H$.
I found in many references the following ...

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### Lagrangian cobordisms from a Legendrian knot to its scaled version

Having a Legendrian knot L in $\mathbb R^3$ and its scale aL (the length of Reeb chords of it are scaled by a>0), are these two Legendrians Legendrian isotopic? Maybe weaker, is there an exact ...

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1
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198
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### Neighborhood theorem for conical Lagrangian

Let $(M,\omega)$ be a compact $2n$ dimensional symplectic manifold and $T$ be a compact smooth $(n-1)$ dimensional manifold.
Let $CT$ be the cone over $T$, i.e. $CT = T\times [0,1] / \sim $ where $\...

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### Overtwisted contact structures on $S^3$

All the isotopy classes of overtwisted contact structures are classified by the Hopf invariant. Are any of these contact structures contactomorphic?
Suppose $d_{3}(\xi_{n}) = n$, then my guess is that ...

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### What is the motivation of contact Hamiltonian equation

I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X_H) = -H\\
\mathrm{d}\eta(X_H,-) = \...

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### Weinstein fillings of a unit cotangent bundle

Given a closed, orientable manifold M, and its unit cotangent bundle $ST^{\ast}M$. I wonder under which conditions $ST^{\ast}M$ admits a subcritical Weinstein filling?

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### Smooth handle attachment vs Weinstein handle attachment

Given a closed smooth manifold $M$ of dimension $n$, to which we attach a $k$-handle $H_k$.
Take $T^{\ast} M$, can one realize $T^{\ast} (M\cup H_k)$ as a result of symplectic or Weinstein handle ...

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343
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### Standard contact forms on the torus

Is there a standard or "simple" contact structure on the $3$-dimensional torus $T^3$, like there are for example for the Eucliden space and the $3$-sphere?
My first thought was to consider a ...

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### Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space

I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it ...

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### Chekanov-Eliashberg Legendrian DGA with positive grading?

I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...

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### On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...

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186
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### Positive vs negative Dehn twist monoids

Given a bounded surface $S$ and a mapping class $h$, construct the open book 3-manifold $(S,h)$. If $h$ lies in the positive monoid of the mapping class group, then $(S,h)$ supports a tight contact ...

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187
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### Are there a geometry behind the singularity formation in solutions to nonlinear ODE's?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.
If we take two apparently simple first order ...

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### Does the blow-up preserve symplectic structure?

Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,...

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178
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### Stein fillable tight contact structures on the 3-torus

Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?

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235
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### Change of Reeb orbits after scaling the contact form

Let $M$ be a contact manifold with a contact form $\theta$ with Reeb vector field $X$ and $f$ be a positive function on $M$. If $\mathcal{L}_X f\neq 0$ the Reeb vector field $X'$ of $\theta'=f \theta$ ...

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### In a contact Lie algebra, when is the Reeb vector a semisimple element?

Below is a question I have come across in my research, and it seems like a question that has been answered (or at least asked) in the past; however, I have been unable to find any references that ...

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### Isotopy of open book supporting same contact structure

In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...

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### Effect of a Lutz twist on Euler number

I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found here, and am trying to ...

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2
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108
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### 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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### 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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114
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### $2$-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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### 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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### 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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### 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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240
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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204
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### $S^3$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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156
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### 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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### «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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### 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 ...