# Questions tagged [contact-geometry]

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

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### 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 ...

**3**

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### 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 $...

**2**

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121 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 ...

**3**

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### 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 ...

**1**

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**0**answers

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### 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 ...

**14**

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**3**answers

803 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?

**3**

votes

**2**answers

163 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\...

**2**

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166 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 ...

**5**

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### 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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66 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 ...

**2**

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**1**answer

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### 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 $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...

**9**

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539 views

### 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}:...

**4**

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107 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 ...

**9**

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**1**answer

405 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) ...

**5**

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### 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 ...