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Questions tagged [contact-geometry]

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

123 questions
0answers
54 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 ...
0answers
42 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 ...
3answers
784 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?
2answers
153 views

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 $\mathcal{L}_X\... 0answers 163 views First Chern Class of Contact Structure which is not Torsion Let$(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 ... 0answers 135 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 ... 0answers 61 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 ... 1answer 76 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$H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ... 2answers 523 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}:...
0answers
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 ...
1answer
338 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) ...
0answers
158 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 ...
1answer
224 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 ...
0answers
157 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 ...
2answers
728 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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