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        Questions tagged [contact-geometry]

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

        3
        votes
        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 ...
        1
        vote
        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 ...
        14
        votes
        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?
        3
        votes
        2answers
        153 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
        votes
        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 ...
        5
        votes
        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 ...
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        vote
        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 ...
        2
        votes
        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 $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...
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        votes
        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}:...
        4
        votes
        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 ...
        9
        votes
        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) ...
        5
        votes
        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 ...
        7
        votes
        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 ...
        3
        votes
        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 ...
        21
        votes
        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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