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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
        71 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 ...
        3
        votes
        0answers
        119 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 $...
        2
        votes
        0answers
        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
        votes
        0answers
        65 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
        46 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
        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
        2answers
        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
        votes
        0answers
        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
        votes
        0answers
        147 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 ...
        1
        vote
        0answers
        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
        votes
        1answer
        78 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 $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...
        9
        votes
        2answers
        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
        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
        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
        votes
        0answers
        161 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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