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        All Questions

        2
        votes
        1answer
        76 views

        Every symplectic submanifold is J-holomorphic

        I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$. The way I am ...
        1
        vote
        1answer
        138 views

        A special non vanishing vector field on odd dimensional compact manifolds

        Edit: According to the comment of Michael Albanese we revise the question. Assume that $n$ is an odd integer and $M\subset \mathbb{R}^{2n}$ is a compact orientable $n$ dimensional submanifold. Does ...
        3
        votes
        2answers
        157 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
        67 views

        Transitivity of Diff on the space of embeddings of balls

        Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...
        5
        votes
        2answers
        318 views

        Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

        I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
        22
        votes
        1answer
        625 views

        What can we say about the Cartesian product of a manifold with its exotic copy?

        Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$. Is it true that $M\times M$ is diffeomorphic to $M\times M^E$? I am ...
        3
        votes
        1answer
        148 views

        What is symplectic cut of a 4-ball?

        Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. ...
        6
        votes
        1answer
        125 views

        Existence of isotopy preserving the action

        Let $\gamma_1$ and $\gamma_2$ be simple closed curves in $R^4.$ Let $\lambda= x_1 dy_1+ x_2dy_2.$ Suppose that $\int_{\gamma_1} \lambda= \int_{\gamma_2} \lambda.$ I am looking for a reference for ...
        4
        votes
        1answer
        276 views

        When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian?

        (Reposted from https://math.stackexchange.com/questions/2589600/when-is-a-divergence-free-vector-field-on-the-tangent-bundle-of-a-riemannian-man) Starting with a closed, connected Riemannian manifold $...
        2
        votes
        1answer
        109 views

        Hamiltonian Group action with infinitely many stabiliser types

        What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types? Infinitely many stabiliser types means that ...
        6
        votes
        1answer
        251 views

        Every contractible smooth loop has a neighbourhood with $H^2=0$

        Let $c: S^1 \to M$ be a smooth contractible loop (not necesarily an embedding, or even an immersion) on the connected, compact symplectic manifold $(M,\omega)$ (if this helps somehow, $c$ is a $1$-...
        22
        votes
        2answers
        493 views

        $(M,\omega)$ not symplectomorphic to $(M,-\omega)$

        Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$. In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...
        11
        votes
        0answers
        466 views

        Third cohomology of symplectic $6$-manifolds

        Suppose that $A$ is a finitely generated abelian group with even rank and without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H^{3}(M,\...
        10
        votes
        1answer
        697 views

        Morse theory in infinite dimensions

        It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts. Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy ...
        7
        votes
        0answers
        210 views

        Higher homotopy of diffeomorphism groups from singularities

        In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth ...

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