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Questions tagged [symplectic-geometry]

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. For more information please consult the Wikipedia page on symplectic geometry.

1282 questions
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Two times of symplectic orthogonal complement of a closed subspace of a symplectic Banach space

Let $W$ be a infinite dimensional Banach space equipped with a symplectic form $\Omega$. Let $E\subset W$ be a closed subspace and $E^\Omega$ the symplectic orthogonal complement of E w.r.t. $\Omega$. I learned from a book that…
Chengbo
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Existence of symplectomorphism that preserves submanifolds

I have the following question: Let $(M,\omega)$ be a symplectic manifold and let $N_1$ and $N_2$ be submanifolds of $M$ such that there is a diffeomorphism $\psi:M \rightarrow M$ such that $\psi(N_1) = N_2$. Does there exist a symplectomorphism…
user353840
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Given two linear symplectic 4-tori. When are they symplectomorphic?

What are symplectic invariants that can be computed for a linear symplectic 4-torus? The symplectic volume is the simplest one. Is it possible to have something else?
gsmirnov
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Symplectic submanifolds

Suppose I have the symplectic manifold $(M, \omega)$. Now consider a function $C: M \rightarrow \mathbb{R}$ whose differential is non-zero. Then restricting to the submanifold of $M$ given by $C=0$ is not a symplectic manifold right? Since the…
Novo
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Topological intuition about a hamiltonian vector field

Could I ask a conceptual question? If you have a symplectic manifold ($M$, $\omega$) and a real valued function $f : M \to \mathbb{R}$, you can define a hamiltonian vector field $X$ corresponding to $f$ by the following equation; $$ i_{X}(\omega) =…
a--
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A vector field on a symplectic submanifold intersecting the symplectic complement

Consider a 4-dim symplectic vector field $X$ on the symplectic manifold $(M, \omega)$ in $\mathbb{R}^4$ with $\omega= \sum_{i=1}^2 dy_i \wedge dx_i$. Moreover, the linear terms of $X$ are given by $y_1 \partial / \partial x_1 - x_1 \partial /…
Novo
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Geometric interpretation of kernel and critical points of a moment map

A moment map $\mu$ is defined when one has a Hamiltonian $G$-action on a symplectic manifold $M$, for some Lie group $G$. My question is, what are the geometric interpretations of the kernel and critical points of the moment map? For the kernel…
Meer Ashwinkumar
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Canonical symplectic structure.

I've got the next question, we have the canonical standard symplectic form, $\omega_{std}$, given in coordinate form $\omega_{std}=\sum_{i=1}^{n} dq_i \wedge dp_i$, and I want to show that it's non-degenrate. I.e, for $x=(x_1,...,x_n) \ \forall…
MathematicalPhysicist
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symplectic surfaces in 4-manifolds

Is it true that for any surface in a symplectic 4-manifold $X$, representing a given homology class of $H_2(X)$, we can assume it is symplectic? I mean for each second homology class, can we find a symplectic surface representing it?
user217565
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Hamiltonian Action of $S^1$ on $\mathbb{C}^n$

Given the n-dimensional complex space, regarded as a symplectic manifold when equipped with the usual symplectic form $\sum_i r_i dr_i \wedge d\theta_i$, we consider the action of $S^1$ defined by multiplication: $$ z\longmapsto t z $$ were $t$…
Brightsun
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What is the total space of this infinite-dimensional vector bundle?

In McDuff-Salamon, the moduli space $\mathcal{M}(A,\Sigma;J)$ of $J$-holomorphic curves $u:(\Sigma,j)\to(M,j)$ satisfying $u_*[\Sigma]=A\in H_2(M;\mathbb Z)$ is thought of as the zero set of a section of some infinite-dimensional vector bundle. In…
boink
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Examples of symplectic automorphisms of $M$ that set-wise preserve a Lagrangian submanifold $L$

I’m interested in finding interesting examples of non-Hamiltonian automorphisms of a symplectic manifold (not necessarily closed - but preferably so) $(M,\omega)$ that set-wise preserve a Lagrangian sub manifold $L$. For example, given a Lagrangian…
Luigi M
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Geometric interpretation for quantum cup product for $\mathbb{C}P^n$

I'm trying to work out Remark 11.1.9 in Salamon in J-holomorphic Curves and Symplectic Topology (Second Edition) by Dusa McDuff and Dietmar Salamon. They say that one should take $L=[\mathbb{C}P^1]\in H^2(\mathbb{C}P^n)$ (I think this is a typo and…
bas
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Symplectic vector field

In symplectic geometry, given a manifold $M$ with closed nondegenerate symplectic 2-form $\omega$, we say that a vector field $X$ is symplectic if $$\mathcal L_X\omega=0\,.$$ I don't know what does it mean to say that $X$ "preserves" the symplectic…
Mira
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Why is the form $\omega = \sum dx_i \wedge d\xi_i$ non degenerate?

Let's consider $M$ to be a smooth manifold with local coordinates $x_1,\ldots,x_n$ on a coordinate chart $U$. Denote by $\partial / \partial x_i$ the dual basis, $dx_j(\partial / \partial x_i)=\delta_{ij}$ and let $\xi_1,\ldots, \xi_n$ be such that…
Mira
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