# 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.

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### An Hamiltonian diffeomorphism is also a Poisson diffeomorphism

Let $(M,\{-,-\})$ be a Poisson manifold. An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
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### Hamiltonian diffeomorphisms on a Poisson manifold

Let $(M,\{-,-\})$ be a Poisson manifold. An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
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### Roadmap for learning symplectic geometry, starting from differential geometry

I am a Physics Major. I have done an undergraduate level course of differential geometry. I want to get into symplectic and Riemannian geometry, but I would like to start over from differential one ...
1 vote
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### A vector field is Poisson iff its flow is a Poisson diffeomorphism

I'm studying Poisson geometry from "Lectures on Poisson Geometry" (Crainic, Fernandes, Marcut). Let $M$ be a Poisson manifold and let $X$ be a vector field on $M$. We define $X$ to be ...
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### On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.3-4. (fubini study form is closed)

In exercise 5.3-4. in Marsden's book I'm asked to prove that $\mathbf d \Omega^{fs} = 0$ on $\mathbb P \mathcal H$ directly, where $\mathbb P \mathcal H$ is an arbitrary projective Hilbert space (...
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### Proof of Square Zero Property for the Total Differential in the de Rham Double Complex of a Lie Groupoid

I am trying to understand the de Rham double complex associated to a Lie groupoid, and I am having trouble proving a fundamental property of the total differential, i.e., that it squares to zero. ...
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### Proving isotropy of a groupoid multiplication graph in a quasi-presymplectic groupoid

I'm studying quasi-presymplectic groupoids and I've come across the following proposition which I'm finding challenging to prove. Any help or guidance would be greatly appreciated. Given a Lie ...
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### When are Infinitesimal $\mathfrak{g}-$actions Lie homomorphisms?

A short background from Smooth Manifolds, by Lee. Given $G$ a Lie group and a right $G$-action on a smooth manifold $M$, we have an infinitesimal $\mathfrak{g}:=Lie(G)$-action over $M$. That is, a Lie ...
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### Reflexivity of infinite dimensional vector spaces and existence of non degenerate bilinear forms

How can an infinite dimensional vector space like (most of) the Lebesgue or Sobolev spaces even be reflexive? If an infinite dimensional vector space cannot be isomorphic to its dual space, the how ...
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### Show that exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$

I need to prove that for $\pi: E → M$ a vector bundle then exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$. I star with definitions: Let be $\pi : E → M$ a vector bundle, $\pi$ is a ...
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### criteria for when the total space of a vector bundle over a toric variety is a toric variety itself

notice that this is not the same as being a toric vector bundle. also, i'm not interested in the projectivization. rather, i'm perfectly fine working with smooth quasi-projective toric varieties. ...
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### Symplectic vector fields form a Lie Algebra

I am trying to prove that symplectic vector fields with the usual Lie bracket of vector fields form a Lie Algebra. Recall that a vector field $X$ on a symplectic manifold $(M,\omega)$ is symplectic ...
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Let $G$ be a connected Lie group acting on a smooth manifold $X$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the induced action on the cotangent space $T^*X$ which is always Hamiltonian ...