# Tagged Questions

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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### What integrability of Hamiltonian system says about non compact orbits?

Suppose that we have an $n$-dimensional manifold $M$ and consider a Hamiltonian system on cotangent bundle which is integrable in the following way: there exist $n$ functions $f_1,...,f_n$, which are ...
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### A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
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### Question about definition of non-compact Calabi-Yau manifolds

Using the following definition: Definition $(X, J, \omega, \Omega)$ is a Calabi-Yau manifold if $g(\cdot, \cdot)= \omega(\cdot, J \cdot)$ and $\Omega$ is a nowhere vanishing holomorphic $(n,0)$-form ...
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### Coordinates for a neighborhood of a totally real submanifold

Motivation: I'm interested in which submanifolds of (half dimension) a complex manifold can be expressed locally in coordinates as the real part of the complex coordinates, and was wondering if the ...
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### Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a complex vector space together with fixed symplectic ...
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### How do we give G/T a symplectic structure

I am new in those staff and I can't find anything introductory explaining those stuff. I am interested in the following. Given $G$ a compact Lie group and consider it's maximal torus $T$. Then I have ...
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### Composition of symplectomorphisms not a symplectomorphism?

Given $R^{2n}=C^{n}$ with its standard symplectic form $\Omega$, and v an arbitrary vector, the individual transvections $\tau(p)=p+\Omega(v,p)v$ and $\sigma(p)=p+\Omega(iv,p)iv$ preserve the ...
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### Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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### Why is the tangent space to the orbit through $p\in\mu^{-1}(0)$ an isotropic subspace of $T_pM$?

I'm reading symplectic geometry notes by Ana Cannas da Silva. The set up is a Hamiltonian action $G\curvearrowright(M,\omega)$ of a Lie group $G$ on the symplectic manifold $(M,\omega)$, with moment ...
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### The fact that surface of spherical segment only depends on its height follows from symplectic geometry

It is quite quite well known that the surface of the piece of a sphere with $z_0<z<z_1$ for some values of $z_0,z_1$ is given by $S = 2\pi R (z_1-z_0)$. So this surface area only depends on ...
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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?
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### Pushforward of Liouville measure on configuration space

Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the ...
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### Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
### Cartesian product of $\mathbb{S}^1$ is symplectic
Prove that the Cartesian products of $\mathbb{S}^1$ for $2n$ times is a symplectic manifold. I have just studied the concepts of symplectic manifold in the class of analytical mechanics. I have not ...