# 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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### Arnold's proof of Liouville's Theorem on integrable systems

My question happens to be almost identical to the one left unanswered/closed here, which gives a bit of background information - it may not be necessary. I hope the reason it was closed on ...
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### How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed?

I am reading the book a guide to quantum groups. I have a question on page 18. How to show that Jacobi identity for $\{,\}$ is equivalent to $\omega$ being closed? Any help will be greatly appreciated!...
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### Pulling back a Kähler structure on a symplectic submanifold

Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
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### Notation for vector fields on cotangent bundle

This is probably an easy question. I have some trouble finding the right notation/words for 2 vector fields. Consider $T^*S^1$, the cotangent bundle of the circle. I know that this is a trivial ...
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### Symplectic manifold, flow $\phi_t$ generated by unique vector fiel $X_f$ preserves symplectic form $\omega$?

Let $M$ be a symplectic manifold with symplectic form $\omega$, let $f$ be a smooth function on $M$, and let $X_f$ be the unique vector field on $M$ so that $df(Y) = \omega(X_f, Y)$ for all vector ...
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### Symplectic manifold $M$, unique vector field $X_f$ on $M$ so that $df(Y)= \omega(X_f, Y)$ for all vector fields $Y$?

Let $M$ be a manifold. A symplectic form is a closed $2$-form $\omega$ suh that $X(p) \to \omega(X,\cdot)(p)$ is an isomorphism from $T_pM$ to $T_p^*M$ for each $p \in M$. A manifold together with a ...
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### Open immersion pulls back symplectic form to symplectic form?

If $M$ is symplectic, and $f: W \to M$ is an open immersion, i.e. an immersion where $W$ and $M$ have the same dimension, does $f$ necessarily pull back a symplectic form on $M$ to a symplectic form ...
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### Symplectically embedding 2-spheres in 6-manifolds

There is a wealth of research that I have found that characterize different classes of compact $4$-manifolds by the ability (or lack thereof, in the case of aspherical) to (symplectically or smoothly) ...
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### Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ ...
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### Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
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### What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
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### Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory. So ...
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### What is a pseudo-Kähler manifold?

I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that ...
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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 ...
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### Liouville's theorem and the Wronskian

Liouville's theorem states that under the action of the equations of motion, the phase volume is conserved. The equations of motion are the flow ODE's generated by a Hamiltonian field $X$ and the ...
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### What is a Hamiltonian in a Poisson algebra?

Classical physics on the phase space $T^* M$ (with $M$ a smooth manifold) is done mostly in the following way: one endows $T^*M$ with a Riemannian structure $g^*$ (that will give the kinetic term) and ...
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### Is there a more formal way to prove Liouville's theorem on the conservation of symplectic volume?

By Liouville's theorem the symplectic volume of a symplectic manifold $(M,\omega)$ is preserved under symplectomorphisms. One usually uses the language of classical mechanics to show that and performs ...
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### Is this the pullback of a Hamiltonian flow?

In this reference just in the beginning the author gives the theorem (Theorem 1) of the conservation of a Hamiltonian flow $\phi_t$. According to it this means that $$\frac{d}{dt}\phi_t^* H = 0$$ I ...
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### Kähler metrics on the coadjoint orbits of a compact Lie group

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, ...
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### Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type $$(1)\quad \ddot x + f(x)=0$$ ...
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### Pullback of a Hamiltonian

I understand that a Hamiltonian vector field $H$ creates a Hamiltonian flow $\phi_t$. Now, in order to prove that the Hamiltonian is conserved one uses the following \begin{eqnarray*} \frac{d}{dt}\...
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### Is the group of symplectomorphisms of a symplectic manifold the symplectic group?

The group of symplectomorphisms of a symplectic manifold $M$ is a subgroup of the group of diffeomorphisms $GL(n)$, actually it is a subgroup of $SL(n)$. My question is whether this group of ...
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### A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
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### Exact sequences of sympletic vector spaces

Is it true that if one has an exact sequence of vector spaces: $0 \to V \to W \to S \to 0$ Such that W is a sympletic vector space of even complex dimension and S has even real dimension and V has ...
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### What's the best place to learn Quantum Homology

I am a PhD student and I am trying to learn quantum homology. I already know some of the analysis, but I am struggling to really find a good readable reference which covers enough of the analysis to ...