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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.

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Duistermaat Heckmann formula

Which physical concepts are related to the Duistermaat-Heckmann formula, and are symplectic schur functions related to it by any chance?
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Integrable system vs lagrangian fibration

Every complete integrable system $I:M \rightarrow \mathbb{R}^n$ is a regular langrangian fibration on a dense subset of the symplectic manifold $M$. It is also known that locally every lagrangian ...
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Euler field for magnectic symplectic structure in $T^*Q$

Let $Q$ be a differentiable manifold, $\pi\colon T^*Q\to Q$ denote its cotangent bundle, and $B \in \Omega^2(Q)$ be a closed form. I'm playing around with some things, and I'm not sure whether it ...
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Why does $g\circ \exp(tv) \circ g^{-1}$ give the one-parameter group of diffeomorphisms generated by $g_* v$?

I have a question regarding the following proof from Cannas da Silva - Introduction to Symplectic and Hamiltonian Geometry: Let $(M, \omega)$ be a symplectic manifold, and let $\alpha$ be a 1-form ...
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Why the Hamiltonian is constant along the integral curves of the hamiltonian vector field?

Let $H$ the Hamiltonian of a system and $\gamma $ an integral curve of the Hamiltonian vector field, i.e. if $\gamma (t)=(q(t),p(t))$ and $H(p,q)$ is the Hamiltonian, then $$\begin{cases} \dot p=-H_q\\...
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Intuition about Poisson bracket

I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket ...
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Constructing diffeomorphisms of moduli spaces of $J$-holomorphic curves

Let $M^{2n}$ be a smooth manifold admitting two almost complex structures $J_0$ and $J_1$. Suppose that $J_0$ and $J_1$ are both regular in the sense that the moduli space $$ \mathcal{M}_i:=\mathcal{...
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Identity Involving Lie Derivative and Local Flows

I'm trying to show, $$ \frac{d}{dt} \varphi_t^* \omega = \varphi_t^* \left( \mathcal{L}_{X_t} \omega \right)$$ but I have another question as well. Every case in which the lie derivative is ...
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introduction to hamiltonian actions and moment maps.

I'm trying to find some good articles or books to learn about moment maps and hamiltonian actions. I ddo have some basic differential geometry (and representation theory) knowledge, but not really ...
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What generalizes the definition of a symplectic manifold?

Reference required for the following: A vector field $v$ on a symplectic manifold $(M,\omega)$ is that preserves the symplectic form $\omega$ along its direction $$\mathcal L_v \omega =0 \,.$$ In ...
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Rescaling a symplectic form and integral cohomology

Let $(M,\omega)$ be a symplectic manifold. I am trying to understand a procedure which seems so obvious that its implications are omitted in any article I could read. I encountered the following: ...
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Deforming antiholomorphic involutions

Let $(M,J)$ be a compact smooth almost complex manifold. We can "deform" $J$ as follows: if$A$ is a smooth section of the endomorphism bundle $\mathrm{End}(TM)\to M$ satisfying $ AJ=-JA, $ it follows ...
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Explicit derivation of the interior product between the Hamiltonian Vector Field and the symplectic two-form

I am trying to understand Example 5.12 of the book Geometry, Topology and Physics by Nakahara. In particular, consider the two-form $\omega=dp_{\mu} \wedge dq^{\mu}$ (1) and the Hamiltonian vector ...
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Symplectomorphisms of a Riemann surface

Let $X$ be a compact Riemann surface (one dimensional complex manifold) of genus $g > 1$, fix a non-degenerate two form $\omega$ on $X$, it is automatically closed by dimension reasons, so it is a ...
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The idea behind Delzant construction of a toric manifold from a convex polytope

I am trying to understand how to visualize a symplectic toric manifold from its moment polytope, following chapter 29.4 in "Lectures on Symplectic Geometry" by Ana Cannas da Silva: https://people.math....
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Recommend one good Symplectic topology textbook

I need to gain some idea of this topic (and holomorphic curves) by the end of next semester. So, if you can, please suggest a textbook or some lecture notes that'll help to build geometric insights.
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subspaces of a symplectic vector spaces are of special forms.

Let $(V,\omega)$ be a symplectic vector space. Let $F \subseteq V$ be a subspace. Show that $V$ admits a symplectic basis $\{e_1,\ldots,e_n,f_1,\ldots,f_n\}$ with the following properties: (1) If $...
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Show that $i^*\omega=-d\alpha$ where $\omega$ is canonical symplectic structure.

I am stuck on the following problem. Let $\omega$ be the canonical symplectic structure. Let $\alpha$ be a differential 1-form on $M$, and let $$L=\{(p,\alpha(p)):p\in M\}\subseteq T^*M.$$ ...
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Vague explanation of Moment maps

I'm orientating myself for some subject for a combined mathematics/physics bachelor project. One of the things I encountered had to do with classical mechanics and symplectic geometry. One important ...
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1answer
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Equivalence of definition of symplectic form

Suppose that $V$ is a vector space of dimension $2n$, and let $\omega \in \Lambda^2(V)$. Prove that the following two statements are equivalent. (1) $\tilde{\omega} : V \rightarrow V^*$ defined ...
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Given any matrix $A$, does there exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal?

Given any $2n\times 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where $$ B=\begin{bmatrix} B_1&0&\cdots&0\\ 0&...
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Standard Symplectic Form

I am stuck at how to make the matrix of standard symplectic form. The given conditions are Definition Let $V$ be a vector space over $\mathbb{R}$. Then $\omega\in \Lambda^2(V)$ is called ...
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Compact simply-connected homogeneous symplectic manifold

I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold have non-zero Euler characteristic. They prove it by quoting a theorem by ...
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Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$ on some symplectic manifold $(M, \omega)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
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Poincaré duality and quantum (co)homology for $S^2 \times S^2$

Consider the symplectic manifold $(M,\omega):=(S^2 \times S^2, \omega_{FS}\oplus \omega_{FS})$. The homology $H_*(M;\mathbb{C})$ has as a basis the 4 non-trivial classes: $[pt],[M],A$ and $B$, where $...
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Hamiltonian and Lagrangian correspondence

I'm trying to clarify how we get a Hamiltonian directly from a Lagrangian using the Legendre transform. Let me give some preliminaries for my question to make sense. A Hamiltonian system is a triple ...
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Computing Hamiltonian vector field

Let $R=(R_1,R_2) \in \mathbb{R}^2$ and consider the product of $2$-spheres $S^2 \times S^2 \subset \mathbb{R}^3 \times \mathbb{R}^3$ with cartesian coordinates $(u_1,u_2)=((x_1,y_1,z_1),(x_2,y_2,z_2))$...
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The defining matrix of a symplectic matrix

Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me. A $2n\times 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition: $$M^T\...
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1answer
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Path components of space of almost complex structures

Let $M$ be a closed oriented smooth 4-manifold which admits an almost complex structure. The Ehresmann-Wu theorem states that a class $c\in H^2(M;\mathbb{Z})$ is realizable as the first Chern class of ...
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Showing that a 2-form on an odd dimensional space is not degenerate

On an odd-dimensional space $\mathbb R^{2n+1}$ with coordinates $x_1...x_n;y_1...y_n;t$ consider the following 2-form: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where $\omega^1$ is any 1-...
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When is the unit ball in $\text{Ham}(M,\omega)$ with respect to the Hofer metric contractible?

Associated to a symplectic manifold $(M^{2n},\omega)$ is the group of (compactly supported) Hamiltonian diffeomorphisms $\text{Ham}(M,\omega)$. This group is equipped with the famous Hofer metric $d_H$...
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Symplectic forms on $\mathbb{C}P^3$

Combining results of Gromov, McDuff, Taubes shows that any two symplectic forms on $\mathbb{C}P^2$ with the same total volume are symplectomorphic (e.g. to the Fubini-Study form $\omega_{FS}$). See ...
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Casimir Functions of Poisson Structure

I have just started to read Poisson Geometry and was working my way through some problems. This may be basic for many, but I am just trying to understand the way to work through it. The problem says: ...
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Symplectic manifolds (a problem in Arnold's classical mechanics book)

I'm working my way through V.I. Arnold's Mathematical Methods of Classical Mechanics. In the process I'm trying to wrap my head around one of the problems in the chapter on Symplectic Manifolds. In ...
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open submanifold of symplectic manifold by considering tangent spaces

This may be a slightly vague/ill-posed question. I apologize in advance. Suppose we have a symplectic manifold $(M,\omega)$ and a smooth submanifold $U$. Suppose at any point $m \in U$ we have that ...
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Does group of symplectomorphisms preserve Hopf fibration

Consider the closed ball of radius c := $B^4(c)$, with the usual symplectic form coming from $\mathbb{R}^4$. In the proof of Lemma 2.1 of the following paper(https://arxiv.org/pdf/math/0207096.pdf) ...
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Infinitesimal generator of a smooth $S^1$-action over $\mathbb C$

Question: Consider the symplectic manifold $(\mathbb C, \omega_0 = \frac{i}{2} {\rm d}z \wedge {\rm d}\bar z)$ and a smooth $\Bbb S^1$-action over $\mathbb C$ given by $$(t,z) \mapsto t^kz$$ for some ...
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Existence of regular values of a toric action.

From a book of M. Audin (p.57) Consider a torus $T$ acting effectively on a compact symplectic manifold $(W,\omega)$ with moment map $\mu\colon W\to\mathfrak t^*$. We know that in this case ...
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What is the connection between these two notions of a “vector field”?

I'm reading a physics text which dives into some mathematical concepts that are new to me. Consider a function $F(q,\dot q)$ (dynamical variable) where here $q$ and $\dot q$ represent sets of local ...
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Is the normal bundle invariant under homotopy?

Let $(M^{2n},J)$ be a complex $n$-manifold and $u:\mathbb{CP}^1\to M$ an immersed $J$-holomorphic curve. Then there exists a short exact sequence of vector bundles $$0\to T\mathbb{CP}^1\to u^{\ast}TM\...
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Equivalent definitions of Hyperkahler manifolds.

I am reading the paper HYPERKAHLER METRICS ON COTANGENT BUNDLES OF ¨ HERMITIAN SYMMETRIC SPACE by OLIVIER BIQUARD AND PAUL GAUDUCHON. Suppose $M$ is a manifold with a triple $(g,I,J)$ where g is a ...
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Definition of bivector field on a manifold.

I am reading this article: http://arxiv.org/abs/1112.5037v1 . In this, it defines a symplectic manifold as a manifold equiped with a nondegenerate bivector field $\pi$ that is Poisson. l want to ...
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Method of generating functions; Do we always get a symplectomorphism?

I am reading this very popular introduction on symplectic geometry by Cannas. The story goes as follows: Having manifolds $X_1, X_2$ , we want to construct symplectomorphisms between $M_1,M_2$ (where $...
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Existence of harmonic symplectic structure on a symplectic manifold (which is cohomologue to the initial symplectic structure)

Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric. Is there a symplectic structure $\omega '$ which is a harmonic $2$-form? Can one choose such ...
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Critical points of action functional are periodic Reeb orbits

Let $(M,\alpha)$ be a contact manifold. It is well-known that the critical points of the action functional, given by \begin{align} A: C^\infty(S^1,M) \to \mathbb{R} \\ \gamma \mapsto \int_\gamma \...
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How do we know the solutions to the Hamilton Jacobi equation are seperable?

So I usually start my questions here with "I'm sure this is a stupid question", but I am even more convinced than usual that it's true this time. We are trying to find a solution $S$ of the Hamilton ...
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1answer
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Two definitions of the space of pointed Riemann surfaces

I was reading Wendl's notes on closed holomorphic curves, and could not seem to figure out the following assertion. I think I am missing some fact or the other about Riemann surfaces. Let $\mathcal{M}...
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Bubbling off in Deligne-Mumford compactification

I'm trying to understand an argument in Casim Abbas' 'An Introduction to Compactness Results in Symplectic Field Theory'. Here's where I stuck: (It's on chapter 3.3.2, Adding additional marked points ...
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Model for symplectic geometry

An almost symplectic structure on a smooth even dimension manifold $M$ can be viewed as a reduction of structure group $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$ for the principal frame ...
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Prerequisites for symplectic reduction and geometric quantization

I am setting out to read Guillemin's "Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-spaces" and am not sure my background is enough. What are the prerequisites of reading this book? ...