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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References on the moduli space of flat connections as a symplectic reduction

In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be ...
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the symplectic version of Gram-Schmidt process and one application of it.

I am trying to understand the following proposition: cf. McDuff and Salamon, Introduction to Symplectic Topology(second edition), Corollary 2.4,page 40 One can assume $\omega_0 =\sum_i dx_i ...
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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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Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It's a tough struggle, given my not-too-great ...
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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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Connectedness of the level sets of Mose-Bott functions

I am reading the proof by McDuff and Salamon of the connectedness of the level sets of Morse-Bott functions with index and coindex different from $1$: connectedness of level sets A smooth function ...
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Problems understanding this proof

$\textbf{Theorem 3.5.10}$ (Arnol'd $[1]$). Suppose $(M,\sigma)$ is a symplectic manifold of dimension $2n$, let $f_1,...,f_n$ be an involution on $M$, and finally assume that the Hamilton fields ...
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Formula about time derivative of pushforward of family of forms: where is it from?

Proving Darboux's theorem, Hofer-Zehnder try to find, given $\omega$ a closed nondegenerate 2-form and $\omega_0$ the canonical symplectic form, a family of diffeomorphisms $\phi^t$ such that for all ...
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Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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Additional References — Symplectic Geometry

Please forgive me if I should ask this question somewhere else. If so, let me know and I will do so. I have never asked a question of this nature before, and so am unsure where an appropriate place ...
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Wedge Products with the Symplectic Form

Let $ \omega $ denote the symplectic form on $ \mathbb{R}^{2n} $, namely $ \omega = dx_1 \wedge dy_1 + \cdots + dx_n \wedge dy_n $. Then let $ T $ be the linear map from $ (n-1) $- forms to $ (n+1) ...
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Momentum Map-Submersion

Let $(M,\omega)$ be a symplectic manifold and $G$ a Lie group acting hamiltonian on $M$, such that the momentum map $\Phi \colon M \to \mathfrak{g}^*$ is $G$-equivariant w.r.t. the coadjoint-action on ...
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Symplectic form on $\mathbb R ^{2n}$

What are all symplectic form $\omega$ on $\mathbb R^{2n}$. Where, a ''symplectic bilinear form'' on $\mathbb R^{2n}$ is . a bilinear form: a map $\omega: \mathbb R^{2n}\times \mathbb R^{2n}\to ...
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22 views

Book or lecture note on presymplectic-manifolds

I'm looking for a book/ lecture note in which presymplectic manifolds and matters relating to them (specially dynamics of Hamiltonian systems) has been fully explained. Does anyone know such a book? ...
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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, ...
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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 ...
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Symplectic Chart

I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following: If $B$ is a zero neighborhood in Banach space. ...
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How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter ...
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Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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Darboux theorem for symplectic manifold of degree 2

Given $p \in M$ and $\alpha \in \Omega^1(M)$ with $\alpha_p \neq 0$, show that there exists a neighbourhood $U$ of $p$ in $M$ and $f,g \in C^{\infty}(U)$ such that $\alpha|_U = f dg$. To show this ...
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Specifics on the Williamson normal form algorithm

I'm looking at the algorithm for Williamson normal form for symplectic diagonalization of positive-definite symmetric real matrices, given on pp.24 here: ...
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Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
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How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
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Exercise about Lagrangian submanifolds

I am trying to solve the following exercise: Let $(M,\omega)$ be a symplectic manifold and $L$ a compact Lagrangian submanifold such that $H^{1}(L)=0$. Let $\{L_{t}\}_{t\in(-1,1)}$ be a smooth family ...
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Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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Poisson manifolds

Poisson manifolds are said to be generalisations of the usual symplectic manifolds. I was wondering in which direction. If every symplectic manifold is a Poisson manifold and the converse is not true, ...
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Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
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Diagonalization of elements of the symplectic algebra.

Let $A$ a symmetric positive definite real matrix of dimension $2n\times 2n$ and $J$ the standard symplectic matrix, with block representation \begin{gather} J= \begin{pmatrix} 0 & -I \\ I ...
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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Infinite dimensional Hamiltonian systems: looking for textbook/general results

Consider an infinite-dimensional phase space $(X,\omega)$, where $X = V \times V'$ with $V$ being a Banach space and $\omega$ a (weak) symplectic form. Let $E : X \to \mathbb{R}$ be a smooth function, ...
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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 ...
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A coisotropic submanifold is locally given by the fibers of a submersion with coordinates in involution

Let $(M, \omega)$ be a symplectic manifold and $Q \subset M$ a coisotropic submanifold of codimension $k$. I'm trying to prove that for every $x \in Q$, there exists an open subset $U \subset M$ ...
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How does Melnikov function for a Hamiltonian change if one considers an augmented symplectic manifold?

Suppose we have a nonautonomous nearly integrable Hamiltonian system, periodic in $t$ with period $2\pi / \omega$ $$H_{\epsilon}(x,y,t)=H_{0}(x,y) + \epsilon H_{1}(x,y,t)$$ with $(x,y,t) \in ...
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Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
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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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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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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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Diffeomorphism group $\text{Diff}_\omega(D^2, \partial D^2)$, exact differential form.

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega = dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of ...
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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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When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
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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*} ...
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Poisson bracket makes $C^\infty(M)$ into a Lie algebra

Let $M$ be a symplectic manifold with symplectic form $\omega$. Define the Poisson bracket of two smooth functions $f$, $g$ by $\{f, g\} := \omega(X_f, X_g)$. How do I see that $X_{\{f, g\}} = [X_f, ...
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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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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 ...