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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Space of almost complex structures on a compact manifold

According to the book by Huybrechts, Complex Geometry: An Introduction, this is a nice space and may be regarded, after some form of completion, as an infinite-dimensional manifold. How is this done, ...
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Reduction along an Orbit for C.-M. systems

I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on ...
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formula in differential-geometry

$(M,\omega)$ is a symplectic manifold, $\omega=d \lambda$, then I want to prove that: $$ i_v\lambda\cdot\omega^n=n\lambda\wedge i_v\omega\wedge\omega^{n-1}. $$
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A problem about symplectic manifolds in Arnold's book

There is a problem in Arnold's Mathematical Methods of Classical Mechanics which says that: Show that the map $A: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ sending $(p, q) \rightarrow (P(p,q), ...
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Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
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Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
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No symplectic structure on $S^{2n},\ n>1$

I am trying to show that there is no symplectic structure on the $2n$-dimensional sphere $S^{2n}$, where $n>1$. I've tried following these steps: (a) Given a compact $2n$-dimensional symplectic ...
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hamiltonian mechanics

In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial ...
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Hamiltonian Action of $S^1$ on $\mathbb{C}^n$

Given the n-dimensional complex space, regarded as a symplectic manifold when equipped with the usual symplectic form $\sum_i r_i dr_i \wedge d\theta_i$, we consider the action of $S^1$ defined by ...
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What does it mean by saying that $u^n, J^n$ “$C^{\infty}$ converges” to u, J?

The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be ...
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Notation for coordinate-free Tautological Form definition

Reading Ana Cannas da Silva's book, I found the following step defining the tautological form (the "$p_i\wedge dq^i$" form) in a coordinate-free manner. Let $X$ be a given manifold, its cotangent ...
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Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
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229 views

Understanding Monodromy by examples

What is the intuition behind "monodromy"? Could you explain with some examples? For instance, what does it mean "monodromy around a singular fiber is a dehn twist" I don't understand what it means ...
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89 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
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Topology on the space of compatible almost complex structures in symplectic geometry

I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first: Let a symplectic manifold $(M,\omega)$ be given. One is ...
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149 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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On the definition/notation for pseudoholomorphic curves

A pseudoholomorphic curve is a map $u:(\Sigma,j) \to (M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J.$ We require ...
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Second Hirzebruch surface as Delzant space associated to trapezoid

I am trying to understand how the second Hirzebruch surface arises as the Delzant space associated to the trapezoid $\Delta \in (\mathbb{R}^2)^\ast$ given by the vertices $(0,0) , (1,0), (1-a,a), ...
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Derivative of a function involving inverse of a matrix-lifting a diffeomorphism to a symplectomorphism

Let $A$ be an $n\times n$ matrix with smooth entries in $x$, such that $A(x)$ is invertible everywhere with smooth entries. Define a function $f\colon \mathbb{R}^n\to\mathbb{R}^n$ by $f (x)=[A ...
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76 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
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151 views

Question about Lie derivative

$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here ...
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The Fubini-Study metric and its associated form.

Assume $ds^2$ is Fubini-Study metric, and $\omega = -\frac{1}{2}Im \{ds^2\}$ is its associated form. Let $V_n = \int_{\mathbb{C}P^n} \omega^n / n!$ the volume of $\mathbb{C}P^n$. Let $V \subset ...
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262 views

Definition of Liouville measure on energy surface of Hamiltonian system

This is a reference request, as I can't for the life of me find anything that answers my question in the literature. If $(M,\omega,H)$ is a Hamiltonian system, we know from Liouvile's theorem that ...
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Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension ...
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Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
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Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an ...
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When a G-Manifold is a Hamiltonian G-manifold

Let G be a lie group, then When a G-Manifold is a Hamiltonian G-manifold and under which condition a manifold is Hamiltonian G manifold for some lie group G
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45 views

tangent and conormal bundles of a Lagrangian

Suppose we have a Lagrangian submanifold $L$ of the symplectic manifold $T^*\mathbb{R}^{n}$ (endowed with symplectic form $\omega$), and a point $p\in L$. I know that there's a map $T_pL\rightarrow ...
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1answer
41 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let (Y,ξ) be a contact manifold, and L⊂(Y,ξ) be a Legendrian knot which is the boundary of a convex surface Σ embedded properly in Y. Why ...
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53 views

M fibers over the circle then construct a symplectic form

I'm trying to prove that if a 3-manifold $M$ fibers over the circle, then $M\times S^1$ admits a symplectic structure. I know that it is an standard result. Probably it is very easy, but I can't see ...
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About symplectic embedding

I never read about the symplectic embeddings. While reading a general math note, I have following question: Does every symplectic manifold $(M,\omega)$ can be symplectically embedded to some ...
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Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
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Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...
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motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
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Unitary trivialization over Riemann surfaces with boundary

I am puzzled with the proof of Proposition 2.66. in the book "Introduction to Symplectic Topology" by Salamon, McDuff. The Proposition states, that every Hermitian vector bundle $E \rightarrow ...
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1answer
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Complex structures on $T^4$

Suppose $(M,\omega)$ is a symplectic manifold, $J(M)$ is the space of all compatible complex structures. How can we show $J(T^4)$ is homotopic to the space of continuous maps $Map(T^4\rightarrow ...
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Fukaya Categories

I was just wondering: what references one might suggest for learning about Fukaya categories (specifically, good references for self-study)? I suppose I should add that any references with an eye ...
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44 views

Completely integrable geodesic flows without any degenerate point

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are ...
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orientation of symplectic manifold and lagrangian submanifolds

A statement: The self-intersection index of lagrangian submanifold $M \subset X$ is equal to Euler characteristic $\chi(M)$. How I should oriented $X$? Let's consider some example. The null-section ...
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Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
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Darboux atlas on a symplectic manifold

Is every finite dimension symplectic manifold admit 'Darboux atlas'. If not can we have counterexample..
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an identity related to moment map

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra, and $X,Y\in \mathfrak{g}$ and also let $\mu:M\to \mathfrak{g^*}$ be moment map($M$ smooth manifold) then prove the following equality ...
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A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...
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Can the system $\partial_x f(x,y) = \dot{y}$, $\partial_y f(x,y) = \dot{x}$ be related to some Hamiltonian system?

If one has found some function $f(x,y): \partial_x f = \dot{y}, \partial_y f = \dot{x}$, is there a simple transformation or change of variables that results in Hamilton's equations $\partial_p H = ...
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Why circle action is not hamiltonian

We have given an action $$S^1\times T^2\to T^2$$ $$(t,(\theta_1,\theta_2))\to (\theta_1+t,\theta_2)$$ Why this action can't be Hamiltonian? May i get some hint, comment suggestion. Thanks.
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self-intersection of lagrangian submanifold

Let's consider lagrangian submanifold $X$ in symplectic manifold $M$. Is it true that self-intersection index of $X$ is equal to the Euler characteristic $\chi(X)$? Can we construct (not canonical) ...
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Some differential geometry computation woes

I'm trying to follow Moser's argument in symplectic geometry, and running into some troubles. Here is a picture (confusing parts are circled in red): For the first one, what happens when $s+t$ ...
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Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on ...
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How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...