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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A difficult question on mathematical physics

Let $TQ^*$ be equipped with its standard symplectic structure and let $X_H$ be a Hamiltonian vector field which is tangent to the fibers of $\pi: TQ^* \to Q.$ I need to show that $$H=h \circ \pi = \pi ...
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Hamiltonian Dynamics and the canonical symplectic form

1- What kind of advantages does one have by having a canonical symplectic form on $T^*M$ apart from the form being exact? Would it for instance provide any advantage to studying Hamiltonian dynamics ...
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What are the elements in $\Gamma(\Lambda^2 TM)$?

In the lecture notes, Proposition 1.19 on page 9, it is said that on every Poisson manifold there is a unique bivector field $\Pi \in \Gamma(\Lambda^2 TM)$ such that $$ \{f, g\} = \langle \Pi, df ...
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Symplectic structures on Hermitian matrices

This is a question taken from Ana Cannas da silva's book on symplectic geometry. Let $\xi\in\mathcal{H}$, the vector space of $n\times n$ hermitian matrix. Define ...
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Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, ...
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What does $e^{\mu}$ mean for a measure $\mu$?

I have seen the notation $\int_M fe^{\mu}$ in some geometry books and I cannot even guess what $e^{\mu}$ might mean for a measure/form $\mu$ on the (symplectic) manifold $M$. Any clarifications are ...
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Help needed in understanding the question

Let $D:=\mathbb{C}^* \to \mathbb{C}^*$ be regarded as an open subset of $\mathbb{C}^2$ which is equipped with its standard (symplectic) $2-$form $$\omega_{std}=\frac{i}{2}(dz_1 \wedge d\bar{z_1}+dz_2 ...
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Condition for local Lipschitzness of pullback map for exterior forms

Given $w\in\Lambda^k(\mathbb{R}^n)$, determine the condition under which the map $T\rightarrow T^*(w)$ is locally Lipschitz, where $T\in GL_n(\mathbb{R})$ and $T^*(w)$ denotes the Pullback of $w$ by ...
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Proof that $S^4$ is not a symplectic manifold

I started learning symplectic geometry, but apparently, i forgot some of my differential geometry. I am studying from the book by Aebischer et al. In it, it is claimed that $S^4$ is not a symplectic ...
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Show that the only invariant is the spectrum

Recall that the symplectic group $$Sp_2(\mathbb{R}):= \{A\in SL_2(\mathbb{R}):A^TJA=J\}, \ \ J= \left[ {\begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} } \right] \ $$ We have its ...
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Writing a two-form as a wedge product

Suppose a differential two-form $\Omega$ on $\mathbb{R}^2$ is defined by $\Omega_p(x, y)=p_2(x_1y_2-x_2y_1)$. Then using coordinates $(p_1, p_2)$ for $\mathbb{R}^2$, this reads ...
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John Lee's book question about symplectic group

Exercise 12-14 in John Lee's book, An introduction to Smooth Manifolds, reads as follows: The real symplectic group is the subgroup $Sp(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$ consisting of ...
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Uniqueness of smooth/symplectic/etc structure

It is well-known that every topological manifold $M$ of dimension $\le 3$ admits a unique smooth structure. That is to say for any choice of atlas on $M$ like $A$ and $B$, the smooth manifolds $(M, ...
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Finding a path of symplectic forms

Suppose $M$ is a symplectic manifold and $\omega_0$ and $\omega_1$ are two symplectic forms belonging to the same de Rham cohomology class. Assume in addition they can be connected via a path of ...
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Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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Do all symplectic transformations give rise to skew symmetric matrices?

Suppose that $ \Delta(x,y) = x^T\Delta y $ where $ \Delta$ is a symplectic matrix of form given in https://en.wikipedia.org/wiki/Symplectic_matrix If I define an inner product $ \alpha(x,y) = ...
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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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Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...
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What is Darboux coordinate?

What is Darboux coordinate? Is it different from coordinates from $\Bbb R^n$ or some smooth manifold? I am familiar with Riemanian manifolds, but at some how Darboux coordinates, came up in some ...
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Symplectic form and volume of parallelepiped

Define the canonical symplectic form $\omega$ on $\mathbb{R}^{2n}$ by $\omega(u,v)=u^TJv$, where $$J=\begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}.$$ I do not understand why the volume ...
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Space of oriented lines in $\mathbb{R}^{n+1}$ as symplectic quotient.

I've been working out a nice example of symplectic reduction, and have come to a solution only after quite a lot of effort. So I was wondering if anyone knew a more straightforward route to the ...
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If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?

Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. We call a relation $U \to V$ a Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus V$, ...
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References for the moduli space of conformal structures on a disk minus boundary points

In an article I'm reading, it is said that the moduli space of conformal structures on a disk minus $k+1$ boundary points is a ($k-2$)-dimensional manifold. I want to understand why and have done some ...
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Constructing lagrangian submanifold of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. To keep it simple, let us take $M = \mathbb{R}^{2n}$ with linear coordinates $(x^1,\ldots,x^n,y^1,\ldots,y^n)$ and the standard symplectic form $\omega = ...
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Is there a nice/clever way to visualize $\mathcal{S}\times \mathbb{R}^2$?

The (velocity) phase space of a double pendulum can be seen as the tangent bundle of its configuration space ($\mathcal{S}^1\times\mathcal{S}^1$), that is: ...
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Symplectomorphism Preserves Cotangent fibrations

Let $M$ be a manifold with local coordinates $x^1,\dots,x^n$ and $T^\ast M$ the cotangent bundle with induced coordinates $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ . Let $\alpha$ be the tautological one form ...
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Time-change of a Hamiltonian diffeomorphism

Let $(X, \omega)$ denote a symplectic manifold and let $\phi : X \to X$ denote a Hamiltonian diffeomorphism, so $\phi = \phi_H^1$ is the time-1 map of the flow $\phi_H^t$ of the vector field $\xi_H$ ...
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Proof of Arnold-Liouville's Thm: movement in angular coordinates conditionally periodic

I'm reading Arnold's book Mathematical Methods in Classical Mechanics and got stuck on the proof of Liouville's theorem on integrable systems. The proof finishes with Problem 11: Show that the motion ...
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Darboux theorem and symplectomorphisms

In the lecture note I am reading there is following claim: Let $(M,\omega)$ be a symplectic manifold, $f,g : M \rightarrow M$ symplectomorphisms, and $L \subset M$ a Lagrangian. Suppose $f(x) = g(x)$ ...
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If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
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Looking for a good book on Morse-Bott functions.

I am looking for a book to study for the first time Morse-Bott functions. Does anyone know one that is easy to follow and detailed? If there is one connecting this subject with symplectic geometry, it ...
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Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
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volume of projective space $\text{Vol}(\mathbb CP^N)$

How can we compute the volume of projective space $$\text{Vol}(\mathbb CP^N)$$
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Definition of Hamiltonian vector fields

Suppose that $(M,\omega)$ is a a symplectic manifold. Since $\omega$ is non-degenerate, it sets up an isomorphism $$\omega:TM\rightarrow T^*M$$ between $TM$ and $T^*M$. Why does non-degeneracy ...
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Moment map of the action of $SU(2)$ on $\mathbb C^{2n}$

Let $SU(2)$ acts on symplectic space $((\mathbb C^2 -\ (0,0))^{n},\omega)$, where $$\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+\cdots+dx_{4n-3}\wedge dx_{4n-2}+dx_{4n-1}\wedge dx_{4n}$$ as ...
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coordinates for $T^*M$, $T^*T^*M$, $T^*T^*… T^*M$

This relates to my previous question on the tautological one form. I'm reading John Lee's Smooth Manifold book. Anyway, I wanted to clear up on some notation, which is confusing at first, but I ...
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the tautological 1 form

My question relates to p317-p318 of John Lee's "Introduction to Smooth Manifolds" discussion about the tautological 1 form. In Proposition 12.24, we have the expression: $\tau_{(x, \xi)} = \pi^* ...
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reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
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the lift of a hamiltonian path

Let $\{f_t\}$ be a Hamiltonian path on a closed manifold $(M,\omega)$, i.e. $f_0=id$, $f_1=f \in Symp(M,\omega)$. And denote by $F$ the Hamiltonian function with $\{f_t\}$. There is a fixed point $x$ ...
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use Floer homology to prove the fixed points

I read paper, in page 21, there is a proposition: Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0$ = id, $f_1= f$ be a Hamiltonian path on M generated by a ...
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A relation between $vol(M)$ and $vol(T^*M)$?

Let $(M,\omega)$ be a symplectic manifold then what is relation between $\operatorname{vol}(M)$ and $\operatorname{vol}(T^*M)$?
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the fixed points of symplectic diffeomorphism

Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0=id$,$f_1=f\ne id$ be a Hamiltonian path on M generated by a Hamiltonian function F. Then how to prove that f has a ...
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Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
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Geodesic of symplectic manifold

Can I define geodesic on a manifold without Riemann structure? To be more specific, how can I define geodesic at symplectic manifold? Let's just look at simple case with symplectic form as ...
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When is symplectic pullback bundle trivial

Suppose $x : S \to M$ is a smooth map, where $M$ is a symplectic manifold and $S$ is a Riemann surface. Consider the pullback bundle $x^*TM \to S$. When is this bundle trivial (as symplectic vector ...
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A complex manifold isn't a sympletic manifold

I want to think about an example of a complex manifold which isn't a sympletic manifold. I consider it in this way: $X=\mathbb{C}^2-\{0\}$, a group $\mathbb{Z}$ acts on X by $(n,z)=2^nz$, then I think ...
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complex structure in odd-dimensional real vector space

If V is an odd-dimensional real vector space, then is there a linear map $J: V \to V$ satisfying $J^2=-1$? i.e. is there a complex structure in odd-dimensional real vector space?
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Moser method for Darboux theorem with parameter

Suppose $(M^{2n}, \omega)$ is symplectic manifold and $ p \in M$. Moser method for proving Darboux theorem is the following: suppose $f : B \subset \mathbb{R}^{2n} \to V \subset M$ is a chart, where ...
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Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
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What is the Lagrangian tori?

I am looking for the definition of lagrangian tori for symplectic manifold $(M,\omega)$ ?