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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Spin structure and rational blow down

If we rationally blow down a spin 4 manifold, is the resulting manifold spin? I don't know if there is any way to determine.
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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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Jacoby identity for inverse matrix

I'm wondering how one can show what Jacobi identity for skewsymmetric matrix $\omega$: $$ \omega^{jk} \partial_{k} \omega^{lm} + \omega^{lk} \partial_{k} \omega^{mj}+ \omega^{mk} \partial_{k} ...
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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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22 views

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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48 views

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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Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
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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 $(M,\omega)$ ?
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Symplectomorphism that's the identity on the zero section

Suppose $\Phi:T^*\mathbb{R}^n\rightarrow T^*\mathbb{R}^n$, $\Phi(x,\xi)=(y,\eta)$ is a symplectomorphism which is the identity when restricted to the zero section $o=\{\xi=0\}$; i.e. $\Phi|_o=Id$. ...
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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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A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
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symplectic coordinate change in tangent space

Given is the Hamiltonian system with energy function $$H(q,p) = \sum_{i = 1}^{2} \frac{p_{i}^{2}}{2m_{i}} + m_{i}V(q_{i}) = H_{0},$$ where $H_{0}$ is some positive constant and the potential energy ...
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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. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...