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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Symplectic reversing diffeomorphisms

Let $(M,\omega)$ be a compact symplectic manifold. Is there a diffeomorphism $f$ on M with $f^{*}\omega =-\omega$?
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Convex boundary of a symplectic manifold

Given a symplectic manifold $(M,\omega)$, suppose that $\partial M$ is of contact type. A Liouville field on a symplectic manifold is a vector field $X$ such that $\mathcal L_X \omega = \omega$. We ...
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symplectic change of variables

I must prove that the change of variables $\Psi: (p,q)\rightarrow(r,\alpha)$ such that $$q=\sqrt{2r}\cos(\alpha), \qquad p=\sqrt{2r}\sin(\alpha)$$ is symplectic. What I know is: $$\dot{q}=p, \qquad ...
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symplectic sructure on a ball

We know by Darboux theorem that any symplectic form on a manifold $W^{2n}$ is locally symplectomorophic to the standard symplectic form $dx\wedge dy$ on $R^{2n}$. Is it true that any symplectic form ...
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Poincaré invariant corresponds to area?

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant $\int p dx = \int_0^{2 ...
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Carathéodory–Jacobi–Lie theorem

I am studying Analytical Mechanics and want to prove the following theorem: Let be $(M, \omega)$ a sympletic manifold, $U \subset M $ open, $f_1,\ \ldots,\ f_n \in C(U)$ such that 1 $\{f_i,f_j\}=0 ...
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Explicit homotopy equivalence of homogeneous spaces $O(2n)/U(n)$ and $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$

Exercise 2.25 of symplectic topology by McDuff and Salamon asks me to prove that $O(2n)/U(n)$ is homotopy equivalent to $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$. They suggest to use the polar ...
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Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
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Any good resources for learning about the moduli space of symplectic structures on a given manifold?

What I can find on the subject are the papers by Fricke and the Habermanns: http://www.researchgate.net/publication/227336993_On_the_geometry_of_moduli_spaces_of_symplectic_structures and ...
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The Classification of almost complex structures (almost) tamed by a quadratic form

Preamble I am currently dealing with the almost complex structures associated with a non-degenerate quadratic form. In particular, I am interested the size of the almost complex structures as a ...
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What is a local invariant?

Let $(M,g)$ be a Riemannian manifold. Then, it is usually said that $M$ has local invariants associated to $g$. For example, the curvature of the Levi-Civita connection associated to $g$. My question ...
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About Lefschetz fibration signature

Does there exists a Lefschetz fibration over $S^2$ for any given number admitting it as a signature? I think it is not possible so I need an counter example.
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rank of the symplectic form

This is a general question about ranks of differential forms. I read in a book the phrase "symplectic form has constant rank..." I understand that the symplectic form is a nondegenerate differential ...
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25 views

covering finite Dimensional vector space

can a finite dimensional symplectic vector space over finite field be covered with mutually transversal Lagrangian planes(maximal Isotropical Subspaces )?
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symplectic manifolds

I know sometimes they use advanced methods to prove a given 4-manifold is not symplectic. for instance by Seiberg-Witten theory. But for a manifold to be symplectic we just need to check that there is ...
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Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
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Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
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Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
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Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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Why is the $ n $ - exterior power of a symplectic form does not vanish?

Let $ V $ be a vector space. Could someone explain to me why is : Any $ \Omega \in \bigwedge^{2} ( V^* ) $ is of the form $ \Omega = e_{1}^* \wedge f_{1}^* + \dots + e_{n}^* \wedge f_{n}^* $ where ...
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Relative line bundle along divisor $D$

Let $X$ and $B$ be a compact Kahler manifolds and $\pi:X\to B$ be a holomorphic surjective map and $D$ be a divisor in $X$ how can we define relative canonical line bundle on $B$ along a divisor $D$? ...
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Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
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Coordinate free definition of the canonical one-form

There is apparently a naturally defined one-form on the cotangent bundle of a smooth manifold $M$: We have the cotangent bundle $\pi:T^*M \rightarrow M$; taking its derivative gives $d \pi:TT^*M ...
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1answer
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What happens to symplectic basis if bilinearity condition is weak

Let $B:V\times V \rightarrow K$ be a (weak!) bilinear form where $K$ is a finite field with base field $F$ and $V$ a vector space over $K$. Let $u,v \in V$ and $\lambda \in F (!)$ $B(u + v, w) = B(u, ...
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Fibered product of symplectic fiber bundles

Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic ...
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symplectic surfaces in 4-manifolds

Is it true that for any surface in a symplectic 4-manifold $X$, representing a given homology class of $H_2(X)$, we can assume it is symplectic? I mean for each second homology class, can we find a ...
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1answer
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Symplectic lie algebra

Can anyone explain me why, in the symplectic lie algebra, which is defined as $ sp(n)=\{X \in gl_{2n}:X^tJ+JX=0\}$ where $J=\begin{pmatrix} 0 & I \\ -I & 0 \\ ...
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Are the following Hamiltonian actions?

Let $(x,y)$ be the coordinate of $\mathbb{C}^2 \subset \mathbb{P}^1\times \mathbb{P}^1$. Is the $S^1$ action on $\mathbb{P}^1\times \mathbb{P}^1$ given by $$ t\cdot(x,y)=(tx,t^{-1}y) $$ Hamiltonian? ...
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According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
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The meaning of variables and derivations in Souriau's book

As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he ...
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Non-commutative symplectic geometry

How is non-commutative symplectic geometry defined? How does it differ from symplectic geometry? Does Darboux's theorem apply also there?
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The best book on symplectic geometry

I want read an introduction to symplectic geometry. Can you suggest me some book on this theory?
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Generators of $Sp(2n)$

Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix ...
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Symplectic manifold question.

Let $M=\mathbb R^2$ with coordinates $x$ and $y$ with a symplectic two form $\omega =dx\wedge dy$. Let $\phi _p :T_pM\mapsto T^*_pM$. My confusion is regarding the initial step in defining the ...
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1answer
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Expression for Hamiltonian vector field!

How does one prove that the Hamiltonian vector field has the following expression, what is the reasoning? \begin{equation} X_H=\sum ^n_{i=1}\frac{\partial H}{\partial q_i}\frac{\partial }{\partial ...
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$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
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$(p,q)$ part of a complex differential form in terms of the complex structure $J$?

Say $M$ is a complex manifold, viewed as real $C^{\infty}$ manifold with an integrable almost complex structure $J$. Let $\omega$ be a complex $r$-form on $M$. Is there a way to express the $(p,q)$ ...
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Lagrangian and Hamiltonian Mechanics

I am interested in how Lagrangian and Hamiltonian mechanics and then symplectic geometry was developed starting from Newtonian mechanics. We can start by assuming that Newtonian mechanics tells us ...
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Invariants under Hamiltonian mechanics?

I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is ...
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acrobatics with $2$-form in $\mathbb{R}^{2n}$ [closed]

In the space $V = \mathbb{R}^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$ consider the $2$-form $\omega = \sum_{i=1}^n x_i \wedge y_i$. Let $A$ be a $n \times n$ matrix. Consider a ...
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What does it mean for a Symplectic Form to be invariant under Group Action?

This should be a very basic question for people familiar with differential manifolds. I'm more or less new to the field so let me apologize in advance for ill-defined questions if arising. I split the ...
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1answer
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How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
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1answer
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computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
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How to find lagrangian submanifolds.

I am quite confused on the definition of a lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$. In particular, I read that $L \subset M$ is lagrangian iff the symplectic form field ...
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wedge product, multilinear algebra in $\mathbb{R}^{2n}$

Denote coordinates in the space $\mathbb{R}^{2n}$ by $(x_1, y_1, \dots, x_n, y_n)$. Consider a $2$-form $$\omega = \sum_{i=1}^n x_i \wedge y_i.$$ (a) Compute$$\underbrace{\omega \wedge \dots \wedge ...
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Help with constructing a symplectic form on a $2-$dimensional torus

Consider the standard two dimensional torus $M=\mathbb{R}^2 / \Gamma$ where $$ \Gamma:=\{(n,m)\in \mathbb{R}^2:n,m\in \mathbb{Z}\}. $$ I want to $1)$ construct a symplectic form $\omega$ on $M$, ...
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1answer
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How to show that geodesics exist for all of time in a compact manifold?

Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form ...
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1answer
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Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple?

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups ...
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Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...
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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 ...