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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Tautological 1-form on the cotangent bundle

I'm trying to understand a little bit about symplectic geometry, in particular the tautological 1-form on the cotangent bundle. I'm following Ana Canas Da Silva's notes. On page 10 she describes the ...
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Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
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356 views

Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
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Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
10
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58 views

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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Floer theory or Floer homology, an introduction for physicists needed

I need an introduction to Floer theory that's suitable for perhaps a beginning math grad student or a 2nd year physics grad student. The wiki article is sufficiently over my head that it reads as ...
7
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208 views

Moment map of the action of $\operatorname{SO}(3)$ on the sphere

The moment map of the action of $\operatorname{SO}(3)$ on the sphere can be thought of as inclusion from $S^2$ into $\mathbb R^3$ by identifying $\mathfrak{so}(3)$ (the Lie algebra of ...
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381 views

Why is the moduli space of flat connections a symplectic orbifold?

In her Lectures on Symplectic Geometry on page 159, Ana Cannas da Silva writes "It turns out that $\mathcal{M}$ is a finite-dimensional symplectic orbifold." Can somebody give me a reference for ...
6
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494 views

Tangent space of Cotangent bundle at zero section?

Let $M$ be a differentiable manifold with cotangent bundle $T^*M$. How can I prove that $T_{(p,0)}T^*M$ is naturally isomorphic to $T_pM\oplus T_pM^*$? If this true, then I think I could prove that ...
6
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316 views

is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
6
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818 views

symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...
6
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161 views

Symplectic geometry as a prequisite for Heegaard Floer homology

I would like to study Heegaard Floer homology in the future in the connection to knot theory. I read a wikipedia article and it seems that I need to first learn a symplectic geometry (topology?). I ...
6
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248 views

is any hamiltonian system with just one degree of freedom completely integrable?

An hamiltonian system with $n$ degree of freedom is said to be completely integrable when there exists an system $f_1,\ldots,f_n$ of first integrals mutually Poisson-commuting, such that ...
6
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193 views

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 ...
5
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289 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
5
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159 views

Why should a symplectic form be closed?

Thanks for reading my question. I'm wonder why a symplectic form should be closed. I found many different answers in the internet, but it sounds like a technical requirement (if we omit this ...
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250 views

compact symplectic manifolds

Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?
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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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141 views

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 ...
5
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117 views

Is the Poisson tensor associated to a left invariant symplectic form, also left invariant?

Given a left invariant symplectic form $\omega$ on a Lie group $G$, the Poisson tensor associated to $\omega$ is given by $$\pi(df,dg)=\omega(X_f,X_g)$$ where $X_f$ is the hamiltonian vector field ...
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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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Scattering of waves by an obstacle

I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. ...
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756 views

Why does symplectic geometry have many applications in mathematics

It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!
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194 views

Self-intersection number of a complex curve in complex projective space

I'm currently trying to get a grip on actually calculating some differential-geometric definitions. I'm looking at the following map from $\mathbb{CP}^{1}$ to $\mathbb{CP}^2$ : ...
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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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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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307 views

Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.

I'm trying to prove that $Sp(2m)$ is a Lie group using this: defining a function $f(A)=A^tJA-J$ and trying to see that this is a submersion. But I've not realized yet what is the domains and the range ...
4
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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^* ...
4
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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. ...
4
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62 views

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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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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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 ...
4
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99 views

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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287 views

basis free volume form for a symplectic vector space

It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$. I'd like to be able to prove it without choosing a ...
3
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Symplectic 2-Torus

Consider the 2-torus $T=S^1\times S^1$ with symplectic form $\omega=d\theta\wedge d\varphi$ and the vector field $X=\partial_\theta$. I wonder if $X$ is hamiltonian. In other words, is $\iota_X\omega$ ...
3
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Symplectic Forms

Let $(M, \omega)$ be a symplectic manifold, so that $\omega$ is a non-degenerate 2-form. If $\dim M = 2n$ why does $\omega$ being non-degenerate imply that $\underbrace{\omega \wedge \ldots \wedge ...
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reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
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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 ...
3
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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 ...
3
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Symplectic Form Preserved by Orthogonal Transformation

I'm trying to prove that the symplectic form $$\omega = d(\cos\theta) \wedge d\phi$$ is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
3
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Symplectic 2-Sphere

Consider the sphere $S^2\subset\mathbb{R}^3$ in cylindrical coordinates $(\theta, z)$ (away from poles $z=\pm 1$) with symplectic structure $\omega=d\theta\wedge dz$. I want to show that the vector ...
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Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
3
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185 views

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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Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.
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Special Kaehler manifolds

If we have complex vector space $V=T^{*}C^{m}$ with standard complex symplectic form $\Omega =\sum_{i=1}^{m}dz^{i}\wedge dw^{i}$, and if $\tau : V\to V$ is standard real structure of $V$ with set of ...
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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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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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156 views

Homology of symplectic manifolds

Could you show me some example of compact symplectic 4-manifold $M$ with the torsion in $H_{2}(M;\mathbb{Z})$
3
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552 views

Symplectic form on a complex manifold

I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So ...
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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 $M$, ...