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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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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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 ...
2
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40 views

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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1answer
91 views

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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96 views

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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72 views

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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108 views

Coordinate-free proof of the hamiltonian character of the geodesic flow

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$ It is well known ...
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265 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 ...
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64 views

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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19 views

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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36 views

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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85 views

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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44 views

Trivialization of a path of tamed almost complex structures

I am wondering if the following result is true: Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
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68 views

Area of flux homomorphism in symplectic topology

Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: ...
2
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120 views

Index of a Symmetric Matrix

In Hansjorg Geiges' introductory textbook on Symplectic Geometry, is defined a projective conic given by $q^tAq=0$ where $A$ is a symmetric matrix of rank 3 and index 2. What does "index 2" mean? I ...
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209 views

Geodesic Flow is the flow of the Hamiltonian Vector field of $|\xi|$

Let $M$ be a complete Riemannian manifold with metric $g$. The geodesic flow is the one-parameter family of diffeomorphisms $\phi_t$ on $T^*M$ defined as follows. If $\xi \in T^* M$ is a vector based ...
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100 views

About $\operatorname{Spin}^\mathbb{C}$ structures

Let $X$ be a orientable 4-manifolds. I know that $X$ can be endowed with $\operatorname{Spin}^\mathbb{C}$ structures by the choice of integral lift of $w_2(X)$ (second Stiefel-Whitney class of ...
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137 views

How to prove that a certain action is hamiltonian?

Reading a paper I had the need to complete a proof, and come up with a certain argument(see below). My question is: could I reduce it to a special case of some theorem? I ask this question in order to ...
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23 views

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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30 views

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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22 views

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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64 views

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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96 views

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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20 views

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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74 views

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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73 views

Torsion-free $G$-Structures

I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...
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68 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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75 views

transformation of symplectic structure by a matrix

Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have $$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$ Let ...
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96 views

A vector field on a symplectic submanifold intersecting the symplectic complement

Consider a 4-dim symplectic vector field $X$ on the symplectic manifold $(M, \omega)$ in $\mathbb{R}^4$ with $\omega= \sum_{i=1}^2 dy_i \wedge dx_i$. Moreover, the linear terms of $X$ are given by ...
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130 views

Energy displacement of a cylinder is at most $\pi r^2$.

I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$. In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
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18 views

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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29 views

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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18 views

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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27 views

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 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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26 views

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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29 views

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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24 views

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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36 views

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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30 views

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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25 views

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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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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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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29 views

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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63 views

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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38 views

Deformation of sympletic manifolds and deformation of complex manifolds

I know that for a complex manifold you can determine the variation of the parameter by picking a $\theta (t) \in H^1(\mathscr{M}, \Theta)$ (where $\Theta$ is the sheaf of vector fields over ...
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77 views

Obstruction to construct the set of vectors in lattice

Lets consider a lattice $\mathbb{Z}^n$ with some unimodular scalar product $\mathbb{Z}^n \times \mathbb{Z}^n \mapsto \mathbb{Z}$ and the set of vectors $e_0,\ldots,e_k$ with conditions: $$ ...
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98 views

Hamiltonian reduction for unit sphere

Let $(M, \omega)$, be Symplectic Vector space and $N\subset M$ be unit sphere. Then why $N/ker\omega\mid _N$ is naturally $\mathbb{P}^{n-1}(\mathbb{C})$ Here ker$\omega$=$\{ y\in M: \omega(x,y)=0, ...
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73 views

a question about pre-symplectic manifold

Let $(M,\omega)$, is pre-symplectic. Then can we say, ker$ \omega$ is subbundle of tangent bundle $TM$?
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71 views

Maslov Index product property.

I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property. Let $\Lambda: ...