Hamiltonian systems, symplectic flows, classical integrable systems

learn more… | top users | synonyms

-1
votes
0answers
110 views

Some question on the paper “Ricci-flat metrics on the complexification of a compact rank one symmetric space”

I am reading the paper "Ricci-flat metrics on the complexification of a compact rank one symmetric space" by Stenzel (here is the ...
1
vote
1answer
209 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 ...
8
votes
0answers
224 views

Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ ...
3
votes
1answer
350 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
6
votes
1answer
532 views

Darboux like theorem for non-degenerate 3-forms in 6-manifolds

we know Darboux theorem for higher-symplectic geometry is not correct in general, but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
0
votes
1answer
197 views

$SU(n)$-structures on a manifold

I have the following question. Consider a $2n$-dimensional almost complex manifold $M$. Assume that on $M$ there exists a complex valued $n$-form $\Omega$ and a $2$-form $\omega$ such that: $\Omega$ ...
5
votes
1answer
323 views

Qustions on R.Bryant's papaer “Calibrated embeddings in the special Lagrangian and coassociative cases”

I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R.Bryant (here the link: http://arxiv.org/abs/math/9912246) and there are certain things that are ...
0
votes
2answers
253 views

Hamiltonian group actions - examples

I want to study about Symplectic group actions and moment map, especially Hamiltonian Group Actions. Can you help me with some concretely example of Hamiltonian group actions ? Where can I find some ...
1
vote
1answer
140 views

finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation $det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$ you can find the definition of effective forms here
4
votes
1answer
368 views

What is the physical interpretation of the canonical relations?

This question arises when studying semi-classical analysis. Since Weinstein's creed claims that "everything is Lagrangian", where a point in the phase space of classical mechanics is just a cotangent ...
3
votes
1answer
507 views

Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal? Being ...
1
vote
3answers
414 views

Reference on generators of subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
7
votes
1answer
292 views

How many “elementary” characterizations of twisted SU(2) representation varieties are known?

If $\Sigma_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma_g) := \{ (A_1, B_1, \ldots, A_g, B_g) \in ...
6
votes
1answer
305 views

Fundamental groups of symplectic leaves

Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$ with $R_0 = \mathbb{C}$ and that the ...
0
votes
1answer
215 views

special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...
1
vote
0answers
171 views

Question in the paper of Robert Bryant “Calibrated embeddings in the special Lagrangian and coassociative cases”

Hallo, I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...
1
vote
1answer
96 views

Is the space of half-dimensional symplectic linear subspaces of $\mathbb{R}^{4n}$ which trivially intersect $\mathbb{R}^{2n}\times\{0\}$ contractible?

I would like to know whether the subspace of the symplectic Grassmanian $Gr_{2n}^{Sp}(\mathbb{R}^{4n})$ consisting of symplectic linear subspaces in $\mathbb{R}^{4n}$ which have dimension $2n$ and ...
5
votes
3answers
771 views

what prevents a manifold to be symplectic?

Are there any obstructions known which prevent an even dimensional orientable manifold from being symplectic? I am a novice in this area so I unfortunately I cannot make the question more precise. ...
1
vote
0answers
52 views

action aifference for cylinders in a symplectic cobordism

Suppose we have a contact manifold (M,$\xi$) and two associated contact forms $\alpha$ and $\beta$ s.t.$\beta=f\alpha$ with $f>0$. Suppose also that we have two almost complex structures $J_\alpha$ ...
4
votes
1answer
408 views

Recognizing the stablizer of a degenerate three forms in six dimension

Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }. we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0} Let $V\cong ...
1
vote
1answer
177 views

Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf. Is there some sense, or some ...
1
vote
2answers
258 views

Proof of Arnold Conjecture for monotone symplectic manifolds

I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf: Namely the author on page 32 ...
1
vote
1answer
149 views

Condition in proof of the Arnold conjecture for monotone manifolds

In http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf the Arnold conjecture from Symplectic Geometry is shown for the case of montone symplectic manifolds $(M, \omega)$ (i.e. we have $\int_{S^2} ...
1
vote
1answer
279 views

What foliations are symplectic foliations?

On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a ...
10
votes
1answer
842 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
5
votes
2answers
370 views

What information is required for SYZ mirror symmetry?

Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...
2
votes
1answer
419 views

why the group $GL(6,V)$ has an open orbit?

N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written ...
8
votes
1answer
478 views

Known size invariant for Riemannian manifolds?

Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under ...
3
votes
2answers
289 views

Uniqueness of Kähler form with same volume

Hallo, Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...
7
votes
2answers
478 views

Orientations for pseudoholomorphic curves with totally real boundary condition

I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition. I believe that Fukaya-Oh-Ohta-Ono have shown that if ...
5
votes
0answers
260 views

Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal ...
5
votes
1answer
278 views

Questions on how SYZ conjectures is deduced from HMS conjeture.

The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) ...
4
votes
1answer
213 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
0
votes
1answer
239 views

$q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define $\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$ by $\perp(\omega)=i_{X_{\Omega}}(\omega)$ here if ...
0
votes
0answers
72 views

Preimage of regular values of a admissible hamiltonian

Hi everyone: I have a question about a proof in Siburg's paper "Symplectic Capacities in Two Dimensions". We define a admissible Hamiltonian $H$ in such a way that near from the boundary of a ...
2
votes
1answer
195 views

Hamiltonian actions and contractible loops

Let $(M, \omega)$ be a symplectic manifold and $G$ be a compact Lie group. Suppose we have a Hamiltonian $G$-action on $M$, with moment map $\mu: M \to {\mathfrak g}^*$. We assume that the moment map ...
4
votes
1answer
199 views

Rotations, Harmonic Oscillators, Gaussians, Ladders

I am trying to understand better the quantization of the Harmonic Oscillator. Here are three ways of thinking about the Harmonic Oscillator. Eigenfunctions of the differential operator: $H = ...
4
votes
2answers
553 views

Applications of Floer homology

Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture. Every answer would be appreciated.
7
votes
3answers
835 views

Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...
3
votes
1answer
260 views

Igusa invariants of genus 2 curves as Siegel modular functions?

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...
12
votes
1answer
251 views

Energy quantization for $J$-holomorphic spheres

Let $(\mathbb{CP}^1, j, g_{\text{FS}})$ be the complex projective line with the standard complex structure and the Fubini-Study metric and let $(M,J,\omega,g)$ be an almost Kähler compact manifold ...
4
votes
1answer
623 views

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that ...
7
votes
0answers
200 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
6
votes
1answer
191 views

Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section. Recall that an ...
1
vote
1answer
140 views

Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

Hallo, Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
6
votes
0answers
181 views

Different complexifications of a real analytic Riemannian manifold

Hi, I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...
1
vote
1answer
207 views

Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo, It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
6
votes
2answers
476 views

A Question on Exterior Forms

For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions ...
1
vote
2answers
280 views

Analytic Lagrangian Submanifolds

Hallo, I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
13
votes
2answers
728 views

Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times ...