Hamiltonian systems, symplectic flows, classical integrable systems

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404 views

### prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...

**5**

votes

**1**answer

250 views

### Riemannian and symplectic structures

Let $(\mathcal M,g)$ be a smooth Riemannian manifold and $\Delta$ be the standard (positive) Laplace operator given in coordinates by the usual
$$
\Delta=-\vert g\vert^{-1/2}\partial_j(\vert ...

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**1**answer

198 views

### choices of connection in prequantization

In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...

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**1**answer

138 views

### pre-symplectic and foliation and its trajectories

Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?

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**0**answers

240 views

### contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?

**5**

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**1**answer

202 views

### Looking for a special rank 2 vector bundle

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$.
By Riemann-Roch theorem,
$$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$
Question: For which $g$, there is such ...

**3**

votes

**1**answer

157 views

### Can symplectic blow up increase symplectic capacities?

Let $N$ be a symplectic submanifold of $M$. Symplectic blow up of $M$ along $N$ is an operation replacing a tubular neighborhood of $N$ with the projectivization of that neighborhood. So it decreases ...

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**1**answer

382 views

### Every real-holomorphic Hamiltonian vector field on a Kähler manifold is Killing (and preserves curvature), yes?

Following the notation of Moroianu's Lectures on Kähler Geometry, we let $(M,g,\Omega,J)$ denote the metric $g$, symplectic form $\Omega$, and complex structure $J$ of a Kähler manifold $M$, ...

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**3**answers

404 views

### Symplectic blow-up

Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the ...

**0**

votes

**1**answer

140 views

### Reference on elements of finite order in principal congruence 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 ...

**11**

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**2**answers

1k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

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**2**answers

473 views

### How to prove Arnold Conjecture without using S^1 localization?

By the Arnold Conjecture, I mean the following statement:
Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# ...

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votes

**0**answers

308 views

### State of research in moduli space of flat connections

I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...

**2**

votes

**1**answer

129 views

### {0,1} Maslov potentials on Legendrian knots

A Legendrian knot is a curve in $\mathbb{R}^3$ on which $dz - ydx$ vanishes identically. Its projection to the $x,z$ plane is called a front diagram; as we can recover $y = dz/dx$ this determines the ...

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**0**answers

144 views

### Examples of non-Kahler symplectic manifolds.

Hi.
I want to find an example of symplectic (smooth compact) manifold $(M,\omega)$ whose Betti numbers are not unimodal, i.e.
$b_i(M) \leq b_{i+2}(M)$ fails for some $i < n$. ($ \dim{M} = 2n$.)
...

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**3**answers

326 views

### Linearization of vector fields

Simply, we discuss things on $C^{2n}$ ($or R^{2n}$) but not on manifolds. Given an anayltic (or real analytic) vector field $V$ on $C^{2n}$ ($or R^{2n}$), with a zero at the origin . And $V_{0}$ is ...

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**1**answer

279 views

### Thom-Gysin Sequences and Stratifications

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...

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**1**answer

229 views

### A regular polytope

For positive integers $m$ and $n$, consider a regular polytope in ${\mathbb R}^{m+n+mn}$ with $2^{m+n}$ vertices, corresponding to each $\sigma \in \{-1,1\}^{m+n}$ as follows. The first $m+n$ ...

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**2**answers

206 views

### Lagrangian submanifolds

Let $\Lambda_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in\Lambda_{n}$. Put $U_{P}= ( Q\in\Lambda_{n} : Q\cap (iP)=0 )$. There is an assertion that the set $U_{P}$ is homeomorphic ...

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**1**answer

190 views

### A basic question related to Hamiltonian isotopy in symplectic geometry

In any standard symplectic geometry/topology textbook, the concept of Hamiltonian isotopy was introduced:
$(M, \omega)$ is a sympplectic manifold. Given a symplectic isotopy
$\phi_t : M \rightarrow ...

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**0**answers

130 views

### SLAGs on elliptic curves are only lines?

Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special Lagrangians on $E$ ...

**3**

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**1**answer

171 views

### Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ...

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**1**answer

246 views

### quasi conformal, area preserving homomorphisms of the disc

Restricting a quasi-conformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasi-conformal homeos of $D^2$) to $QS(S^1)$ (quasi-symmetric homeos of the ...

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**0**answers

160 views

### A question on the SYZ mirror symmetry

A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form ...

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**3**answers

924 views

### Intuition for Levi-Civita connection via Hamiltonian flows

A Euclidean metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?

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**1**answer

650 views

### What is known about the strong Arnold conjecture?

Here are the two versions of Arnold's conjecture on Hamiltonian orbits:
Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate ...

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votes

**0**answers

84 views

### Ricci-flat Kähler metrics on symmetric varieties

Hallo,
I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: ...

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votes

**0**answers

105 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 ...

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**1**answer

194 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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**0**answers

222 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**

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**1**answer

346 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 ...

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**1**answer

521 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?

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**1**answer

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

**1**answer

314 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 ...

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**2**answers

235 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**

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**1**answer

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**

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**1**answer

362 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

**1**answer

455 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 ...

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**3**answers

403 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

**1**answer

285 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**

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**1**answer

297 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

**1**answer

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 ...

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**0**answers

163 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

**1**answer

93 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 ...

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**3**answers

724 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. ...

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vote

**0**answers

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 $J_\alpha$ ...

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**1**answer

407 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 ...

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vote

**1**answer

170 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 ...

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**2**answers

255 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 ...

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**1**answer

148 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} ...