1
vote
1answer
32 views

What is the Lagrangian tori?

I am looking for the definition of lagrangian tori for symplectic manifold $(M,\omega)$ ?
2
votes
0answers
27 views

formula in differential-geometry

$(M,\omega)$ is a symplectic manifold, $\omega=d \lambda$, then I want to prove that: $$ i_v\lambda\cdot\omega^n=n\lambda\wedge i_v\omega\wedge\omega^{n-1}. $$
1
vote
1answer
33 views

What does it mean by saying that $u^n, J^n$ “$C^{\infty}$ converges” to u, J?

The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be ...
2
votes
1answer
24 views

Notation for coordinate-free Tautological Form definition

Reading Ana Cannas da Silva's book, I found the following step defining the tautological form (the "$p_i\wedge dq^i$" form) in a coordinate-free manner. Let $X$ be a given manifold, its cotangent ...
0
votes
0answers
25 views

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 ...
0
votes
0answers
59 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
0
votes
0answers
35 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 ...
0
votes
1answer
64 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
3
votes
1answer
66 views
+200

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 ...
2
votes
2answers
45 views

On the definition/notation for pseudoholomorphic curves

A pseudoholomorphic curve is a map $u:(\Sigma,j) \to (M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J.$ We require ...
1
vote
1answer
36 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
2
votes
1answer
125 views

Question about Lie derivative

$(M,w)$ is symplectic manifold. $f_t : M\to M$ is a symplectic isotopy between $f_0=id$ and $f_1$. Let X_t be the vector field on M satisfying $d(f_t)/dt=X_t(f_t)$ Now I differentiate $(f_t)^*w$. Here ...
1
vote
0answers
77 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 ...
3
votes
1answer
58 views

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 ...
2
votes
0answers
73 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 ...
3
votes
2answers
136 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 ...
0
votes
1answer
38 views

When a G-Manifold is a Hamiltonian G-manifold

Let G be a lie group, then When a G-Manifold is a Hamiltonian G-manifold and under which condition a manifold is Hamiltonian G manifold for some lie group G
0
votes
1answer
286 views

Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...
2
votes
1answer
50 views

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 ...
1
vote
2answers
33 views

Unitary trivialization over Riemann surfaces with boundary

I am puzzled with the proof of Proposition 2.66. in the book "Introduction to Symplectic Topology" by Salamon, McDuff. The Proposition states, that every Hermitian vector bundle $E \rightarrow ...
0
votes
0answers
38 views

Completely integrable geodesic flows without any degenerate point

Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are ...
0
votes
0answers
41 views

Darboux atlas on a symplectic manifold

Is every finite dimension symplectic manifold admit 'Darboux atlas'. If not can we have counterexample..
1
vote
1answer
50 views

an identity related to moment map

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra, and $X,Y\in \mathfrak{g}$ and also let $\mu:M\to \mathfrak{g^*}$ be moment map($M$ smooth manifold) then prove the following equality ...
1
vote
0answers
39 views

A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...
5
votes
1answer
71 views

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 $s+t$ ...
10
votes
1answer
144 views

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 ...
0
votes
1answer
102 views

lagrangian subspace and Heisenberg group

Let $(V,\omega)$ be a symplectic vector space. Also we assume $L\subset V$ be a Lagrangian subspace., and $H(V)$ be Heisenberg group, then why $L\bigoplus U(1)\subset H(V)$ is maximal abelian ...
1
vote
1answer
49 views

Existence of prequantization on a simply connected manifold

Let $M$ be a simply connected manifold. Then when, $M$ has a unique pre-quantization and when there is no pre-quantization on $M$.
0
votes
0answers
89 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, ...
0
votes
0answers
62 views

a question about pre-symplectic manifold

Let $(M,\omega)$, is pre-symplectic. Then can we say, ker$ \omega$ is subbundle of tangent bundle $TM$?
2
votes
1answer
62 views

Hamiltonian reduction in symplectic geometry

If $V$ is symplectic and $W^\perp \subseteq W\subseteq V$, then why is $W$ a pre-symplectic vector space? Why is $W/W^\perp$ symplectic?
2
votes
2answers
264 views

why symplectic form should be closed when we work on a manifold

For defining the symplectic space $(V, \omega)$ where $V$ is a vector space, it doesn't necessary to add the condition $d\omega=0$. But, when we work on a manifold instead of vector space, then we ...
1
vote
2answers
211 views

Application of the implicit function theorem

Assume that the equation $F(x,y,p)=0$ defines a regular submanifold $M$ of $R^3$. Consider the projection $\pi :M \rightarrow R^2$, given by $\pi (x,y,p)=(x,y)$. By the implicit function theorem, in ...
4
votes
2answers
125 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 $\mathbb{CP}^2$ : ...
3
votes
2answers
108 views

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 ...
2
votes
0answers
43 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 ...
2
votes
1answer
50 views

Lagrangian subspaces

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 ...
3
votes
1answer
78 views

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 ...
5
votes
1answer
123 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 ...
1
vote
0answers
65 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 ...
1
vote
1answer
430 views

Darboux's theorem in the symplectic geometry

From the Darboux's theorem in the symplectic geometry, we know that symplectic manifolds with the same dimension is locally "equivalence". I have a little puzzle with the meaning of "equivalence". ...
1
vote
0answers
60 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
6
votes
0answers
250 views

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. ...
1
vote
1answer
79 views

Visualizing diffeomorphisms

This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
3
votes
1answer
82 views

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 ...
3
votes
1answer
139 views

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 ...
0
votes
1answer
80 views

$Sp(V)$ acts transitively on $V^*-\{0\}$ where $\Omega$ here is symplectic 2 form

Let $\dim(V)=6$. Show that $Sp(V,\Omega)$ acts transitively on $V^*-\{0\}$, where $\Omega$ here is a symplectic 2 form on $V$. ($V^*$ here is algebraic dual of $V$)
9
votes
1answer
400 views

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 ...
4
votes
1answer
110 views

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. ...
1
vote
2answers
278 views

Regarding Legendre transform from tangent bundle to cotangent bundle

(I'm a complete beginner at differential geometry) I'm studying about constrained systems, in which we "map a Lagrangian system from a tangent to a cotangent bundle. Hamiltonian dynamics then appears ...