0
votes
0answers
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, ...
0
votes
0answers
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 $T^*G$.
1
vote
1answer
64 views

Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
0
votes
0answers
28 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
2answers
57 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
1
vote
1answer
54 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
1answer
57 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 $M$.
2
votes
1answer
49 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why folloing sequence is exact? $0\to \mathbb{R} \to C^\infty (M)\to A\to 0$ Which $A$ here is the set of Global Hamiltonian vector fields.
0
votes
0answers
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, ...
1
vote
1answer
59 views

integrability of ker $\omega$ in symplectic case

How can we prove that if $(M,\omega)$, is pre-symplectic and d$\omega=0$ then ker$\omega$ is integrable?.
0
votes
0answers
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 $TM$?
1
vote
0answers
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 ...
5
votes
1answer
235 views

compact symplectic manifolds

Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?
4
votes
4answers
701 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!