0
votes
0answers
19 views

hamiltonian mechanics

In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial ...
1
vote
1answer
54 views

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 ...
2
votes
1answer
61 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?
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: ...
2
votes
0answers
182 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 ...
6
votes
1answer
245 views

is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
3
votes
0answers
102 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 ...
5
votes
1answer
365 views

Tangent space of Cotangent bundle at zero section?

Let $M$ be a differentiable manifold with cotangent bundle $T^*M$. How can I prove that $T_{(p,0)}T^*M$ is naturally isomorphic to $T_pM\oplus T_pM^*$? If this true, then I think I could prove that ...