Polarisation in a nighbourhood of a Lagrangian submanifold

2012-11-13T13:31:44Z

Hallo,

Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such that $\alpha = 0$ of $TX|_{M}$. I am interested in the following questions:

Is there a unique polarisation defined on $X$ near $M$ which is transversal to $M$ and whose one form is $\alpha$ ? By polarisation I mean the following: A polarisation of a symplectic manifold $X$, with symplectic form $\omega$, is a smooth assignment of a Lagrangian subspace of $T_{x}X$ to each $x \in X$ in such a way that this assignment is integrable.
If 1. is true, is there a symplectic diffeomorpism $\Phi$ of a neigbourhood of $M$ is $X$ with a neigbourhood of $M$ in its cotangent bundle which carries the leaves of the polarisation into the standard cotangent fibration of $T^{*}M$ ?

Actually I know that these results are true. I would like to see the proof of them. Are there any references where I can look them up? If so, can you please tell me where these references can be found? Thanks a lot!

hapchiu

Answer by Francois Ziegler for Polarisation in a nighbourhood of a Lagrangian submanifold

2012-11-13T16:14:15Z

Arnol'd (p. 3314) puts it that way:

Weinstein's Theorem. Some neighborhood of any Lagrangian submanifold in any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance in its own cotangent bundle space.

(The resulting neighborhood then has the obvious transverse polarization by fibers of the cotangent bundle.) Unless I am mistaken, Weinstein proves this in Symplectic manifolds and their lagrangian submanifolds, Theorem 6.1 and Corollary 6.2 (which he points out goes back to Souriau).

Polarisation in a nighbourhood of a Lagrangian submanifold - MathOverflow

Polarisation in a nighbourhood of a Lagrangian submanifold

Answer by Francois Ziegler for Polarisation in a nighbourhood of a Lagrangian submanifold