In a symplectic manifold, can I always find a darboux chart (x,y) so that x is also the coordinates of a submanifold?

Question

Given a symplectic manifold $M$, darboux' theorem guarantees that I can always find a local chart $\varphi$ of the manifold, in which the symplectic form can be written as $$ \omega = \sum_i dx_i \wedge dy_i $$ Now my question is: Given a halfdimensional submanifold $S$ of the symplectic manifold, can I always find the darboux-chart (x, y) in such a way that x can be thought of as the coordinates of the submanifold?

Answer 1

This is true if and only if $S$ is Lagrangian.

If $S\subset (M,\omega)$ is Lagrangian, then a neighborhood of $S$ in $(M,\omega)$ is symplectomorphic with a neighborhood of $S$ in $(T^{*}S,\omega_{can})$. This is Weinstein's Lagrangian neighborhood theorem. So without loss of generality, we can work on $(T^{*}S,\omega_{can})$ instead. In cotangent coordinates $(x,y)$, we have $$ \omega_{can}=\sum_{i}dx_{i}\wedge dy_{i},\\ S\leftrightarrow \{y=0\}, $$ so these coordinates are as desired.

Conversely, if around any point in $S$ you can find such Darboux coordinates $(x,y)$ adapted to $S$, then the pullback of $\omega$ to $S$ is zero (since the $y$-coordinates are constant on $S$). So $S$ is isotropic, and since $S$ is half-dimensional it must be Lagrangian.