Weinstein's neighbourhood theorem

In symplectic geometry, a branch of mathematics, Weinstein's neighbourhood theorem refers to a few distinct but related theorems, involving the neighbourhoods of submanifolds in symplectic manifolds and generalising the classical Darboux's theorem. They were proved by Alan Weinstein in 1971.

This statement is a direct generalisation of Darboux's theorem, which is recovered by taking a point as ${\displaystyle X}$.

Its proof employs Moser's trick.

The statement (and the proof) of Darboux-Moser-Weinstein theorem can be generalised in presence of a symplectic action of a Lie group.

In particular, taking again ${\displaystyle X}$as a point, one obtains an equivariant version of the classical Darboux theorem.

This statement is proved using the Darboux-Moser-Weinstein theorem, taking ${\displaystyle X=L}$a Lagrangian submanifold, together with a version of the Whitney Extension Theorem for smooth manifolds.

Weinstein's result can be generalised by weakening the assumption that ${\displaystyle L}$is Lagrangian.

While Darboux's theorem identifies locally a symplectic manifold ${\displaystyle M}$with ${\displaystyle T^{*}L}$, Weinstein's theorem identifies locally a Lagrangian ${\displaystyle L}$with the zero section of ${\displaystyle T^{*}L}$. More precisely

This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem.