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.
[1] 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
be a smooth manifold of dimension
two symplectic forms on
Consider a compact submanifold
.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.
be a smooth manifold of dimension
two symplectic forms on
be a compact Lie group acting on
Consider a compact and
-invariant submanifold
as a point, one obtains an equivariant version of the classical Darboux theorem.
be a smooth manifold of dimension
two symplectic forms on
Consider a compact submanifold
which is a Lagrangian submanifold of both
.This statement is proved using the Darboux-Moser-Weinstein theorem, taking
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
be a smooth manifold of dimension
two symplectic forms on
Consider a compact submanifold
which is a coisotropic submanifold of both
While Darboux's theorem identifies locally a symplectic manifold
, Weinstein's theorem identifies locally a Lagrangian
be a symplectic manifold and
a Lagrangian submanifold.
.This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem.