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.

Darboux-Moser-Weinstein theorem

This statement is a direct generalisation of Darboux's theorem, which is recovered by taking a point as .<blockquote>Let be a smooth manifold of dimension , and and two symplectic forms on . Consider a compact submanifold such that . Then there exist

two open neighbourhoods and of in ;
a diffeomorphism ;

such that and .</blockquote>Its proof employs Moser's trick.

Generalisation: equivariant Darboux theorem

The statement (and the proof) of Darboux-Moser-Weinstein theorem can be generalised in presence of a symplectic action of a Lie group.<blockquote>Let be a smooth manifold of dimension , and and two symplectic forms on . Let also be a compact Lie group acting on and leaving both and invariant. Consider a compact and -invariant submanifold such that . Then there exist

two open -invariant neighbourhoods and of in ;
a -equivariant diffeomorphism ;

such that and .</blockquote>In particular, taking again as a point, one obtains an equivariant version of the classical Darboux theorem.

Weinstein's Lagrangian neighbourhood theorem

<blockquote>Let be a smooth manifold of dimension , and and two symplectic forms on . Consider a compact submanifold of dimension which is a Lagrangian submanifold of both and , i.e. . Then there exist

two open neighbourhoods and of in ;
a diffeomorphism ;

such that and .</blockquote>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.

Generalisation: Coisotropic Embedding Theorem

Weinstein's result can be generalised by weakening the assumption that is Lagrangian.<blockquote>Let be a smooth manifold of dimension , and and two symplectic forms on . Consider a compact submanifold of dimension which is a coisotropic submanifold of both and , and such that . Then there exist

two open neighbourhoods and of in ;
a diffeomorphism ;

such that and .</blockquote>

Weinstein's tubular neighbourhood theorem

While Darboux's theorem identifies locally a symplectic manifold with , Weinstein's theorem identifies locally a Lagrangian with the zero section of . More precisely<blockquote>Let be a symplectic manifold and a Lagrangian submanifold. Then there exist

an open neighbourhood of in ;
an open neighbourhood of the zero section in the cotangent bundle ;
a symplectomorphism ;

such that sends to .</blockquote>

Proof

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