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 .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 .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.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 .In particular, taking again as a point, one obtains an equivariant version of the classical Darboux theorem.
Weinstein's Lagrangian neighbourhood theorem 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 .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.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 .
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 preciselyLet 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 .
Proof This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem.
References