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

$X$

.

Let

$M$

be a smooth manifold of dimension

$2n$

, and

${\omega }_1$

and

${\omega }_2$

two symplectic forms on

$M$

. Consider a compact submanifold

$i:X\hookrightarrow M$

such that

$i^{*}{\omega }_1=i^{*}{\omega }_2$

. Then there exist

• two open neighbourhoods $U_1$ and $U_2$ of $X$ in $M$;
• a diffeomorphism $f:U_1\to U_2$;

such that

$f^{*}{\omega }_2={\omega }_1$

and

$f{|}_X={\mathrm{i}\mathrm{d}}_X$

.

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

$M$

be a smooth manifold of dimension

$2n$

, and

${\omega }_1$

and

${\omega }_2$

two symplectic forms on

$M$

. Let also

$G$

be a compact Lie group acting on

$M$

and leaving both

${\omega }_1$

and

${\omega }_2$

invariant. Consider a compact and

$G$

-invariant submanifold

$i:X\hookrightarrow M$

such that

$i^{*}{\omega }_1=i^{*}{\omega }_2$

. Then there exist

• two open $G$-invariant neighbourhoods $U_1$ and $U_2$ of $X$ in $M$;
• a $G$-equivariant diffeomorphism $f:U_1\to U_2$;

such that

$f^{*}{\omega }_2={\omega }_1$

and

$f{|}_X={\mathrm{i}\mathrm{d}}_X$

.

In particular, taking again

$X$

as a point, one obtains an equivariant version of the classical Darboux theorem.

Weinstein's Lagrangian neighbourhood theorem

Let

$M$

be a smooth manifold of dimension

$2n$

, and

${\omega }_1$

and

${\omega }_2$

two symplectic forms on

$M$

. Consider a compact submanifold

$i:L\hookrightarrow M$

of dimension

$n$

which is a Lagrangian submanifold of both

$(M,{\omega }_1)$

and

$(M,{\omega }_2)$

, i.e.

$i^{*}{\omega }_1=i^{*}{\omega }_2=0$

. Then there exist

• two open neighbourhoods $U_1$ and $U_2$ of $L$ in $M$;
• a diffeomorphism $f:U_1\to U_2$;

such that

$f^{*}{\omega }_2={\omega }_1$

and

$f{|}_L={\mathrm{i}\mathrm{d}}_L$

.

This statement is proved using the Darboux-Moser-Weinstein theorem, taking

$X=L$

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

$L$

is Lagrangian.

Let

$M$

be a smooth manifold of dimension

$2n$

, and

${\omega }_1$

and

${\omega }_2$

two symplectic forms on

$M$

. Consider a compact submanifold

$i:L\hookrightarrow M$

of dimension

$k$

which is a coisotropic submanifold of both

$(M,{\omega }_1)$

and

$(M,{\omega }_2)$

, and such that

$i^{*}{\omega }_1=i^{*}{\omega }_2$

. Then there exist

• two open neighbourhoods $U_1$ and $U_2$ of $L$ in $M$;
• a diffeomorphism $f:U_1\to U_2$;

such that

$f^{*}{\omega }_2={\omega }_1$

and

$f{|}_L={\mathrm{i}\mathrm{d}}_L$

.

Weinstein's tubular neighbourhood theorem

While Darboux's theorem identifies locally a symplectic manifold

$M$

with

$T^{*}L$

, Weinstein's theorem identifies locally a Lagrangian

$L$

with the zero section of

$T^{*}L$

. More precisely

Let

$(M,\omega )$

be a symplectic manifold and

$L$

a Lagrangian submanifold. Then there exist

• an open neighbourhood $U$ of $L$ in $M$;
• an open neighbourhood $V$ of the zero section $L_0$ in the cotangent bundle $T^{*}L$;
• a symplectomorphism $f:U\to V$;

such that

$f$

sends

$L$

to

$L_0$

.

Proof

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