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 $i^{\ast }{\omega }_1=i^{\ast }{\omega }_2$. 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 $f^{\ast }{\omega }_2={\omega }_1$ 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 $i^{\ast }{\omega }_1=i^{\ast }{\omega }_2$. 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 $f^{\ast }{\omega }_2={\omega }_1$ 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. $i^{\ast }{\omega }_1=i^{\ast }{\omega }_2=0$. 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 $f^{\ast }{\omega }_2={\omega }_1$ 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 $i^{\ast }{\omega }_1=i^{\ast }{\omega }_2$. 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 $f^{\ast }{\omega }_2={\omega }_1$ and $f{|}_L={\mathrm{i}\mathrm{d}}_L$.

Weinstein's tubular neighbourhood theorem

While Darboux's theorem identifies locally a symplectic manifold $M$ with $T^{\ast }L$, Weinstein's theorem identifies locally a Lagrangian $L$ with the zero section of $T^{\ast }L$. 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 $T^{\ast }L$;
• 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.