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 $f|_{X}=\mathrm {id} _{X}$.

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 $f|_{X}=\mathrm {id} _{X}$.

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 $f|_{L}=\mathrm {id} _{L}$.

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 $f|_{L}=\mathrm {id} _{L}$.

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.