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.^[1] They were proved by Alan Weinstein in 1971.^[2]

This statement is a direct generalisation of Darboux's theorem, which is recovered by taking a point as ${\displaystyle X}$.^[1][2]

Let ${\displaystyle M}$ be a smooth manifold of dimension ${\displaystyle 2n}$, and ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ two symplectic forms on ${\displaystyle M}$. Consider a compact submanifold ${\displaystyle i:X\hookrightarrow M}$ such that ${\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}}$. Then there exist

• two open neighbourhoods ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ of ${\displaystyle X}$ in ${\displaystyle M}$;
• a diffeomorphism ${\displaystyle f:U_{1}\to U_{2}}$;

such that ${\displaystyle f^{*}\omega _{2}=\omega _{1}}$ and ${\displaystyle f|_{X}=\mathrm {id} _{X}}$.

Its proof employs Moser's trick.^[3][4]

The statement (and the proof) of Darboux-Moser-Weinstein theorem can be generalised in presence of a symplectic action of a Lie group.^[2]

Let ${\displaystyle M}$ be a smooth manifold of dimension ${\displaystyle 2n}$, and ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ two symplectic forms on ${\displaystyle M}$. Let also ${\displaystyle G}$ be a compact Lie group acting on ${\displaystyle M}$ and leaving both ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ invariant. Consider a compact and ${\displaystyle G}$-invariant submanifold ${\displaystyle i:X\hookrightarrow M}$ such that ${\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}}$. Then there exist

• two open ${\displaystyle G}$-invariant neighbourhoods ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ of ${\displaystyle X}$ in ${\displaystyle M}$;
• a ${\displaystyle G}$-equivariant diffeomorphism ${\displaystyle f:U_{1}\to U_{2}}$;

such that ${\displaystyle f^{*}\omega _{2}=\omega _{1}}$ and ${\displaystyle f|_{X}=\mathrm {id} _{X}}$.

In particular, taking again ${\displaystyle X}$ as a point, one obtains an equivariant version of the classical Darboux theorem.

Let ${\displaystyle M}$ be a smooth manifold of dimension ${\displaystyle 2n}$, and ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ two symplectic forms on ${\displaystyle M}$. Consider a compact submanifold ${\displaystyle i:L\hookrightarrow M}$ of dimension ${\displaystyle n}$ which is a Lagrangian submanifold of both ${\displaystyle (M,\omega _{1})}$ and ${\displaystyle (M,\omega _{2})}$, i.e. ${\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}=0}$. Then there exist

• two open neighbourhoods ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ of ${\displaystyle L}$ in ${\displaystyle M}$;
• a diffeomorphism ${\displaystyle f:U_{1}\to U_{2}}$;

such that ${\displaystyle f^{*}\omega _{2}=\omega _{1}}$ and ${\displaystyle f|_{L}=\mathrm {id} _{L}}$.

This statement is proved using the Darboux-Moser-Weinstein theorem, taking ${\displaystyle X=L}$ a Lagrangian submanifold, together with a version of the Whitney Extension Theorem for smooth manifolds.^[1]

Weinstein's result can be generalised by weakening the assumption that ${\displaystyle L}$ is Lagrangian.^[5][6]

Let ${\displaystyle M}$ be a smooth manifold of dimension ${\displaystyle 2n}$, and ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ two symplectic forms on ${\displaystyle M}$. Consider a compact submanifold ${\displaystyle i:L\hookrightarrow M}$ of dimension ${\displaystyle k}$ which is a coisotropic submanifold of both ${\displaystyle (M,\omega _{1})}$ and ${\displaystyle (M,\omega _{2})}$, and such that ${\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}}$. Then there exist

• two open neighbourhoods ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ of ${\displaystyle L}$ in ${\displaystyle M}$;
• a diffeomorphism ${\displaystyle f:U_{1}\to U_{2}}$;

such that ${\displaystyle f^{*}\omega _{2}=\omega _{1}}$ and ${\displaystyle f|_{L}=\mathrm {id} _{L}}$.

While Darboux's theorem identifies locally a symplectic manifold ${\displaystyle M}$ with ${\displaystyle T^{*}L}$, Weinstein's theorem identifies locally a Lagrangian ${\displaystyle L}$ with the zero section of ${\displaystyle T^{*}L}$. More precisely

Let ${\displaystyle (M,\omega )}$ be a symplectic manifold and ${\displaystyle L}$ a Lagrangian submanifold. Then there exist

• an open neighbourhood ${\displaystyle U}$ of ${\displaystyle L}$ in ${\displaystyle M}$;
• an open neighbourhood ${\displaystyle V}$ of the zero section ${\displaystyle L_{0}}$ in the cotangent bundle ${\displaystyle T^{*}L}$;
• a symplectomorphism ${\displaystyle f:U\to V}$;

such that ${\displaystyle f}$ sends ${\displaystyle L}$ to ${\displaystyle L_{0}}$.

This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem.^[1]