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.