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]

Darboux-Moser-Weinstein theorem

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]}

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. ^[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.

Weinstein's Lagrangian neighbourhood 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]

Generalisation: Coisotropic Embedding Theorem

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

Weinstein's tubular neighbourhood theorem

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 ${\displaystyle L_{0}}$.

Proof

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

References

1. Cannas Silva, Ana (2008). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN   978-3-540-42195-5.
2. Weinstein, Alan (1971-06-01). "Symplectic manifolds and their lagrangian submanifolds". Advances in Mathematics . 6 (3): 329–346. doi: 10.1016/0001-8708(71)90020-X . ISSN   0001-8708.
3. Moser, Jürgen (1965). "On the volume elements on a manifold". Transactions of the American Mathematical Society . 120 (2): 286–294. doi: 10.1090/S0002-9947-1965-0182927-5 . ISSN   0002-9947.
4. McDuff, Dusa; Salamon, Dietmar (2017-06-22). Introduction to Symplectic Topology. Vol. 1. Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN   978-0-19-879489-9.
5. Gotay, Mark J. (1982). "On coisotropic imbeddings of presymplectic manifolds". Proceedings of the American Mathematical Society . 84 (1): 111–114. doi: 10.1090/S0002-9939-1982-0633290-X . ISSN   0002-9939.
6. Weinstein, Alan (1981-01-01). "Neighborhood classification of isotropic embeddings". Journal of Differential Geometry . 16 (1). doi: 10.4310/jdg/1214435995 . ISSN   0022-040X.