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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[edit]

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

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.^[3]^[4]

Generalisation: equivariant Darboux theorem[edit]

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 $M$ 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[edit]

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.^[1]

Generalisation: Coisotropic Embedding Theorem[edit]

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

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[edit]

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[edit]

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

References[edit]

^ ^a ^b ^c ^d Cannas Silva, Ana (2008). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.
^ ^a ^b ^c 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.
^ 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.
^ 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.
^ 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.
^ Weinstein, Alan (1981-01-01). "Neighborhood classification of isotropic embeddings". Journal of Differential Geometry. 16 (1). doi:10.4310/jdg/1214435995. ISSN 0022-040X.

[:0-1] Cannas Silva, Ana (2008). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.

[:1-2] 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.

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