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

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

Let   be a smooth manifold of dimension  , and   and   two symplectic forms on  . Consider a compact submanifold   such that  . Then there exist

  • two open neighbourhoods   and   of   in  ;
  • a diffeomorphism  ;

such that   and  .

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

Generalisation: equivariant Darboux theorem

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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   be a smooth manifold of dimension  , and   and   two symplectic forms on  . Let also   be a compact Lie group acting on   and leaving both   and   invariant. Consider a compact and  -invariant submanifold   such that  . Then there exist

  • two open  -invariant neighbourhoods   and   of   in  ;
  • a  -equivariant diffeomorphism  ;

such that   and  .

In particular, taking again   as a point, one obtains an equivariant version of the classical Darboux theorem.

Weinstein's Lagrangian neighbourhood theorem

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Let   be a smooth manifold of dimension  , and   and   two symplectic forms on  . Consider a compact submanifold   of dimension   which is a Lagrangian submanifold of both   and  , i.e.  . Then there exist

  • two open neighbourhoods   and   of   in  ;
  • a diffeomorphism  ;

such that   and  .

This statement is proved using the Darboux-Moser-Weinstein theorem, taking   a Lagrangian submanifold, together with a version of the Whitney Extension Theorem for smooth manifolds.[1]

Generalisation: Coisotropic Embedding Theorem

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

Let   be a smooth manifold of dimension  , and   and   two symplectic forms on  . Consider a compact submanifold   of dimension   which is a coisotropic submanifold of both   and  , and such that  . Then there exist

  • two open neighbourhoods   and   of   in  ;
  • a diffeomorphism  ;

such that   and  .

Weinstein's tubular neighbourhood theorem

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While Darboux's theorem identifies locally a symplectic manifold   with  , Weinstein's theorem identifies locally a Lagrangian   with the zero section of  . More precisely

Let   be a symplectic manifold and   a Lagrangian submanifold. Then there exist

  • an open neighbourhood   of   in  ;
  • an open neighbourhood   of the zero section   in the cotangent bundle  ;
  • a symplectomorphism  ;

such that   sends   to  .

Proof

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

References

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  1. ^ 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.
  2. ^ 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.
  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.