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 [math]\displaystyle{ X }[/math].[1][2]
Let [math]\displaystyle{ M }[/math] be a smooth manifold of dimension [math]\displaystyle{ 2n }[/math], and [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \omega_2 }[/math] two symplectic forms on [math]\displaystyle{ M }[/math]. Consider a compact submanifold [math]\displaystyle{ i: X \hookrightarrow M }[/math] such that [math]\displaystyle{ i^* \omega_1 = i^* \omega_2 }[/math]. Then there exist
- two open neighbourhoods [math]\displaystyle{ U_1 }[/math] and [math]\displaystyle{ U_2 }[/math] of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ M }[/math];
- a diffeomorphism [math]\displaystyle{ f: U_1 \to U_2 }[/math];
such that [math]\displaystyle{ f^* \omega_2 = \omega_1 }[/math] and [math]\displaystyle{ f |_X = \mathrm{id}_X }[/math].
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 [math]\displaystyle{ M }[/math] be a smooth manifold of dimension [math]\displaystyle{ 2n }[/math], and [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \omega_2 }[/math] two symplectic forms on [math]\displaystyle{ M }[/math]. Let also [math]\displaystyle{ G }[/math] be a compact Lie group acting on [math]\displaystyle{ M }[/math] and leaving both [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \omega_2 }[/math] invariant. Consider a compact and [math]\displaystyle{ G }[/math]-invariant submanifold [math]\displaystyle{ i: X \hookrightarrow M }[/math] such that [math]\displaystyle{ i^* \omega_1 = i^* \omega_2 }[/math]. Then there exist
- two open [math]\displaystyle{ G }[/math]-invariant neighbourhoods [math]\displaystyle{ U_1 }[/math] and [math]\displaystyle{ U_2 }[/math] of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ M }[/math];
- a [math]\displaystyle{ G }[/math]-equivariant diffeomorphism [math]\displaystyle{ f: U_1 \to U_2 }[/math];
such that [math]\displaystyle{ f^* \omega_2 = \omega_1 }[/math] and [math]\displaystyle{ f |_X = \mathrm{id}_X }[/math].
In particular, taking again [math]\displaystyle{ X }[/math] as a point, one obtains an equivariant version of the classical Darboux theorem.
Weinstein's Lagrangian neighbourhood theorem
Let [math]\displaystyle{ M }[/math] be a smooth manifold of dimension [math]\displaystyle{ 2n }[/math], and [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \omega_2 }[/math] two symplectic forms on [math]\displaystyle{ M }[/math]. Consider a compact submanifold [math]\displaystyle{ i: L \hookrightarrow M }[/math] of dimension [math]\displaystyle{ n }[/math] which is a Lagrangian submanifold of both [math]\displaystyle{ (M,\omega_1) }[/math] and [math]\displaystyle{ (M, \omega_2) }[/math], i.e. [math]\displaystyle{ i^* \omega_1 = i^* \omega_2 = 0 }[/math]. Then there exist
- two open neighbourhoods [math]\displaystyle{ U_1 }[/math] and [math]\displaystyle{ U_2 }[/math] of [math]\displaystyle{ L }[/math] in [math]\displaystyle{ M }[/math];
- a diffeomorphism [math]\displaystyle{ f: U_1 \to U_2 }[/math];
such that [math]\displaystyle{ f^* \omega_2 = \omega_1 }[/math] and [math]\displaystyle{ f |_L = \mathrm{id}_L }[/math].
This statement is proved using the Darboux-Moser-Weinstein theorem, taking [math]\displaystyle{ X = L }[/math] 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 [math]\displaystyle{ L }[/math] is Lagrangian.[5][6]
Let [math]\displaystyle{ M }[/math] be a smooth manifold of dimension [math]\displaystyle{ 2n }[/math], and [math]\displaystyle{ \omega_1 }[/math] and [math]\displaystyle{ \omega_2 }[/math] two symplectic forms on [math]\displaystyle{ M }[/math]. Consider a compact submanifold [math]\displaystyle{ i: L \hookrightarrow M }[/math] of dimension [math]\displaystyle{ k }[/math] which is a coisotropic submanifold of both [math]\displaystyle{ (M,\omega_1) }[/math] and [math]\displaystyle{ (M, \omega_2) }[/math], and such that [math]\displaystyle{ i^* \omega_1 = i^* \omega_2 }[/math]. Then there exist
- two open neighbourhoods [math]\displaystyle{ U_1 }[/math] and [math]\displaystyle{ U_2 }[/math] of [math]\displaystyle{ L }[/math] in [math]\displaystyle{ M }[/math];
- a diffeomorphism [math]\displaystyle{ f: U_1 \to U_2 }[/math];
such that [math]\displaystyle{ f^* \omega_2 = \omega_1 }[/math] and [math]\displaystyle{ f |_L = \mathrm{id}_L }[/math].
Weinstein's tubular neighbourhood theorem
While Darboux's theorem identifies locally a symplectic manifold [math]\displaystyle{ M }[/math] with [math]\displaystyle{ T^*L }[/math], Weinstein's theorem identifies locally a Lagrangian [math]\displaystyle{ L }[/math] with the zero section of [math]\displaystyle{ T^*L }[/math]. More precisely
Let [math]\displaystyle{ (M,\omega) }[/math] be a symplectic manifold and [math]\displaystyle{ L }[/math] a Lagrangian submanifold. Then there exist
- an open neighbourhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ L }[/math] in [math]\displaystyle{ M }[/math];
- an open neighbourhood [math]\displaystyle{ V }[/math] of the zero section [math]\displaystyle{ L_0 }[/math] in the cotangent bundle [math]\displaystyle{ T^*L }[/math];
- a symplectomorphism [math]\displaystyle{ f: U \to V }[/math];
such that [math]\displaystyle{ f }[/math] sends [math]\displaystyle{ L }[/math] to [math]\displaystyle{ L_0 }[/math].
Proof
This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem.[1]
References
- ↑ 1.0 1.1 1.2 1.3 Cannas Silva, Ana (2008) (in en). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5. https://link.springer.com/book/10.1007/978-3-540-45330-7.
- ↑ 2.0 2.1 2.2 Weinstein, Alan (1971-06-01). "Symplectic manifolds and their lagrangian submanifolds" (in en). Advances in Mathematics 6 (3): 329–346. doi:10.1016/0001-8708(71)90020-X. ISSN 0001-8708. https://www.sciencedirect.com/science/article/pii/000187087190020X.
- ↑ Moser, Jürgen (1965). "On the volume elements on a manifold" (in en). Transactions of the American Mathematical Society 120 (2): 286–294. doi:10.1090/S0002-9947-1965-0182927-5. ISSN 0002-9947. https://www.ams.org/tran/1965-120-02/S0002-9947-1965-0182927-5/.
- ↑ McDuff, Dusa; Salamon, Dietmar (2017-06-22) (in en). Introduction to Symplectic Topology. 1. Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879489-9. https://academic.oup.com/book/43512.
- ↑ Gotay, Mark J. (1982). "On coisotropic imbeddings of presymplectic manifolds" (in en). Proceedings of the American Mathematical Society 84 (1): 111–114. doi:10.1090/S0002-9939-1982-0633290-X. ISSN 0002-9939. https://www.ams.org/proc/1982-084-01/S0002-9939-1982-0633290-X/.
- ↑ Weinstein, Alan (1981-01-01). "Neighborhood classification of isotropic embeddings". Journal of Differential Geometry 16 (1). doi:10.4310/jdg/1214435995. ISSN 0022-040X. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-16/issue-1/Neighborhood-classification-of-isotropic-embeddings/10.4310/jdg/1214435995.full.
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