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Darboux's theorem (symplectic geometry)
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(Theorem)
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If $(M,\omega)$ is a $2n$ -dimensional symplectic manifold, and $m\in M$ , then there exists a neighborhood $U$ of $m$ with a coordinate chart $$x=(x_1,\ldots,x_{2n}):U\to\R^{2n},$$ such that $$\omega=\sum_{i=1}^ndx_{i}\wedge dx_{n+i}.$$ These are called canonical or Darboux coordinates. On $U$ , $\omega$ is the pullback by $X$ of the standard symplectic form on $\R^{2n}$ , so $x$ is a symplectomorphism. Darboux's theorem implies that there are no local invariants in symplectic geometry, unlike in Riemannian geometry, where there is curvature.
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"Darboux's theorem (symplectic geometry)" is owned by bwebste.
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| Other names: |
Darboux coordinates |
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Cross-references: curvature, geometry, invariants, implies, theorem, symplectomorphism, symplectic form, pullback, canonical, coordinate chart, neighborhood, symplectic manifold
There are 4 references to this entry.
This is version 3 of Darboux's theorem (symplectic geometry), born on 2002-12-12, modified 2007-06-24.
Object id is 3738, canonical name is DarbouxsTheoremSymplecticGeometry.
Accessed 5401 times total.
Classification:
| AMS MSC: | 53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general) |
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Pending Errata and Addenda
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