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Darboux's theorem

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Contents

Idea

Darboux’s theorem states that a smooth manifold XX equipped with a differential 1-form θ\theta which is sufficiently non-degenerate admits local coordinate charts ϕ i: 2nX\phi_i \colon\mathbb{R}^{2n} \to X on which θ\theta takes the canonical form ϕ i *θ= kx 2kdx 2k+1\phi_i^\ast \theta = \sum_{k} x^{2k} \mathbf{d} x^{2k+1}.

In particular

  • for (X,ω)(X, \omega) a symplectic manifold there are local charts in which the symplectic form ω\omega takes the form ϕ i *ω= kdx 2kdx 2k+1\phi_i^\ast \omega = \sum_{k} \mathbf{d} x^{2k} \wedge \mathbf{d} x^{2k+1};

  • similarly, for contact manifolds there are adapated local charts.

(e.g. Arnold 78, p. 362)

For the case of symplectic manifolds the Darboux theorem may also be read as saying that a G-structure for G=Sp(2n)G = Sp(2n) the symplectic group (hence an almost symplectic structure) is an integrable G-structure already when it is first-order integrable, i.e. torsion-free, i.e. symplectic. See at integrability of G-structures – Examples – Symplectic structure.

References

Lecture notes include

  • Andreas Čap, section 1.8 of Differential Geometry 2, 2011/2012 (pdf)

  • Federica Pasquotto, Linear GG-structures by example (pdf)

Textbook accounts include

See also

A version for supergeometry is due to

  • Bertram Kostant, in Lecture Notes in Mathematics vol.570, 177, (Bleuler, K. and Reetz, A.eds), Proc. Conf. on Diff. Geom. Meth. in Math. Phys., Bonn 1975., Springer-Verlag, Berlin, 1977

A version for prequantized symplectic manifolds (prequantum bundles) is discussed in

  • Frédéric Faure, Masato Tsujii, around prop. 2.15 of Prequantum transfer operator for symplectic Anosov diffeomorphism (arXiv:1206.0282)

A version for shifted symplectic structures (BV-brackets) on stacks is discussed in

Last revised on November 17, 2017 at 13:15:27. See the history of this page for a list of all contributions to it.