Darboux’s theorem (symplectic geometry)

If (M,ω) is a 2n-dimensional symplectic manifoldMathworldPlanetmath, and mM, then there exists a neighborhoodMathworldPlanetmath U of m with a coordinate chartMathworldPlanetmath


such that


These are called canonical or Darboux coordinates. On U, ω is the pullback by X of the standard symplectic form on 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 curvaturePlanetmathPlanetmath.

Title Darboux’s theorem (symplectic geometry)
Canonical name DarbouxsTheoremsymplecticGeometry
Date of creation 2013-03-22 13:15:31
Last modified on 2013-03-22 13:15:31
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 6
Author bwebste (988)
Entry type Theorem
Classification msc 53D05
Synonym Darboux coordinates