Suppose that is a differential 1-form on an -dimensional manifold, such that has constant rank . Then
- if everywhere, then there is a local system of coordinates in which
- if everywhere, then there is a local system of coordinates in which
Darboux's original proof used induction on and it can be equivalently presented in terms of distributions[3] or of differential ideals.[4]
Suppose that is a symplectic 2-form on an -dimensional manifold . In a neighborhood of each point of , by the Poincaré lemma, there is a 1-form with . Moreover, satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart near in which
Taking an exterior derivative now shows
The chart is said to be a Darboux chart around .[5] The manifold can be covered by such charts.
To state this differently, identify with by letting . If is a Darboux chart, then can be written as the pullback of the standard symplectic form on :
A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick.[5][6]
Comparison with Riemannian geometry
Darboux's theorem for symplectic manifolds implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares of coordinate differentials.
The difference is that Darboux's theorem states that can be made to take the standard form in an entire neighborhood around . In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.