Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.
One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cn with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.
In particular, suppose that ω is a symplectic 2-form on an n=2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ=ω. Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which
To state this differently, identify R2m with Cm by letting zj = xj + i yj. If φ : U → Cn is a Darboux chart, then ω is the pullback of the standard symplectic form ω0 on Cn:
This result 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.
It should be emphasized that the difference is that Darboux's theorem states that ω can be made to take the standard form in an entire neighborhood around p. 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.