Darboux's theorem

In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux^[1] who established it as the solution of the Pfaff problem.^[2]

It is a foundational result in several fields, the chief among them being symplectic geometry. Indeed, one of its many consequences is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every ${\displaystyle 2n}$-dimensional symplectic manifold can be made to look locally like the linear symplectic space ${\displaystyle \mathbb {C} ^{n}}$ with its canonical symplectic form.

There is also an analogous consequence of the theorem applied to contact geometry.