Darboux's theorem
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 2"n"-dimensional symplectic manifold can be made to look locally like the linear symplectic  space C with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry. Statement and first consequences. The precise statement is as follows. Suppose that formula_1 is a differential 1-form on an "n" dimensional manifold, such that formula_2 has constant rank "p". If then there is a local system of coordinates formula_4 in which If, on the other hand, then there is a local system of coordinates 'formula_4 in which Note that if formula_6 everywhere and formula_10 then formula_11 is a contact form. In particular, suppose that formula_12 is a symplectic  2-form on an "n"=2"m" dimensional manifold "M". In a neighborhood of each point "p" of "M", by the Poincaré lemma, there is a 1-form formula_1 with formula_14. Moreover, formula_1 satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart "U" near "p" in which Taking an exterior derivative now shows The chart "U" is said to be a Darboux chart around "p". The manifold "M" can be covered by such charts. To state this differently, identify formula_18 with formula_19 by letting formula_20. If formula_21is a Darboux chart, then formula_22 is  the pullback of the standard symplectic form formula_23 on formula_24: Comparison with Riemannian geometry. 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 of coordinate differentials. 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.