Darboux's theorem

This article is about Darboux's theorem in differential geometry. For Darboux's theorem related to the intermediate value theorem, see Darboux's theorem (analysis).

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 -dimensional symplectic manifold can be made to look locally like the linear symplectic space with its canonical symplectic form.

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

  1. ^ Darboux, Gaston (1882). "Sur le problème de Pfaff" [On the Pfaff's problem]. Bull. Sci. Math. (in French). 6: 14–36, 49–68. JFM 05.0196.01.
  2. ^ Pfaff, Johann Friedrich (1814–1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi" [A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables]. Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin (in Latin): 76–136.