Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Théorème de Darboux (géométrie)
Résumé
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 who established it as the solution of the Pfaff problem.
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.
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 or of differential ideals.
Darboux's theorem for ensures the any 1-form such that can be written as in some coordinate system .
This recovers one of the formulation of Frobenius theorem in terms of differential forms: if is the differential ideal generated by , then implies the existence of a coordinate system where is actually generated by .
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 . 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.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.