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
Signature operator
Résumé
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator.
Let be a compact Riemannian manifold of even dimension . Let
be the exterior derivative on -th order differential forms on . The Riemannian metric on allows us to define the Hodge star operator and with it the inner product
on forms. Denote by
the adjoint operator of the exterior differential . This operator can be expressed purely in terms of the Hodge star operator as follows:
Now consider acting on the space of all forms .
One way to consider this as a graded operator is the following: Let be an involution on the space of all forms defined by:
It is verified that anti-commutes with and, consequently, switches the -eigenspaces of
Consequently,
Definition: The operator with the above grading respectively the above operator is called the signature operator of .
In the odd-dimensional case one defines the signature operator to be acting
on the even-dimensional forms of .
If , so that the dimension of is a multiple of four, then Hodge theory implies that:
where the right hand side is the topological signature (i.e. the signature of a quadratic form on defined by the cup product).
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:
where is the Hirzebruch L-Polynomial, and the the Pontrjagin forms on .
Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.
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.