1301.2016 (Carlos Batista)

On the Weyl Tensor Classification in All Dimensions and its Relation
with Integrability Properties    [PDF]

Carlos Batista

In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well known Petrov classification emerging as a special case. Then it is shown that the integrability condition of maximally isotropic distributions can be described in terms of the invariance of certain subspaces under these operators. Here it is also proved a new generalization of the Goldberg-Sachs theorem, valid in all even dimensions, stating that the existence of an integrable maximally isotropic distribution imposes restrictions on the shear matrix. Finally the higher-dimensional versions of the self-dual manifolds are defined and investigated.

View original: http://arxiv.org/abs/1301.2016