1205.4666 (Carlos Batista)

A Generalization of the Goldberg-Sachs Theorem and its Consequences    [PDF]

Carlos Batista

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endued with torsion-free connection compatible with the metric, the treatment includes all signatures as well the complex manifolds. It is shown that when Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that when the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and admits a solution for the source-free Einstein-Maxwell equations.

View original: http://arxiv.org/abs/1205.4666