Pseudo-Riemannian metrics Hermitian Geometry General Relativity Department of Complex Analysis
Issue Date:
Sep-1996
Publisher:
Institute of Mathematics with Computer Center at the Bulgarian Academy of Sciences
Citation:
Preprint
Series/Report no.:
1996;8
Abstract:
It has been recently observed that the Generalized Goldberg-Sachs Theorem in General Relativity as well as some of its corollaries admit appropriate Riemannian versions.
In this paper we use the formalism of spinors to give alternative proofs of these results clarifying the analogy between the positive Hermitian structures of oriented Riemannian four-manifolds and the shear-free congruences of oriented Lorentzian four-manifolds. We also prove similar results for the oriented pseudo-Rimennian four-manifolds when the metric is of zero signature. This allows us to describe the compact oriented four-manifolds possibly addmiting a pseudo-Riemannian Einstein metric of zero signature whose positive Weyl tensor has two distinctinct eigenvalues corresponding to non-isotropic eigenspaces.1991 MSC: 53C50; 53C55.
