A Generalization of the Goldberg-Sachs Theorem and its Consequences

Abstract

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the 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 if 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.

14 pages. This version matches the published one

References in corpus (2)

• On a five-dimensional version of the Goldberg-Sachs theorem
• Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures

Cited by in corpus (8)

• Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures
• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• Spinors and the Weyl Tensor Classification in Six Dimensions
• A Class of Integrable Metrics
• Pure Subspaces, Generalizing the Concept of Pure Spinors
• A Class of Integrable Metrics and Gauge Fields
• A Class of Integrable Metrics II
• On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem