General Relativity and Gravitation

, Volume 45, Issue 7, pp 1411–1431 | Cite as

A Generalization of the Goldberg–Sachs theorem and its consequences

  • Carlos Batista
Research Article


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.


Goldberg–Sachs theorem Weyl tensor Integrable distributions  Petrov classification General relativity 



I want to thank Bruno G. Carneiro da Cunha for the encouragement and for the manuscript revision. This research was supported by CNPq(Conselho Nacional de Desenvolvimento Científico e Tecnológico).


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de PernambucoRecifeBrazil

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