Sharp version of the Goldberg-Sachs theorem

2009/11/17 by A Rod Gover, A. Rod Gover, Gover, A Rod +6 · 1 citation
Mathematics · Physics and Astronomy · #83C05 #83C60 #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.DG #msc:83C05 #msc:83C60

paper · pdf · doi:10.48550/arxiv.0911.3364

In this version we made a minor change in Remark 5.5 and simplified Section 6, starting at Theorem 6.5

openalex publication_date 2009/11/17 · arxiv created 2009/12/11 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)⇒(algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes we associate a natural connection, in terms of which these conditions have a very simple form.

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