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Robinson Manifolds as the Lorentzian Analogs of Hermite Manifolds

Pawel Nurowski·Andrzej Trautman
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Abstract

A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature ((2n +1, 1)). A Robinson manifold is a Lorentzian manifold (M) of dimension (\geqslant 4) with a subbundle (N) of the complexification of (TM) such that the fibers of (N\to M) are maximal totally null (isotropic) and ([\Sec N, \Sec N]\subset \Sec N). Robinson manifolds are close analogs of the proper Riemannian, Hermite manifolds. In dimension 4, they correspond to space-times of general relativity, foliated by a family of null geodesics without shear. Such space-times, introduced in the 1950s by Ivor Robinson, played an important role in the study of solutions of Einstein's equations: plane and sphere-fronted waves, the Gödel universe, the Kerr solution, and their generalizations, are among them. In this survey article, the analogies between Hermite and Robinson manifolds are presented in considerable detail.

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Pawel Nurowski, Andrzej Trautman. 2002-01-28. Robinson Manifolds as the Lorentzian Analogs of Hermite Manifolds. https://arxiv.org/abs/math/0201266

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