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Differential Geometry and its Applications, Vol. 17 (2-3) (2002) pp. 175-195
© 2002 Elsevier Science B.V. All rights reserved.
PII: S0926-2245(02)00106-7

Robinson manifolds as the Lorentzian analogs of Hermite manifolds

Pawel Nurowski and Andrzej Trautman *

Instytut Fizyki Teoretycznej, Hoza 69, 00681 Warszawa, Poland
Plenary lecture at the 8th DGA Conference, Opava, Czech Republic, August 27-31, 2001

Communicated by D. Krupka


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 >=4 with a subbundle N of the complexification of TM such that the fibers of N\toM are maximal totally null (isotropic) and [SecN,SecN]\subsetSecN. 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.

MSC: 32C81; 53B30; 32V30; 83C20

Keywords: Lorentz manifolds; Robinson manifolds; Hermite manifolds; Twistor bundles

*Corresponding author.

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