Robinson manifolds as the Lorentzian analogs of Hermite manifolds

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 ⩾4 with a subbundle N of the complexification of TM such that the fibers of N→M are maximal totally null (isotropic) and [SecN,SecN]⊂SecN. 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 Godel 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.