Robinson manifolds and Cauchy–Riemann spaces

Abstract

A Robinson manifold is defined as a Lorentz manifold (M, g) of dimension 2n ≥ 4 with a bundle N ⊂ ℂ ⊗ TM such that the fibres of N are maximal totally null and there holds the integrability condition [Sec N, Sec N] ⊂ Sec N. The real part of N ∩ N̄ is a bundle of null directions tangent to a congruence of null geodesics. This generalizes the notion of a shear-free congruence of null geodesics (SNG) in dimension 4. Under a natural regularity assumption, the set ℳ of all these geodesics has the structure of a Cauchy–Riemann manifold of dimension 2n − 1. Conversely, every such CR manifold lifts to many Robinson manifolds. Three definitions of a CR manifold are described here in considerable detail; they are equivalent under the assumption of real analyticity, but not in the smooth category. The distinctions between these definitions have a bearing on the validity of the Robinson theorem on the existence of null Maxwell fields associated with SNGs.This paper is largely a review intended to recall the major influence that Ivor Robinson exerted on the development of this subject.