Robinson manifolds and Cauchy-Riemann spaces

A Robinson manifold is defined as a Lorentz manifold (M, g) of dimension 2n greater than or equal to 4 with a bundle N subset of C x TM such that the fibres of N are maximal totally null and there holds the integrability condition [SecN, Sec N] C SecN. The real part of N boolean AND 9 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.