Einstein manifolds with optical geometries of Kerr type

We classify the Ricci flat Lorentzian n-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that if n > 4, there is no Lorentzian manifold satisfying the considered conditions, while for n = 4 there are two large classes of such manifolds. Each class consists of manifolds fibering over open Riemann surfaces, equipped with a metric of constant Gaussian curvature = 1 or = -1. The first class properly includes a three parameter family of metrics admitting real analytic extensions to (R³ \0\) R = (S² R_+) R (all others develop singularities) and all such extensions are isometric to the well-known Kerr metrics. The three parameters correspond to the three space-like components of the angular momentum of the gravitational field. The second class contains a subclass of metrics defined on (D R_+) R, where D is the Lobachevsky Poincar\'e disc. This subclass is in bijection with the holomorphic functions on D satisfying an appropriate open condition. These and other results are obtained as consequences of a very simple method of generating a number of explicit examples of Ricci flat Lorentzian manifolds.