Shearfree Lorentzian Geometry and CR Geometry

Abstract

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call FRT-metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of 3-dimensional CR-structures in terms of the Ricci curvature of the FRT-metrics in the spirit of the results by Lewandowski et al. in [37] and also Hill et al. in [25]. We also study higher dimensional versions of shearfree null congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure.