Intrinsic geometry of oriented congruences in three dimensions

Starting from the classical notion of an oriented congruence (i.e. a foliation by oriented curves) in R3R3, we abstract the notion of an oriented congruence structure  . This is a 3-dimensional CR manifold (M,H,J)(M,H,J) with a preferred splitting of the tangent space TM=V⊕HTM=V⊕H. We find all local invariants of such structures using Cartan’s equivalence method refining Cartan’s classification of 3-dimensional CR structures. We use these invariants and perform Fefferman like constructions, to obtain interesting Lorentzian metrics in four dimensions, which include explicit Ricci-flat and Einstein metrics, as well as not conformally Einstein Bach-flat metrics.