On a conformal Lorentzian 4-manifold, there are certain foliations of null geodesics, known as shearfree congruences of null geodesics (SCNG), which are of central importance in the study of solutions of Einstein's equations. It is well-known that their generators must be principal null directions of the Weyl tensor. What is more, their leaf space is endowed with the structure of a CR manifold. In this talk I will give the integrability condition for the existence of SCNGs in dimension greater than four, and show that remarkably, in even dimension, the connection between SCNGs and (almost) CR structures subsist under relatively mild curvature conditions on the Weyl tensor. CR invariants can then easily be read off from the Weyl tensor. It turns out that under an additional curvature condition, Einstein's equations reduce to purely CR data. This generalises a construction of Einstein metrics due to Lewandowski and Nurowski in dimension four.
Speaker: Dr. Arman Taghavi-Chabert