3-folds CR-embedded in 5-dimensional real hyperquadrics

Abstract

E. Cartan’s method of moving frames is applied to 3-dimensional manifolds M which are CR-embedded in 5-dimensional real hyperquadrics Q in order to classify M up to CR symmetries of Q given by the action of one of the Lie groups SU(3,1) or SU(2,2). In the latter case, the CR structure of M derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed. In both cases, we compute which homogeneous CR 3-folds appear in Q.