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

TL;DR

This paper classifies 3-dimensional CR manifolds embedded in 5-dimensional hyperquadrics using Cartan's moving frames, revealing connections to relativity and identifying homogeneous CR 3-folds within these quadrics.

Contribution

It applies Cartan's method to classify CR-embedded 3-folds in hyperquadrics and explores their relation to shear-free null geodesic congruences and Lorentzian geometry.

Findings

Classification of CR 3-folds in hyperquadrics

Identification of homogeneous CR 3-folds

Connection to shear-free null geodesic congruences

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 $Q$.