Conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space

Abstract

We study conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space R 1 n + 1 , which is the projectivized light cone R ̂ 1 n + 1 ⊂ R P n + 2 induced from R 2 n + 3 . We establish a Lorentzian version of the local classification theorem of Cartan, in terms of branched channel hypersurfaces for n ≥ 4 , and for n = 3 , in terms of the conformal fundamental forms. For hypersurfaces whose shape operator has complex eigenvalues, we give a necessary condition for being conformally flat in terms of local integrability of distributions.