Normalized null hypersurfaces in nonflat Lorentzian space forms satisfying Lrx = Ux + b

Abstract

In the present work, we classify normalized null hypersurfaces x : (M, g,N) → Qn+2 1 (c) immersed into one of the two real standard nonflat Lorentzian space-forms and satisfying the equation Lrx = Ux + b for some field of screen constant matrices U and some field of screen constant vectors b ∈ Rn+2 , where Lr is the linearized operator of the (r + 1)−mean curvature of the normalized null hypersurface for r = 0, ..., n. We show that if the immersion x is a solution of the equation Lrx = Ux + b for 1 ≤ r ≤ n and the normalization N is quasi-conformal, then M is either an (r +1)−maximal null hypersurface, or a totally umbilical (or geodesic) null hypersurface or an almost isoparametric normalized null hypersurface with at most two non-zero principal curvatures. We also show that a null hypersurface M, of a real standard semi-Riemannian nonflat space form Qn+2 t (c) , admits a totally umbilical screen distribution (and then M is totally umbilical or totally geodesic) if and only if M is a section of Qn+2 t (c) by a hyperplane of Rn+3 . In particular a null hypersurface M → Qn+2 t (c) is totally geodesic if and only if M is a section of Qn+2 t (c) by a hyperplane of Rn+3 passing through the origin.