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 ) → Q n+2 1 (c) immersed into one of the two real standard non-flat Lorentzian space-forms and satisfying the equation Lrx = U x + b for some field of screen constant matrices U and some field of screen constant vectors b ∈ R n+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 = U x + b for 1 ≤ r ≤ n and the normalization N is quasiconformal, 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 non-flat space form Q n+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 Q n+2 t (c) by a hyperplane of R n+3 . In particular a null hypersurface M → Q n+2 t (c) is totally geodesic if and only if M is a section of Q n+2 t (c) by a hyperplane of R n+3 passing through the origin.