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

Abstract

In the present work, we classify normalized null hypersurfaces $x:(M,g,N)\to Q^{n+2}_1(c)$ immersed into one of the two real standard nonflat Lorentzian space-forms and satisfying the equation $L_r x=\mathcal U x+b$ for some field of screen constant matrices $\mathcal U$ and some field of screen constant vectors $b\in\mathbb{R}^{n+2}$, where $L_r$ 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 $L_r x=\mathcal U x+b$ for $1\leq r\leq n$ 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 $Q_t^{n+2}(c)$, admits a totally umbilical screen distribution (and then $M$ is totally umbilical or totally geodesic) if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$. In particular a null hypersurface $M\to Q_t^{n+2}(c)$ is totally geodesic if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$ passing through the origin.