HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

Abstract

We study hypersurfaces either in the De Sitter space $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or in the anti De Sitter space $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$ whose position vector $\psi$ satisfies the condition $L_k \psi = A \psi + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface, for a fixed $k = 0,\dots,n-1$, $A$ is an $(n+2) \times (n+2)$ constant matrix and $b$ is a constant vector in the corresponding pseudo-Euclidean space. For every $k$, we prove that when $A$ is self-adjoint and $b = 0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of standard pseudo-Riemannian products in $\mathbb{S}_1^{n+1}$ ($\mathbb{S}_1^m(r) \times \mathbb{S}^{n-m}(\sqrt{1-r^2})$, $\mathbb{H}^m(-r) \times \mathbb{S}^{n-m}$ $(\sqrt{1+r^2})$, $\mathbb{S}_1^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r)$, $\mathbb{H}^m(-\sqrt{r^2-1}) \times \mathbb{S}^{n-m}(r)$), open pieces of standard pseudo-Riemannian products in $\mathbb{H}_1^{n+1}$ ($\mathbb{H}_1^m(-r) \times \mathbb{S}^{n-m}(\sqrt{r^2-1})$, $\mathbb{H}^m(-\sqrt{1+r^2}) \times \mathbb{S}_1^{n-m}(r)$, $\mathbb{S}_1^m(\sqrt{r^2-1}) \times \mathbb{H}^{n-m}(-r)$, $\mathbb{H}^m(-\sqrt{1-r^2}) \times \mathbb{H}^{n-m}(-r)$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}_{c}^{n+1} \;|\; = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $t^2+at+b$, $a^2-4b \leq 0$, and $\mathbb{M}_{c}^{n+1}$ stands for $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$. When the $k$-th mean curvature is constant and $b$ is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2).