HYPERSURFACES IN NON-FLAT PSEUDO-RIEMANNIAN SPACE FORMS SATISFYING A LINEAR CONDITION IN THE LINEARIZED OPERATOR OF A HIGHER ORDER MEAN CURVATURE

Abstract

We study hypersurfaces either in the pseudo-Riemannian De Sitter space $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ orin the pseudo-Riemannian anti De Sitter space $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$ 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 $H_k$ is constant, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of a totally umbilical hypersurface in $\mathbb{S}^{n+1}_t$ ($\mathbb{S}^n_{t-1}(r)$, $r \gt 1$; $\mathbb{S}^n_t(r)$, $0 \lt r \lt 1$; $\mathbb{H}^n_{t-1}(-r)$, $r \gt 0$; $\mathbb{R}^n_{t-1}$), open pieces of a totally umbilical hypersurface in $\mathbb{H}^{n+1}_t$ ($\mathbb{H}^n_t(-r)$, $r \gt 1$; $\mathbb{H}^n_{t-1}(-r)$, $0 \lt r \lt 1$; $\mathbb{S}^n_t(r)$, $r \gt 0$; $\mathbb{R}^n_t$), open pieces of a standard pseudo-Riemannian product in $\mathbb{S}_t^{n+1}$ ($\mathbb{S}_u^m(r) \times \mathbb{S}^{n-m}_v(\sqrt{1-r^2})$, $\mathbb{H}^m_{u-1}(-r) \times \mathbb{S}^{n-m}_v(\sqrt{1+r^2})$, $\mathbb{S}^m_u(r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{r^2-1})$), open pieces of a standard pseudo-Riemannian product in $\mathbb{H}_t^{n+1}$ ($\mathbb{H}_u^m(-r) \times \mathbb{S}^{n-m}_v(\sqrt{r^2-1})$, $\mathbb{S}_u^m(r) \times \mathbb{H}^{n-m}_v(-\sqrt{1+r^2})$, $\mathbb{H}^m_u(-r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{1-r^2})$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}^{n+1}_t(c) \mid \langle Rx,x \rangle = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $\mu_R(z) = z^2 + az + b$, $a^2 - 4b \leq 0$, and $\mathbb{M}^{n+1}_t(c)$ stands for $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ or $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$.