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We study hypersurfaces either in the De Sitter space$\S_1^{n+1}\subset\R_1^{n+2}$ or in the anti De Sitter space$\H_1^{n+1}\subset\R_2^{n+2}$ whose position vector $\psi$ satisfies thecondition $L_k\psi=A\psi+b$, where $L_k$ is the linearized operator of the$(k+1)$-th mean curvature of the hypersurface, for a fixed $k=0,...,n-1$, $A$is an $(n+2)\times(n+2)$ constant matrix and $b$ is a constant vector in thecorresponding pseudo-Euclidean space. For every $k$, we prove that when $A$ isself-adjoint and $b=0$, the only hypersurfaces satisfying that condition arehypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th meancurvature, open pieces of standard pseudo-Riemannian products in $\S_1^{n+1}$($\S_1^m(r)\times\S^{n-m}(\sqrt{1-r^2})$,$\H^m(-r)\times\S^{n-m}(\sqrt{1+r^2})$,$\S_1^m(\sqrt{1-r^2})\times\S^{n-m}(r)$,$\H^m(-\sqrt{r^2-1})\times\S^{n-m}(r)$), open pieces of standardpseudo-Riemannian products in $\H_1^{n+1}$($\H_1^m(-r)\times\S^{n-m}(\sqrt{r^2-1})$,$\H^m(-\sqrt{1+r^2})\times\S_1^{n-m}(r)$,$\S_1^m(\sqrt{r^2-1})\times\H^{n-m}(-r)$,$\H^m(-\sqrt{1-r^2})\times\H^{n-m}(-r)$) and open pieces of a quadratichypersurface $\{x\in\mathbb{M}_{c}^{n+1}\;|\;Rx,x=d\}$, where $R$ is aself-adjoint constant matrix whose minimal polynomial is $t^2+at+b$,$a^2-4b\leq 0$, and $\mathbb{M}_{c}^{n+1}$ stands for$\S_1^{n+1}\subset\R_1^{n+2}$ or $\H_1^{n+1}\subset\R_2^{n+2}$. When $H_k$ isconstant and $b$ is a non-zero constant vector, we show that the hypersurfaceis totally umbilical, and then we also obtain a classification result (seeTheorem 2).

Hypersurfaces in non-flat Lorentzian space forms satisfying $L_k\psi=A\psi+b$