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

We study hypersurfaces either in the De Sitter space S-1(n+1) C R-1(n+2) or in the anti De Sitter space H-1(n+1) subset of 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, ... , n-1, A is an (n+2) x (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 S-1(n+1) (S-1(m)(r)xS(n-m)(root 1 - r(2)), H-m (-r) x Sn-m (root 1 + r(2)), S-1(m)(root 1 - r(2)) x Sn-m(root r(2) - 1) x Sn-m(r)), open pieces of standard pseudo-Riemannian products in H-1(n+1) (H-1(m) x Sn-m(root r(2)-1), H-m(-root 1+ r(2)) x S-1(n-m)(r), S-1(m)(root r(2) - 1) x Hn-m(-r), H-m(-root 1-r(2)) x Hn-m(-r) and open pieces of a quadratic hypersurface m {x is an element of M-c(n+1) < Rx, x > = d}, where R is a self-adjoint constant matrix whose minimal polynomial is t(2) + at + b, a(2) - 4b <= 0, and M-c(n+1) stands for S-1(n+1) subset of R-1(n+2) or H-1(n+1) subset of 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).