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

We study hypersurfaces either in the pseudo-Riemannian De Sitter space S-t(n+1) subset of R-t(n+2) or in the pseudo-Riemannian anti De Sitter space H-t(n+1) subset of 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,.., n-1, A is an (m+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 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 S-t(n+1) (S-t-1(n) (r), r > 1; S-t(n) (r), 0 < r < 1; H-t-1(n) (-r), r > 0; R-t-1(n)), open pieces of a totally umbilical hypersurface in H-t(n+1) (H-t(n) (-r), r > 1; H-t-1(n) (-r), 0 < r < 1; S-t(n)(r), r > 0; R-t(n)), open pieces of a standard pseudo-Riemannian product in S-t(n+1) (S-u(m) (r) x S-v(n-m) (root 1-r(2)), H-u-1(m) (-r) x S-v(n-m) (root 1+r(2)), S-u(m)(r) x H-v-1(n-m) (-root r(2)-1)), open pieces of a standard pseudo-Riemannian product in H-t(n+1) (H-u(m) (-r) x S-v(n-m)(root r(2)-1), S-u(m) (r) x H-v(n-m) (-root 1+r(2)), H-u(m) (-r) x H-v-1(n-m) (-root 1-r(2))) and open pieces of a quadratic hypersurface {x is an element of M-t(n+1) (c) vertical bar < Rx, x > = d}, where R is a self-adjoint constant matrix whose minimal polynomial is mu(R)(z) = z(2) + az + b, a(2)-4b <= 0, and M-t(n+1) (c) stands for S-t(n+1) subset of R-t(n+2) or H-t(n+1) subset of R-t+1(n+2).