Timelike Hypersurfaces in the Standard Lorentzian Space Forms Satisfying L k x = Ax + b

In this paper, we study connected orientable timelike hypersurfaces isometrically immersed by ${x : M_1^{n} rightarrow {tilde{M}}_1^{n+1}(c)}$ into the de Sitter space ${mathbb{S}_1^{n+1}}$ (when c = 1) or anti-de Sitter space ${mathbb{H}_1^{n+1}}$ (when c = ?1) satisfying the condition L _k x = Ax + b, where the second order differential operator L _k is the linearized operator associated with the first normal variation of the (k + 1)-th mean curvature of M for an integer k, 0 ≤ k < n, A is a matrix and b is a vector. We characterize these hypersurfaces when b = 0 or when the k-th mean curvature of M is constant.