Timelike Hypersurfaces in the Standard Lorentzian Space Forms Satisfying Lkx = Ax + b

In this paper, we study connected orientable timelike hypersurfaces isometrically immersed by x:M1n→M~1n+1(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x : M_1^{n} \rightarrow {\tilde{M}}_1^{n+1}(c)}$$\end{document} into the de Sitter space S1n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{S}_1^{n+1}}$$\end{document} (when c = 1) or anti-de Sitter space H1n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{H}_1^{n+1}}$$\end{document} (when c = −1) satisfying the condition Lkx = Ax + b, where the second order differential operator Lk 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.