Surfaces in Lorentzian space forms satisfying the condition Δx = Ax + B Luis J. Al´ ıas, Angel Ferr´ andez and Pascual Lucas Geometry and Topology of Submanifolds, vol. VI, pp. 3–15, 1993, World. Sci. Publ. Co. (Partially supported by DGICYT grant PB91-0705) 1. Introduction In [2] the authors have obtained a classification of surfaces in the 3-dimensional Lorentz- Minkowski space L 3 satisfying the condition Δx = Ax + B, where x stands for the isometric immersion, A is an endomorphism of L 3 and B is a constant vector. That condition was originally introduced by Dillen, Pas and Verstraelen in [5] for surfaces in the 3-dimensional Euclidean space and it has been studied by several authors for hypersurfaces in Riemannian space forms, [4], [6] and [8], who have obtained some interesting classification theorems. It should be noticed that those results obtained in the Riemannian cases strongly depend on the diagonalizability of the shape operator. However, a surface in a Lorentzian space can be endowed with a Riemannian or Lorentzian metric, and in the last case its shape operator does not need to be diagonalizable. Therefore, it is worth bringing that condition to the non-flat Lorentzian space forms, that is, the De Sitter space S 3 1 ⊂ R 4 1 and anti De Sitter space H 3 1 ⊂ R 4 2 , and it seems natural to hope for finding new classes of examples having no Riemannian counterpart. Moreover, in this new situation the codimension of the surface in the corresponding pseudo-Euclidean space is two and the proofs given in [2] do not work here, even so we follow the techniques developed there. In this paper we are going to classify the surfaces in S 3 1 and H 3 1 with isometric immersion x satisfying the condition Δx = Ax + B, where A is an endomorphism of the corresponding 4-dimensional pseudo-Euclidean space and B is a constant vector. The classification is given by showing that the asked condition is a constant mean curvature condition and, under non-minimality hypothesis, it yields a flat surface with parallel second fundamental form in the pseudo-Euclidean space. We point out that in contrast to the case of surfaces in L 3 , examples of surfaces in H 3 1 satis- fying that condition and having non-diagonalizable shape operator can be found (see Examples 5.1 and 5.2). 2. Preliminaries Let us denote by M 3 1 (c) the standard model of a 3-dimensional Lorentz space with constant cur- vature c =1, −1, say the De Sitter space S 3 1 = {x ∈ R 4 1 : 〈x, x〉 =1} and the anti De Sitter space H 3 1 = {x ∈ R 4 2 : 〈x, x〉 = −1}, respectively, 〈, 〉 standing for the indefinite inner product in the corresponding pseudo-Euclidean space R 4 q , q =1, 2, where M 3 1 (c) is lying. 1