Three-dimensional Lorentz metrics and curvature homogeneity of order one

In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k is an element of N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.