Examples of curvature homogeneous Lorentz metrics

Abstract

Examples of a three- and a four-dimensional Lorentz manifold are presented which are curvature homogeneous up to order one, without being locally homogeneous, in contrast to the situation in the Riemannian case, where a curvature homogeneity up to order one implies local homogeneity in the three- and four-dimensional cases. It is further shown that these manifolds satisfy the property that all scalar curvature invariants vanish identically, i.e. are those of a flat Lorentz manifold. As an immediate consequence, we also obtain examples of Lorentz manifolds whose curvature invariants are all constant, but which are not locally homogeneous, again in contrast to the Riemannian case where such manifolds are always locally homogeneous.