Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

THE CURVATURE HOMOGENEITY BOUND FOR LORENTZIAN FOUR-MANIFOLDS

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH3for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, CH2manifolds that are not homogeneous. The resulting metrics belong to the class of null electromagnetic radiation, type N solutions on an anti-de Sitter background. These findings prove that the four-dimensional Lorentzian Singer number k1,3= 3, falsifying some recent conjectures [1]. We also prove that invariant classification for these proper CH2solutions requires ∇(7)R, and that these are the unique metrics requiring the seventh order.