THE CURVATURE HOMOGENEITY BOUND FOR LORENTZIAN FOUR-MANIFOLDS

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH3 for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, CH2 manifolds that are not homogeneous. The resulting metrics belong to the class of null electromagnetic radiation, type N solutions on an antide Sitter background. These. findings prove that the four-dimensional Lorentzian Singer number k(1,3) = 3, falsifying some recent conjectures [1]. We also prove that invariant classification for these proper CH2 solutions requires del((7)) R, and that these are the unique metrics requiring the seventh order.