The curvature homogeneity bound for Lorentzian four-manifolds

Understand

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH_3 for short, is necessarily locally homogeneous.

• We also exhibit and classify four-dimensional Lorentzian, CH_2 manifolds 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 $k_{1,3}=3$, falsifying some recent conjectures by Gilkey.