The curvature homogeneity bound for Lorentzian four-manifolds

Abstract

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. We also prove that invariant classification for these proper CH_2 solutions requires ${\nabla }^{(7)}R$, and that these are the unique metrics requiring the seventh order.

arXiv comment24 pages, streamlined version

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Cited by in corpus (2)

• Spacetimes characterized by their scalar curvature invariants
• Lorentzian spacetimes with constant curvature invariants in four dimensions