The curvature homogeneity bound for Lorentzian four-manifolds

Details The curvature homogeneity bound for Lorentzian four-manifolds The curvature homogeneity bound for Lorentzian four-manifolds

2007-11-24

arxiv.org/abs/0711.3851v3

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

24 pages, streamlined version

Scientific paper

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.

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