The curvature homogeneity bound for Lorentzian four-manifolds
نویسندگان
چکیده
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 anti-de Sitter background. These findings prove that the four-dimensional Lorentzian Singer number k1,3 = 3, falsifying some recent conjectures[25]. We also prove that invariant classification for these proper CH2 solutions requires ∇(7)R, and that these are the unique metrics requiring the seventh order. PACS numbers: 04.20, 02.40 AMS classification scheme numbers: 53C50
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