On curvature homogeneous three-dimensional Lorentzian manifolds
type: Item
instance of: scholarly article
title: On curvature homogeneous three-dimensional Lorentzian manifolds
zbMATH Open document ID: 0881.53055
DOI: 10.1016/S0393-0440(96)00037-X
author: Peter Bueken
published in: Journal of Geometry and Physics
publication date: 1997-07-17T00:00:00Z
review text: A pseudo-Riemannian manifold \((M,g)\) is said to be curvature homogeneous if, for every \(p,q\in M\), there exists a linear isometry \(F:T_pM \to T_qM\) such that \(F^*R_q = R_p\) where \(R\) denotes the Riemannian curvature tensor. All locally homogeneous spaces \((M,g)\) are trivial examples. Moreover, a curvature homogeneous space \((M,g)\) is said to have the same curvature tensor as a homogeneous space \((\overline M, \overline g)\) if, for any pair of points \(m\in M\) and \(\overline m\in\overline M\), there exists a linear isometry \(\varphi: T_m M \to T_{\overline m} \overline M\) such that \(\varphi^* \overline R_{\overline m} =R_m\). In this case, \((\overline M, \overline g)\) is said to be a (homogeneous) model space for \((M,g)\). Curvature homogeneous Riemannian spaces have been studied in detail by several people, and a lot of examples, with or without a homogeneous model, have been discovered. Note that, in dimension three, \((M,g)\) is curvature homogeneous if and only if the Ricci operator has constant eigenvalues. We refer to [\textit{E. Boeckx}, \textit{O. Kowalski} and \textit{L. Vanhecke}, `Riemannian manifolds of conullity two' (World Scientific, River Edge, NJ) (1996)] for a survey about the results and for further references. A similar study has already been started for pseudo-Riemannian manifolds, in particular for Lorentz spaces. It turns out that there are remarkable differences mainly due to the fact that the Ricci operator is not always diagonalizable. In another paper, the author studied in detail the diagonalizable case for three-dimensional Lorentz spaces when the constant eigenvalues satisfy \(\rho_1= \rho_2\neq \rho_3\) and determined a complete local classification. In this paper he continues this study and treats the case when the non-diagonalizable Ricci operator has three equal constant eigenvalues whose associated eigenspace is two-dimensional. He proves the existence of such metrics and studies when they are locally isometr…
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1036752
zbMATH Keywords: curvature homogeneous Lorentz 3-manifolds
zbMATH Keywords: classification up to local isometry
zbMATH Keywords: non-diagonalizable Ricci operator
zbMATH Keywords: homogeneous model
MaRDI profile type: Publication
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Wikidata QID: Q126781275
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description: scientific article; zbMATH DE number 1036752