Four-dimensional Einstein-like manifolds and curvature homogeneity
type: Item
instance of: scholarly article
title: Four-dimensional Einstein-like manifolds and curvature homogeneity
zbMATH Open document ID: 0835.53056
DOI: 10.1007/BF01265339
author: Andrea Spiro
author: Fabio Podestà
published in: Geometriae Dedicata
publication date: 1995-11-21T00:00:00Z
review text: A Riemannian manifold \((M,g)\) is said to be curvature homogeneous if for any two points \(p, q \in M\), there exists a linear isometry \(F : T_p M \to T_q M\) such that \(F^* R = R_p\), where \(R\) denotes the Riemannian curvature tensor. In dimension two and three this is equivalent to the constancy of the eigenvalues of the Ricci operator \(Q\). Three-dimensional curvature homogeneous spaces have been studied in detail by \textit{O. Kowalski} and \textit{F. Prüfer} [see J. Anal. Anwend. 14, 43-58 (1995; Zbl 0821.53036) for a short survey]. In dimension four a result of Derdzinkski yields that a curvature homogeneous Einstein space is locally symmetric. The main purpose of the authors is to consider four-dimensional spaces \((M,g)\) which have constant Ricci eigenvalues (a condition which is satisfied in the curvature homogeneous case) and such that the Ricci tensor \(\rho\) of type (0,2) is a Codazzi tensor or is cyclic-parallel. For the first class they prove that when \((M,g)\) is not an Einstein space then it is locally symmetric. Together with Derdzinski's result, this implies that any four-dimensional curvature homogeneous manifold with Codazzi tensor \(\rho\) is locally symmetric. Further, they also consider the second class and provide a classification when not all the Ricci eigenvalues are distinct. This restriction is due to the complexity of the system of equations involved in the determination of these manifolds by means of their method of attack.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 806208
zbMATH Keywords: curvature homogeneous Riemannian manifold
zbMATH Keywords: eigenvalues of the Ricci operator
zbMATH Keywords: locally symmetric
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
cites work: Einstein-like manifolds which are not Einstein
cites work: Homogeneous Einstein spaces of dimension four
cites work: Q3767975
cites work: The fundamental equations of a submersion
cites work: Curvature homogeneous riemannian manifolds
cites work: A characterization of locally homogeneous Riemann manifolds of dimension 3
full work available at URL: https://doi.org/10.1007/bf01265339
OpenAlex ID: W2079313003
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description: scientific article; zbMATH DE number 806208