Classification of Riemannian 3-manifolds with distinct constant principal Ricci curvatures
type: Item
instance of: scholarly article
title: Classification of Riemannian 3-manifolds with distinct constant principal Ricci curvatures
zbMATH Open document ID: 0942.53025
author: Zdeněk Vlášek
author: Oldřich Kowalski
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1999-04-06T00:00:00Z
review text: A Riemannian manifold \((M,g)\) is said to be curvature homogeneous if its curvature tensor is ``the same'' at every point, that is, for every pair of points \(p,q \in M\), there exists a linear isometry \(\varphi : T_pM \to T_qM\) such that \(\varphi* R_q = R_p\). For three-dimensional Riemannian manifolds it can be shown that curvature homogeneity is equivalent to Ricci curvature homogeneity, that is, the constancy of the three principal Ricci curvatures (the eigenvalues \(\varrho_1,\varrho_2,\varrho_3\) of the Ricci operator). It is easily seen that all locally homogeneous Riemannian manifolds are curvature homogeneous, but many examples are known of manifolds which are curvature homogeneous without being locally homogeneous. It is therefore an interesting problem to construct a complete classification (up to local isometries) of all curvature homogeneous manifolds which are not locally homogeneous. In the three-dimensional case, this amounts to the classification of Riemannian manifolds with constant principal Ricci curvatures. If all three eigenvalues of the Ricci operator are equal (and constant), the manifold has constant sectional curvature and hence it is locally homogeneous. If the manifold admits two equal eigenvalues (\(\varrho_1 = \varrho_2 \neq \varrho_3\)), the isometry classes of such manifolds turn out to depend on two arbitrary functions of one variable. The present paper completely solves the classification problem for the case of three distinct (constant) principal Ricci curvatures (\(\varrho_1\neq \varrho_2 \neq \varrho_3 \neq \varrho_1\)). It had already been shown in [\textit{A. Spiro} and \textit{F. Tricerri}, J. Math. Pures Appl., IX. Sér. 74, No. 3, 253-271 (1995; Zbl 0851.53022)] that, in this case, the local isometry classes depend on an infinite number of parameters. The main result of the present paper states that the isometry classes of germs of three-dimensional (real analytic) Riemannian metrics with distinct constant principal Ricci…
reviewed by: Peter Bueken
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 1266834
zbMATH Keywords: curvature homogeneous manifold
zbMATH Keywords: constant principal Ricci curvatures
zbMATH Keywords: nonhomogeneous manifold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1266834