Classification of Riemannian 3-manifolds with distinct constant principal Ricci curvatures
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 curvatures are parametrized by three functions of two variables.
scientific article; zbMATH DE number 1266834