On Riemannian 3-manifolds with distinct constant Ricci eigenvalues
A 3-dimensional Riemannian manifold \((M,g)\) is curvature homogeneous if and only if the eigenvalues \(\rho_ i\), \(i=1,2,3\), of the Ricci endomorphism are constant. The classification of this class of spaces has been treated by the first author for the case \(\rho_ 1= \rho_ 2 \neq\rho_ 3\) and a lot of examples are given. For the case of different Ricci eigenvalues, Yamato provided already some explicit complete examples on \(\mathbb{R}^ 3\) and the first author gave some modified noncomplete examples. Moreover, Spiro and Tricerri proved the existence of such spaces for each triplet of prescribed distinct constant Ricci eigenvalues by using the technique of formally integrable differentiable systems but they did not provide explicit solutions. In the present paper the authors also prove this existence but now by providing explicit examples for every choice of \(\rho_ 1> \rho_ 2> \rho_ 3\).
scientific article; zbMATH DE number 683130