0 10 90 08 v 1 4 S ep 2 00 1 On the Space of 3 – dimensional Homogeneous Riemannian

We answer the following question: Let λ, μ, ν be arbitrary real numbers. Does there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues of the Ricci tensor are just λ, μ and ν ? It is essential to observe that inspite of the fact that the curvature of a 3–manifold is uniquely determined by the Ricci tensor, nevertheless, the constancy of all its three eigenvalues does not imply the manifold to be locally homogeneous, see Ref. 1 for examples. The answer to the question posed in the abstract does not alter if we replace the word “homogeneous” by the notion “locally homogeneous”. In principle, it could have been answered since decades already. But the answer seems not to be published until this Conference. And, accidentally, the answer was found independently by Kowalski/Nikcevic, Ref. 2, on the one hand, and by Rainer/Schmidt, Ref. 3, on the other hand. ∗1991 Mathematics Subject Classification 53 B20.