*(English)*Zbl 1186.53058

The only known examples of noncompact homogeneous Riemannian manifolds which admit invariant Einstein metric are simply connected solvable Lie groups with a left invariant Riemannian metric (solvmanifolds). There is a conjecture that these examples exhaust all the possibilities and the conjecture is true up to dimension 5.

The paper gives a detailed report on the present status of the study of Einstein solvmanifolds.

A generalization of Einstein metrics is the notion of Ricci soliton. If $S$ is an Einstein solvmanifolds, then the metric restricted to the submanifold $N=[S,S]$ is a Ricci soliton and conversely, any Ricci soliton left invariant metric on a nilpotent Lie group $N$ (a nilsoliton) can be extended to an Einstein solvmanifold. The problem of classification of Einstein solvmanifolds is equivalent to the classification of Ricci solitons on nilpotent Lie groups.