This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.

Ricci-flat and Einstein pseudoriemannian nilmanifolds

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property.

Negative eigenvalues of the Ricci operator of solvable metric Lie algebras

We study the Ricci curvature of completely solvablemetric Lie algebras. In particular,we prove that the Ricci operator of every completely solvable nonunimodular or every noncommutative nilpotent metric Lie algebra has at least two negative eigenvalues.

The Ricci operator of completely solvable metric lie algebras

We prove that the Ricci operator of any nonunimoular solvable metric Lie algebra having a two-step nilpotent derived Lie algebra of dimension 6 has at least two negative eigenvalues.

On the Ricci curvature of solvable metric lie algebras with two-step nilpotent derived algebras

The Ricci curvature of solvable metric Lie algebras is studied. In particular, we prove that the Ricci operator of any metric nonunimodular solvable Lie algebra of dimension not exceeding 6 has at least two negative eigenvalues, that generalizes the known results.

M. S. Chebarykov

Papers

On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension