Aula Seminari, Department of Mathematics
A pseudo-Riemannian metric is said to be Einstein if its Ricci tensor is a constant multiple of the metric. Bi-invariant metrics on simple Lie groups are examples; more generally, there are several known constructions to obtain homogeneous Einstein metrics, i.e. invariant under the transitive action of a group.
A recent result of Böhm and Lafuente states that every homogeneous, Einstein Riemannian manifold with negative scalar curvature is isometric to a solvmanifold. The structure of Riemannian solvmanifolds is well known, thanks to the work of Lauret and Heber.
In the case of indefinite signature, this theory can be adapted to construct vast classes of examples, but more flexibility occurs. In particular, it is possible to construct nonflat, Ricci-flat homogeneous metrics, nonsymmetric bi-invariant metrics, and Einstein nilmanifolds with nonzero scalar curvature.
After introducing the general context, I will illustrate some constructive results for these types of metrics, obtained jointly with Federico A. Rossi and Viviana del Barco.