In the Riemannian setting, the theory of Einstein solvmanifolds is well established and extensively understood. In this seminar, we will initially review the Riemannian case, providing a foundation for our subsequent exploration of the indefinite setting.

Then, we will switch to indefinite metrics, highlighting with examples the differences with the Riemannian case. However, we will describe analogous techniques for constructing pseudo-Riemannian Einstein solvmanifolds using indefinite nilsolitons, emphasizing the role of pseudo-Iwasawa decomposition and nice nilpotent Lie algebras.

Next, we will introduce two distinct methods for constructing Einstein solvmanifolds, which are not based on nilsolitons and are unique to the indefinite setting. The first approach involves contact symplectic reduction, a technique characteristic of pseudo-Sasaki geometry. The second, a more recent development, is based on solving the generalized nilsoliton equation and presents a new methodological framework. Both methods produce examples of Einstein solvmanifolds that are not isometric to any of the pseudo-Iwasawa type.

Time permitting, we will also discuss the interplay between Einstein solvmanifolds and related geometric structures, such as Sasaki, pseudo-Kähler, and para-Kähler geometries.

This presentation is based on joint work with D. Conti and R. Segnan Dalmasso.