Vaisman solvmanifolds and relations with other geometric structures

TL;DR

This paper characterizes unimodular solvable Lie algebras with Vaisman structures, explores their relations with other geometric structures, and constructs new examples of solvmanifolds with invariant Vaisman structures.

Contribution

It provides a new algebraic characterization of Vaisman structures on solvable Lie algebras and constructs explicit examples of solvmanifolds with these structures.

Findings

Algebraic restrictions for Vaisman structures identified

Relations established with Sasakian, coK"ahler, and left-symmetric structures

New families of solvmanifolds with invariant Vaisman structures constructed

Abstract

We characterize unimodular solvable Lie algebras with Vaisman structures in terms of K\"ahler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sasakian, coK\"ahler and left-symmetric algebra structures. Applying these results we construct families of Lie algebras and Lie groups admitting a Vaisman structure and we show the existence of lattices in some of these families, obtaining in this way many examples of new solvmanifolds equipped with invariant Vaisman structures.