Asian Journal of Mathematics

Volume 24 (2020)

Number 1

Vaisman solvmanifolds and relations with other geometric structures

Pages: 117 – 146

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n1.a5

Authors

A. Andrada (FaMAF-CIEM, Ciudad Universitaria, Universidad Nacional de Córdoba, Argentina)

M. Origlia (KU Leuven Kulak, Kortrijk, Belgium; and FaMAF-CIEM, Universidad Nacional de Córdoba, Argentina)

Abstract

We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler 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, co‑Kähler 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.

Keywords

locally conformally Kähler structure, Vaisman structure, solvable Lie group, lattice, solvmanifold

2010 Mathematics Subject Classification

22E25, 22E40, 53B35, 53C25

Full Text (PDF format)

This work was partially supported by CONICET, SECyTUNC and ANPCyT (Argentina) and the Research Foundation Flanders (Project G.0F93.17N).

Received 4 October 2017

Accepted 7 May 2019

Published 21 August 2020