Homogeneous quaternionic Kähler structures on Alekseevskian \(\mathcal{W}\)-spaces (Q2880820)

A quaternionic Kähler manifold which admits a simply transitive and real solvable group of isometries is called Alekseevskian space. It is well known that an Alekseevskian space is symmetric if and only if its sectional curvature is non-positive. In the paper of \textit{V. Cortés} [Differ. Geom. Appl. 6, No. 2, 129--168 (1996; Zbl 0846.53030)], the rank of an Alekseevskian space is defined as the rank of its Alekseevskian Lie algebra and three difference cases are distinguished according to the type of the Lie algebra and among those, the type 1 class has the rank 4 Alekseevskian space, \({W(p,q)}\)- and \({V}\)-spaces.NEWLINENEWLINE Moreover, homogeneous quaternionic structures were classified by \textit{A. Fino} [Math. J. Toyama Univ. 21, 1--22 (1998; Zbl 0980.53060)] into five basic geometric types \({QK}_{1},\dots,{QK}_{5}\). In this paper, the authors provide the expressions of the homogeneous quaternionic Kähler structures on \({W(p,q)}\) and also show that it has a non zero component in each basic Fino Type (Theorem 4.1).