LEFT-INVARIANT METRIC f-STRUCTURES ON THREE-DIMENSIONAL SOLVABLE LIE GROUPS

Abstract

In the paper, we investigate three-dimensional solvable Lie groups from the point of view of the generalized Hermitian geometry. The corresponding three-dimensional solvable Lie algebras were firstly classified by G. M. Mubarakzyanov in 1963. Using the classification in somewhat different notations, we construct basic left-invariant metric f-structures of rank 2 on all three-dimensional solvable Lie groups equipped with the standard left-invariant Riemannian metric. It was proved that all the considered f-structures belong to one or several classes of generalized almost Hermitian structures. As a result, it gives the opportunity to present new examples of left-invariant Killling, nearly Kähler, generalized nearly Kähler and Hermitian f-structures on solvable Lie groups.