On three-parametric Lie groups as quasi-Kähler manifolds with Killing Norden metric

It is a fundamental fact that on an almost complex manifold with Hermitian metric (almost Hermitian manifold), the action of the almost complex structure on the tangent space at each point of the manifold is isometry. There is another kind of metric, called a Norden metric or a B-metric on an almost complex manifold, such that the action of the almost complex structure is anti-isometry with respect to the metric. Such a manifold is called an almost complex manifold with Norden metric or with B-metric. See also Ref. 5 for generalized B-manifolds. It is known that these manifolds are classified into eight classes. The purpose of the present paper is to exhibit, by construction, almost complex structures with Norden metric on Lie groups as 6-manifolds, which are of a certain class, called quasi-Kähler manifold with Norden metric. It is proved that the constructed 6-manifold is isotropic Kählerian if and only if it is scalar flat or it has zero holomorphic sectional curvatures.