On the geometry of the $B$-connection on quasi-K\"ahler manifolds with Norden metric

Abstract

The $B$-connection on almost complex manifolds with Norden metric is an analogue of the first canonical connection of Lihnerovich in Hermitian geometry. In the present paper it is considered a $B$-connection in the class of the quasi-K\"ahler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be K\"ahlerian. Curvature properties for this connection are obtained. Conditions are given for the considered manifolds to be isotropic-K\"ahler.