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

TL;DR

This paper investigates the properties of the $B$-connection on quasi-K"ahler manifolds with Norden metric, focusing on curvature conditions and criteria for isotropic-K"ahler structures.

Contribution

It provides necessary and sufficient conditions for the curvature tensor to be K"ahlerian and explores curvature properties and isotropic-K"ahler conditions for these manifolds.

Findings

Derived conditions for the curvature tensor to be K"ahlerian.

Established curvature properties specific to the $B$-connection.

Identified criteria for manifolds to be isotropic-K"ahler.

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.