Almost contact B-metric manifolds with curvature tensors of K\"ahler type

TL;DR

This paper characterizes the form of K"ahler-type curvature tensors on 5-dimensional almost contact B-metric manifolds, linking associated 1-forms to scalar curvatures for specific classes with closed 1-forms.

Contribution

It determines the structure of K"ahler-type tensors on these manifolds and relates associated 1-forms to scalar curvatures in main classes with closed 1-forms.

Findings

Form of K"ahler-type tensors explicitly determined

Associated 1-forms expressed via scalar curvatures

Results apply to main classes with closed 1-forms

Abstract

On 5-dimensional almost contact B-metric manifolds, the form of any K\"ahler-type tensor (i.e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvatures of the K\"ahler-type tensor for the canonical connection on the manifolds from the main classes with closed 1-forms.