A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

TL;DR

This paper studies a special class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics, introducing a linear connection with totally skew-symmetric torsion and exploring its properties and curvature characteristics.

Contribution

It introduces a new linear connection with parallel structure and totally skew-symmetric torsion on these manifolds, analyzing its properties and curvature implications.

Findings

The connection has a D-parallel torsion.

The connection is weak if not flat.

Curvature properties of these manifolds are characterized.

Abstract

The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied.