Natural Connections with Totally Skew-Symmetric Torsion on Manifolds with Almost Contact 3-Structure and Metrics of Hermitian-Norden Type

TL;DR

This paper investigates manifolds with almost contact 3-structures and metrics of Hermitian-Norden type, establishing conditions for the existence of special connections with totally skew-symmetric torsion that preserve these structures.

Contribution

It introduces associated Nijenhuis tensors for these structures and characterizes their vanishing as equivalent to the existence of compatible connections with skew-symmetric torsion.

Findings

Vanishing of Nijenhuis tensors characterizes the existence of special connections.

Provides an explicit example of a 7-dimensional manifold with such connections.

Establishes geometric interpretation of torsion-preserving conditions.

Abstract

It is considered a differentiable manifold equipped with a pseudo-Riemannian metric and an almost contact 3-struc\-ture so that an almost contact metric structure and two almost contact B-metric structures are generated. There are introduced the so-called associated Nijenhuis tensors for the studied structures. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of linear connections with totally skew-symmetric torsions preserving the structure. An example of a 7-dimensional manifold with connections of the considered type is given.