A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

TL;DR

This paper explores a special linear connection on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics, studying their curvature, torsion, and conformal properties, and providing a concrete 4D example.

Contribution

It introduces a new linear connection that preserves the manifold's structure and analyzes curvature and torsion properties, including a specific class of conformally related manifolds.

Findings

Identification of a linear connection preserving the structure

Analysis of curvature properties of these manifolds

Construction of a 4-dimensional example

Abstract

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.