Metric connections with parallel twistor-free torsion

TL;DR

This paper classifies certain Riemannian manifolds with metric connections that have parallel torsion, specifically those with a non-zero vectorial component and no twistorial component, expanding understanding of torsion structures.

Contribution

It provides a classification of complete simply connected Riemannian manifolds with parallel torsion having a vectorial component and no twistorial component.

Findings

Classification of manifolds with specified torsion properties

Identification of conditions for parallel torsion with vectorial component

Insights into the structure of metric connections with particular torsion types

Abstract

The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, one of vectorial type, and one of twistorial type. In this paper we classify complete simply connected Riemannian manifolds carrying a metric connection whose torsion is parallel, has non-zero vectorial component and vanishing twistorial component.