Metric connections with parallel skew-symmetric torsion

TL;DR

This paper investigates the structure of Riemannian manifolds with a metric connection that has parallel skew-symmetric torsion, providing a local classification and constructing new examples within this geometric framework.

Contribution

It offers a local structural analysis of manifolds with parallel skew-symmetric torsion and classifies cases where the submersion forms a principal bundle, including new examples.

Findings

Natural splitting of tangent bundle leads to Riemannian submersions.

Existing examples fit into the proposed structural pattern.

Complete local classification for principal bundle cases.

Abstract

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly K\"ahler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-K\"ahler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.