Submersion constructions for geometries with parallel skew torsion

TL;DR

This paper develops a framework to analyze geometries with parallel skew torsion using submersion techniques, extending classifications and revealing structure in related geometric manifolds.

Contribution

It introduces a unified approach to view geometries with parallel skew torsion as Riemannian submersions, extending existing classifications and characterizing special cases.

Findings

Extended the classification of irreducible geometries with skew torsion.

Characterized cases with larger stabilizer of torsion.

Derived structure results for Gray, nearly parallel G2, and Sasaki manifolds.

Abstract

In the absence of a de Rham decomposition theorem for geometries with torsion, we develop and unify ways to view a geometry with parallel skew torsion as the total space of a locally defined, not necessarily unique Riemannian submersion with totally geodesic fibers. We complete and extend the Cleyton-Swann classification of irreducible such geometries and characterize the cases where the stabilizer of the torsion is larger than the holonomy. As a byproduct, we obtain structure results on Gray manifolds, nearly parallel G$_2$-manifolds and Sasaki manifolds with reducible holonomy.