Geometries with parallel, skew-symmetric and closed torsion

TL;DR

This paper classifies Riemannian manifolds with a special metric connection having parallel, skew-symmetric, and closed torsion, revealing their local product structure and analyzing related geometric structures.

Contribution

It provides a complete local classification of PSCT manifolds and explores their associated G-structures, especially almost Hermitian types.

Findings

PSCT manifolds locally split into known factors

Complete local classification achieved

Analysis of G-structures and Gray--Hervella classes

Abstract

We study Riemannian manifolds carrying a metric connection with parallel, skew-symmetric and closed torsion, which we call in short PSCT manifolds. We prove that PSCT manifolds always locally split into a product of well-understood factors, allowing a complete local classification. Further, we investigate various $G$-structures of PSCT type, with a focus on almost Hermitian structures and their possible Gray--Hervella classes.