Reducible Holonomy in Closed Torsion Geometries

TL;DR

This paper explores how connections with closed skew-symmetric torsion and reducible holonomy lead to locally defined Riemannian submersions, connecting known structures like SKT manifolds and flag manifolds.

Contribution

It demonstrates the existence of Riemannian submersions in closed torsion geometries and extends known results to new contexts such as homogeneous SKT structures.

Findings

Existence of Riemannian submersions in closed torsion geometries.

Holonomy decomposition for homogeneous SKT structures on semi-simple Lie groups.

Identification of holomorphic submersions over generalized flag manifolds.

Abstract

The purpose of this note is to show that a connection with closed skewsymmetric torsion and reducible holonomy admits a locally defined Riemannian submersion together with a projected geometry on the base. We reframe known submersion results for non-K\"ahler Bismut Hermite Einstein manifolds and sHKT structures in this context. For homogeneous SKT structures on semi-simple Lie groups we obtain the holonomy decomposition leading to holomorphic submersions over generalized flag manifolds.