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Reducible Holonomy in Closed Torsion Geometries

Leander Stecker

TL;DR

The work studies connections with closed torsion and reducible holonomy, showing that a holonomy-invariant orthogonal split TM = H ⊕ V yields a local Riemannian submersion with fibers along V and a projected torsion on the base. It formalizes a Submersion Theorem and derives curvature and integrability relations that link the total, base, and fiber geometries, allowing the base to inherit a skew torsion connection. Specializing to SKT manifolds, it identifies conditions under which the almost complex structure projects and the base carries the Bismut connection, recovering known submersion results for BHE and sHKT. In homogeneous SKT spaces on semisimple groups with Samelson complex structures, it decomposes the Bismut holonomy and constructs holomorphic submersions to generalized flag manifolds G^C/P_I, establishing explicit holonomy constraints on the base.

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-Kähler 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.

Reducible Holonomy in Closed Torsion Geometries

TL;DR

The work studies connections with closed torsion and reducible holonomy, showing that a holonomy-invariant orthogonal split TM = H ⊕ V yields a local Riemannian submersion with fibers along V and a projected torsion on the base. It formalizes a Submersion Theorem and derives curvature and integrability relations that link the total, base, and fiber geometries, allowing the base to inherit a skew torsion connection. Specializing to SKT manifolds, it identifies conditions under which the almost complex structure projects and the base carries the Bismut connection, recovering known submersion results for BHE and sHKT. In homogeneous SKT spaces on semisimple groups with Samelson complex structures, it decomposes the Bismut holonomy and constructs holomorphic submersions to generalized flag manifolds G^C/P_I, establishing explicit holonomy constraints on the base.

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-Kähler 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.
Paper Structure (4 sections, 16 theorems, 61 equations)

This paper contains 4 sections, 16 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Submersion Theorem
  3. Submersions in SKT geometry
  4. Submersions on SKT Samelson Spaces

Key Result

Theorem 1.1

Suppose $\nabla$ is a metric connection with closed skew torsion $T$ and $TM=\mathcal{H}\oplus\mathcal{V}$ splits orthogonally as a representation of the holonomy group $\mathrm{Hol}(\nabla)$. Assume further that Then

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Theorem 2.1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • ...and 28 more