Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonical Submersions in Nearly Kähler Geometry

Leander Stecker

TL;DR

The paper develops a canonical submersion framework for metric connections with parallel skew torsion under reducible holonomy, enabling a local Riemannian submersion with geodesic fibers and a projected torsion on the base. This machinery is then applied to parallel $3$-$(\alpha,\delta)$-Sasaki manifolds with $\delta=2\alpha$, showing that a submersion along a Reeb field yields a nearly Kähler orbifold base whose geometric data are compatible with a parallel skew torsion connection. A secondary application reinterprets a known construction to produce quaternionic Kähler structures on the base in a Nagy-type setting, via a local reduction when the torsion and holonomy satisfy specific scalar-operator conditions. Together, these results bridge torsionful holonomy with classical special geometries, offering a unified method to obtain nearly Kähler and quaternionic Kähler bases from canonical submersions of high-dimensional, torsionful manifolds.

Abstract

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application we show that parallel 3-$(α,δ)$-Sasaki manifolds admit 1-dimensional submersions onto nearly Kähler orbifolds. As a secondary application we reprove that a certain class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives an direct expression for the quaternionic structure on the base.

Canonical Submersions in Nearly Kähler Geometry

TL;DR

The paper develops a canonical submersion framework for metric connections with parallel skew torsion under reducible holonomy, enabling a local Riemannian submersion with geodesic fibers and a projected torsion on the base. This machinery is then applied to parallel --Sasaki manifolds with , showing that a submersion along a Reeb field yields a nearly Kähler orbifold base whose geometric data are compatible with a parallel skew torsion connection. A secondary application reinterprets a known construction to produce quaternionic Kähler structures on the base in a Nagy-type setting, via a local reduction when the torsion and holonomy satisfy specific scalar-operator conditions. Together, these results bridge torsionful holonomy with classical special geometries, offering a unified method to obtain nearly Kähler and quaternionic Kähler bases from canonical submersions of high-dimensional, torsionful manifolds.

Abstract

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application we show that parallel 3--Sasaki manifolds admit 1-dimensional submersions onto nearly Kähler orbifolds. As a secondary application we reprove that a certain class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives an direct expression for the quaternionic structure on the base.
Paper Structure (4 sections, 11 theorems, 47 equations)

This paper contains 4 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Canonical Submersions
  3. Parallel 3-(a,d)-Sasaki and Nearly Kähler Manifolds
  4. Nearly Kähler orbifolds from parallel 3-(a,d)-Sasaki manifolds

Key Result

Theorem 2.1

Suppose $\nabla$ is a metric connection with parallel skew torsion $T$ on $(M,g)$ and $TM=\mathcal{H}\oplus\mathcal{V}$ splits orthogonally as representation of the reduced holonomy group $\mathrm{Hol}_0(\nabla)$. Assume further that i.e. the $\Lambda^2\mathcal{V}\wedge\mathcal{H}$-part of $T$ vanishes. Then there exists a locally defined Riemannian submersion $\pi\colon (M,g)\to (N,g_N)$ with to

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Definition 1: AgrDil
  • ...and 14 more