Canonical Submersions in Nearly K\"ahler Geometry

TL;DR

This paper investigates submersions in nearly K"ahler geometry, extending existing theorems and demonstrating new submersion structures in specific manifolds, with applications to Sasaki and quaternionic K"ahler manifolds.

Contribution

It extends submersion theorems to broader contexts and provides new explicit descriptions of structures on the base manifolds in nearly K"ahler geometry.

Findings

Parallel 3-$(eta, au)$-Sasaki manifolds admit 1-dimensional submersions onto nearly K"ahler orbifolds.

A class of nearly K"ahler manifolds submerges onto quaternionic K"ahler manifolds with explicit structure expressions.

New proof techniques simplify understanding of submersions in nearly K"ahler geometry.

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-$(\alpha,\delta)$-Sasaki manifolds admit 1-dimensional submersions onto nearly K\"ahler orbifolds. As a secondary application we reprove that a certain class of nearly K\"ahler manifolds submerges onto quaternionic K\"ahler manifolds. This new proof gives an direct expression for the quaternionic structure on the base.