Conformal flatness, non-Abelian Kaluza-Klein and the Quaternions

TL;DR

This paper explores the relationship between conformally flat spaces and non-Abelian Kaluza-Klein reductions, revealing that special quaternionic holonomy manifolds can be derived from conformally flat higher-dimensional spaces.

Contribution

It establishes a connection between conformally flat spaces and quaternionic holonomy manifolds via non-Abelian Kaluza-Klein reduction, highlighting a natural correspondence.

Findings

Maximally symmetric manifolds with quaternionic holonomy arise from conformally flat spaces.

Special manifolds with constant curvature relate to conformally flat, non-Abelian Kaluza-Klein spaces.

The work links geometric structures with Kaluza-Klein reduction techniques.

Abstract

Maximally symmetric manifolds with holonomy in the unitary quaternionic group Sp(d/4) emerge from the non-Abelian Kaluza-Klein reduction of conformally flat spaces. Thus, all special manifolds with constant properly `holonomy-related' sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza-Klein spaces.