On $G_2$-manifolds and geometry in dimensions $6$ and $8$

TL;DR

This paper explores the geometric structures of $G_2$-manifolds and related dimensions, focusing on orbit spaces of Killing vector fields and classifying harmonic morphisms to Einstein manifolds.

Contribution

It provides new classifications of harmonic morphisms from $G_2$-manifolds and analyzes the geometry of orbit spaces under specific group actions.

Findings

Classification of harmonic morphisms with 1D fibers from $G_2$-manifolds

Analysis of geometry on orbit spaces of Killing vector fields

Insights into $G_2$ and Spin(7) geometries

Abstract

We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G={\rm Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.