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

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.