The conformal-Killing equation on G2 and Spin7 structures

TL;DR

This paper characterizes when G2 and Spin7 structures on 7- and 8-manifolds are conformal-Killing, showing that for G2 structures it occurs precisely when they are nearly parallel, and for Spin7 structures when they are parallel.

Contribution

It provides a complete characterization of conformal-Killing G2 and Spin7 structures, linking them to nearly parallel and parallel structures respectively.

Findings

G2-structure is conformal-Killing iff nearly parallel

Spin7-structure is conformal-Killing iff parallel

Characterization of conformal-Killing structures on special holonomy manifolds

Abstract

Let M be a 7-manifold with a G2-structure defined by \phi \in\Omega^{3}_{+}(M). We prove that {\phi} is conformal-Killing with respect to the associated metric g(\phi) if and only if the G2-structure is nearly parallel. Let M be an 8-manifold with a Spin7-structure defined by \psi \in\Omega^{4}_{+}(M). We prove that {\psi} is conformal-Killing with respect to the associated metric g(\psi) if and only if the Spin7-structure is parallel.