Instability of some Riemannian manifolds with real Killing spinors

TL;DR

This paper demonstrates the instability of certain Riemannian manifolds with real Killing spinors, including specific Einstein metrics on Aloff-Wallach spaces and various homogeneous Einstein spaces, expanding understanding of their geometric stability.

Contribution

It proves the instability of a broad class of manifolds with real Killing spinors, including invariant Einstein metrics and homogeneous Einstein spaces, which was previously not well understood.

Findings

Invariant Einstein metrics on Aloff-Wallach spaces are unstable.

Most simply connected non-symmetric homogeneous Einstein spaces are unstable.

Nearly Kähler manifolds in the studied classes are also shown to be unstable.

Abstract

We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly K\"ahler ones (except ${\rm G}_2/{\rm SU}(3)$).