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# Communications in Analysis and Geometry

## Volume 30 (2022)

### Number 8

### Instability of some Riemannian manifolds with real Killing spinors

Pages: 1895 – 1931

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n8.a9

#### Authors

#### 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 $N_{k,l} = \mathrm{SU}(3) / i_{k,l} (S^1)$ (which are all nearly parallel $\mathrm{G}_2$ 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 Kähler ones (except $\mathrm{G}_2 / \mathrm{SU}(3)$).

Received 25 July 2019

Accepted 24 February 2020

Published 13 July 2023