Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

Instability of some Riemannian manifolds with real Killing spinors

Pages: 1895 – 1931



Changliang Wang (School of Mathematical Sciences and Institute for Advanced Study, Tongji University, Shanghai, China)

McKenzie Y. Wang (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada)


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