Stability of Riemannian manifolds with Killing spinors

TL;DR

This paper proves the stability of manifolds with imaginary Killing spinors using a new approach and explores the instability of certain Sasaki-Einstein manifolds with real Killing spinors, providing new examples.

Contribution

It offers a new proof of stability for manifolds with imaginary Killing spinors and identifies conditions leading to instability in Sasaki-Einstein manifolds with real Killing spinors.

Findings

Complete manifolds with imaginary Killing spinors are strictly stable.

Regular Sasaki-Einstein manifolds over product Kähler-Einstein bases are unstable.

New examples of unstable manifolds with real Killing spinors are provided.

Abstract

Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kr\"{o}ncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a K\"{a}hler-Einstein manifold. We prove that if the base space is a product of two K\"{a}hler-Einstein manifolds then the regular Sasaki-Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.