Stability of Einstein Metrics on Fiber Bundles

We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian product structures on the base. As an application, we estimate the coindex of the Einstein metrics constructed in Wang and Ziller (J Differ Geom 31:215–248, 1990) and Wang (Math Z 210:305–326, 1992). Finally, we investigate more closely the linear stability of Einstein metrics from circle bundle constructions and obtain a rigidity result for linearly stable Einstein metrics of this type.