Linear instability of Sasaki Einstein and nearly parallel G2 manifolds

In this paper, we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel [Formula: see text] manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly, we prove that nearly parallel [Formula: see text] manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space [Formula: see text] which is a [Formula: see text]-dimensional homology sphere with a proper nearly parallel [Formula: see text] structure.