Stability of the non-symmetric space \(\operatorname{E}_7 / \operatorname{PSO}(8)\) (Q6049887)

This paper introduces the first known example of a stable non-symmetric Einstein metric of positive scalar curvature. The authors show that the normal metric on \(\mathrm{E}_7/\mathrm{PSO}(8)\) is stable in the class of all Riemannian metrics. Their proof is based on estimates of the Lichnerowicz Laplacian on the space of divergence-free, trace-free symmetric 2-tensors. The paper also contains general formulas that relate the curvature endomorphisms of the Levi-Civita connection of a Riemannian manifold with the curvature endomorphism of another linear connection with parallel skew-torsion. When this relationship is applied to the reductive connection of the normal homogeneous space \(\mathrm{E}_7/\mathrm{PSO}(8)\), the Casimir operator appears.