Stability of the non-symmetric space \(\operatorname{E}_7 / \operatorname{PSO}(8)\). (English) Zbl 1529.53039
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.
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.
Reviewer: Viviana del Barco (Campinas)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C30 | Differential geometry of homogeneous manifolds |
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