Curvature of an exotic 7-sphere

Journal article, 2025

We study the geometry of the Gromoll–Meyer sphere, one of Milnor's exotic 7-spheres. We focus on a Kaluza–Klein Ansatz, with a round S⁴ as base space, unit S³ as fibre, and k=1,2 SU(2) instantons as gauge fields, where all quantities admit an elegant description in quaternionic language. The metric's moduli space coincides with the k=1,2 instantons' moduli space quotiented by the isometry of the base, plus an additional R⁺ factor corresponding to the radius of the base, r. We identify a “center” of the k=2 instanton moduli space with enhanced symmetry. This k=2 solution is used together with the maximally symmetric k=1 solution to obtain a metric of maximal isometry, SO(3)×O(2), and to explicitly compute its Ricci tensor. This allows us to put a bound on r to ensure positive Ricci curvature, which implies various energy conditions for an 8-dimensional static space-time. This construction then enables a concrete examination of the properties of the sectional curvature.