Curvature of an exotic 7-sphere

Abstract

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 $S^4$ as base space, unit $S^3$ as fibre, and $k=1,2$ $SU\left(2\right)$ 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, $SO\left(3\right)\times O\left(2\right)$, 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.