Curvature of an exotic 7-sphere

TL;DR

This paper investigates the geometry and curvature properties of the Gromoll-Meyer exotic 7-sphere using a Kaluza-Klein framework with instanton gauge fields, identifying conditions for positive Ricci curvature and analyzing sectional curvature.

Contribution

It introduces a novel geometric construction of the Gromoll-Meyer sphere using quaternionic instantons and explicitly computes its Ricci tensor and curvature bounds.

Findings

Identified a symmetry-enhanced point in the instanton moduli space.

Derived bounds on the base radius for positive Ricci curvature.

Explicitly computed the Ricci tensor and sectional curvature properties.

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 $k=1,2$ $SU(2)$ 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 $\mathbb{R}^+$ 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, $SO(3)\times O(2)$, 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.