Almost positive curvature on the Gromoll-Meyer sphere

TL;DR

This paper demonstrates that a family of metrics on the Gromoll-Meyer sphere, an exotic 7-sphere, exhibits almost positive curvature, with strictly positive sectional curvature outside certain subvarieties.

Contribution

It provides explicit metrics on the Gromoll-Meyer sphere with almost positive curvature, expanding understanding of curvature properties on exotic spheres.

Findings

Metrics with almost positive curvature on the Gromoll-Meyer sphere

Positive curvature outside specific subvarieties

Explicit description of subvarieties of codimension ≥ 1

Abstract

Gromoll and Meyer have represented a certain exotic 7-sphere $\Sigma^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $\Sigma^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ which we describe explicitly.