An exotic sphere with positive curvature almost everywhere

Almost positive curvature on the Gromoll-Meyer sphere

“…In fact the curvature is strictly positive on some nonempty open subset of M . However, as was observed by F. Wilhelm [7], there is also an open subset with zero curvature planes in the tangent space of each of its points. But Wilhelm constructed another U -invariant metric on Sp(2) (neither left nor right invariant) for which the curvature of M is strictly positive outside a subset of measure zero in M ("almost positive curvature").…”

Exotic spheres and curvature

“…In [Wi2], Wilhelm studied the Gromoll-Meyer metric and deformations of the Gromoll-Meyer metric in some detail. Gromoll and Meyer had claimed that their metric had positive sectional curvature almost everywhere.…”

Examples of Manifolds with Non-negative Sectional Curvature

“…In [82] it was shown that the Gromoll Meyer sphere Σ 7 = Sp(2)// Sp(1) admits a metric with almost positive curvature as well. See [29,30] for a simpler proof for a slightly different metric on Σ 7 .…”