An exotic sphere with positive curvature almost everywhere. (Q1608505)

Since Milnor's discovery of exotic spheres, an interesting problem in Riemannian geometry has been whether there are exotic spheres with positive curvature. The Gromoll-Meyer sphere gives an example of a nonnegatively curved exotic \(7\)-dimensional sphere [\textit{D. Gromoll}, and \textit{W. Meyer}, Ann. Math. (2) 100, 401--406 (1974; Zbl 0293.53015)]. This was the first example of an exotic sphere with such a property. In the present paper, the author shows that the Gromoll-Meyer metric has zero curvature on an open set of points. Moreover, the Gromoll-Meyer metric can be perturbed to one with positive curvature almost everywhere and admitting an effective \(SO(3)\)-action.