, Volume 11, Issue 3, pp 519-560

An exotic sphere with positive curvature almost everywhere

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Abstract

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere.

In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.