An exotic sphere with positive curvature almost everywhere
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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 . 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.
Math Subject Classifications53C20
Key Words and Phrasesexotic sphere positive curvature
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