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.
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Wilhelm, F. An exotic sphere with positive curvature almost everywhere. J Geom Anal 11, 519–560 (2001). https://doi.org/10.1007/BF02922018
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DOI: https://doi.org/10.1007/BF02922018