An exotic sphere with positive curvature

Abstract

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive sectional curvature on a homotopy (but not diffeomorphic) $RP^7$.

This paper has been withdrawn by the authors. As pointed out to us by Chao Qian, Martin Kerin and Burkhard Wilking, Theorem 3.1 is incorrect, namely, the zero locus should be larger than that in Theorem 3.1. We are sincerely grateful to them for their valuable comments. Nevertheless, the metrics we constructed have positive sectional curvature almost everywhere on the Gromoll-Meyer sphere and on the homotopy (not diffeomorphic) RP^7