A metric on $S^2 \times S^2$ with positive sectional curvature

Abstract

We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-M\"uter metric. We then consider a suitable third order perturbation of this Cheeger-M\"uter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.