A metric on S² x S² with positive sectional curvature

We describe a construction of a positively curved metric on $S^2\times S^2$. 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-Mueter metric. We then consider a suitable third order perturbation of this Cheeger-Mueter metric and show that the perturbed metrics have positive sectional curvature. This is joint work with Simon Brendle.