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-Müter metric. We then consider a suitable third order perturbation of this Cheeger-Müter 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.

It’s usually the task of the modeler to make their assumptions fit the real world as closely as is practical. It was now our task to make a real-world drum that conformed to the modeling assumptions.