Positive sectional curvature on $S^2\times S^2$

Does $S^2\times S^2$ admit a Riemannian metric of strictly positive sectional curvature? The question asks for a metric $g$ in which every tangent $2$-plane, at every point, curves positively. It crystallized in the mid-1960s — the date is approximate, marking the problem's emergence rather than a single securely dated proposal — and was given wide circulation through the problem literature of the following decades, notably Yau's collection [Yau1982OpenProblemsGeometry].

The difficulty sits exactly at the border between nonnegative and positive curvature. The product of two round spheres has $K\ge 0$, but any mixed plane spanned by one direction from each factor is flat, so the product metric narrowly fails. At the same time no known obstruction rules out a positively curved metric on $S^2\times S^2$. Positively curved examples remain scarce in general, as documented by Ziller [Ziller2007ExamplesNonnegative] and Grove [Grove2010RiemannianPositive], and despite the many constructions in nonnegative curvature, none has been pushed to strict positivity on $S^2\times S^2$.

The problem is open in both directions: no positively curved metric has been constructed, and no obstruction ruling one out is known.