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 emerged in the mid-1960s — the date is approximate rather than a securely dated first proposal — and was given wide circulation through later problem lists, notably Yau's collection [Yau1982OpenProblemsGeometry].

The product of two round spheres has $K\ge0$, but mixed tangent planes are flat, placing the manifold exactly at the border between nonnegative and positive curvature. Surveys by Ziller and Grove explain both the abundance of nonnegatively curved constructions and the scarcity of known closed positively curved manifolds [Ziller2007ExamplesNonnegative] [Grove2010RiemannianPositive]. No accepted topological obstruction rules out strict positivity on $S^2\times S^2$.

On August 19, 2026, Brendle and Hung posted a preprint claiming a complete positive-curvature construction on $S^2\times S^2$, using a Cheeger-deformed metric, a third-order perturbation, and accompanying symbolic calculations [BrendleHung2026PositiveS2S2]. The claim addresses the full existence problem, not a weaker pinching or almost-positive variant, but its crucial analytic and computer-assisted identities still await independent end-to-end verification. The catalog therefore keeps the problem operationally open while recording the full claim as pending.