Does the manifold \(S^2 \times S^2\) admit a Riemannian metric of strictly positive sectional curvature? Equivalently: is the standard product of two 2-spheres, which carries nonnegative curvature with flat planes, deformable to everywhere-positive curvature?
Going back to Heinz Hopf in the 1930s, from the early days of global Riemannian geometry: which topologies can carry everywhere-positive curvature? The tools that settled neighbouring questions — Gauss–Bonnet in dimension two, Synge's theorem, the sphere theorems — all stop short here. No metric of positive sectional curvature on \(S^2 \times S^2\) has ever been exhibited, and no topological obstruction ruling one out has ever been found; the product metric has nonnegative curvature with flat directions, and every perturbation studied so far creates some negative curvature somewhere.
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