A Metric with Positive Sectional Curvature on $S^3\times S^3$

Abstract

We construct a smooth Riemannian metric with strictly positive sectional curvature on \(S^3\times S^3\). Since \(S^3\times S^3\) is even-dimensional and has Euler characteristic zero, our result disproves the positive-curvature case of Hopf's sign conjecture. The metric and the proof were discovered by the Odin Automatic AI Research Agent.