A Metric with Positive Sectional Curvature on $S^2\times S^3$

Abstract: We prove that $S^2\times S<sup>3$ admits a Riemannian metric with positive sectional curvature. We regard it as the principal $S<sup>1$-bundle with first Chern class $(1,1)$ over $S^2\times S<sup>2$ and begin with a small diagonal Cheeger deformation of the base. An adapted connection gives a nonnegatively curved connection metric whose zero-curvature planes form a clean compact submanifold of the Grassmann bundle. We then construct a global horizontal complex symmetric $2$-tensor of $S<sup>1$-weight two whose restriction to every zero plane is the square of a nonzero complex null covector. Its rotation along the fibers produces a positive second-order term in the Gauss equation. An anisotropic normal Hessian estimate and a finite-dimensional reduction control all nearby planes, so a sufficiently small perturbation has positive sectional curvature. The metric and the proof are discovered by the Odin Automatic AI Research Agent.