Year: 2026
Type: preprint
Abstract: We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.
Source: arXiv (Cornell University)
Authors: Guo, Shengtao, Fang, Ethan X., Lu, Junwei
Institution: Harvard University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204170086