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arXiv · 2609.11980

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

Shengtao Guo·Ethan X. Fang·Junwei Lu
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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.

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Shengtao Guo, Ethan X. Fang, Junwei Lu. 2026-09-06. A Metric with Positive Sectional Curvature on $S^3\times S^3$. https://arxiv.org/abs/2609.11980

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