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arXiv · 2609.29426

Positive Sectional Curvature on All Smooth 7-spheres

Yang-Hui He·Ziran Liu·Shing-Tung Yau
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Abstract

We construct a smooth Riemannian metric with strictly positive sectional curvature on every smooth homotopy seven-sphere. For each smooth structure, we construct positively curved metrics on two seven-dimensional disks whose induced boundary metrics agree under the prescribed attaching map. Under this identification, a gluing theorem then yields the required smooth metric on the closed manifold. The construction applies to all 28 oriented diffeomorphism classes.

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Yang-Hui He, Ziran Liu, Shing-Tung Yau. 2026-09-24. Positive Sectional Curvature on All Smooth 7-spheres. https://arxiv.org/abs/2609.29426

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