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Positive Sectional Curvature on All Smooth 7-spheres
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.
Related
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BibTeX
@misc{he2026positive,
title = {{Positive Sectional Curvature on All Smooth 7-spheres}},
author = {Yang-Hui He and Ziran Liu and Shing-Tung Yau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29426},
eprint = {2609.29426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29426},
}
RIS
TY - UNPB
TI - Positive Sectional Curvature on All Smooth 7-spheres
AU - Yang-Hui He
AU - Ziran Liu
AU - Shing-Tung Yau
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29426
UR - https://arxiv.org/abs/2609.29426
AB - 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.
ER -