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A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere
Abstract
We construct an explicit one-parameter family of smooth metrics on the Gromoll-Meyer exotic seven-sphere, converging in C∞ to a fixed further Cheeger deformation of the Eschenburg-Kerin metric and having strictly positive sectional curvature for all sufficiently small positive parameter values. The first perturbation preserves the totally geodesic flats of one zero-plane family while making curvature positive near the other; the second removes the remaining zero curvature. The metric and the proof were discovered by the Odin Automatic AI Research Agent.
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BibTeX
@misc{guo2026twostage,
title = {{A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere}},
author = {Shengtao Guo and Ethan X. Fang and Junwei Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12882},
eprint = {2609.12882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12882},
}
RIS
TY - UNPB
TI - A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere
AU - Shengtao Guo
AU - Ethan X. Fang
AU - Junwei Lu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12882
UR - https://arxiv.org/abs/2609.12882
AB - We construct an explicit one-parameter family of smooth metrics on the Gromoll-Meyer exotic seven-sphere, converging in C^∞ to a fixed further Cheeger deformation of the Eschenburg-Kerin metric and having strictly positive sectional curvature for all sufficiently small positive parameter values. The first perturbation preserves the totally geodesic flats of one zero-plane family while making curvature positive near the other; the second removes the remaining zero curvature. The metric and the proof were discovered by the Odin Automatic AI Research Agent.
ER -