PreprintarXiv licencearXiv
A positively curved metric on the Gromoll-Meyer sphere
Abstract
We construct an explicit deformation of the bi-invariant metric on Sp(2) whose induced metrics on the Gromoll-Meyer exotic 7-sphere have positive sectional curvature for all sufficiently small positive parameters. The deformation is determined by a fixed self-adjoint operator compatible with the quotient action. The proof combines a uniform curvature estimate with an algebraic analysis of the horizontal commuting planes. On the critical set where the lower-order positive terms vanish, we establish an explicit positive lower bound for the cubic coefficient of the curvature numerator. Uniform control of the fourth-order remainder and a compactness argument then show that every sectional curvature is positive
Related
- A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere
- Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
- Positively Lin-Lu-Yau curved graphs with no K2,t minor
- Timelike ideal boundary of non-positively curved Lorentzian spaces
- Singular Limits of Frame Bundles of Noncollapsing Positively Curved Manifolds
BibTeX
@misc{ouyang2026positively,
title = {{A positively curved metric on the Gromoll-Meyer sphere}},
author = {Zexuan Ouyang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11484},
eprint = {2609.11484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11484},
}
RIS
TY - UNPB
TI - A positively curved metric on the Gromoll-Meyer sphere
AU - Zexuan Ouyang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11484
UR - https://arxiv.org/abs/2609.11484
AB - We construct an explicit deformation of the bi-invariant metric on Sp(2) whose induced metrics on the Gromoll-Meyer exotic 7-sphere have positive sectional curvature for all sufficiently small positive parameters. The deformation is determined by a fixed self-adjoint operator compatible with the quotient action. The proof combines a uniform curvature estimate with an algebraic analysis of the horizontal commuting planes. On the critical set where the lower-order positive terms vanish, we establish an explicit positive lower bound for the cubic coefficient of the curvature numerator. Uniform control of the fourth-order remainder and a compactness argument then show that every sectional curvature is positive
ER -