Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres
Abstract
We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells--Kuiper invariants form the even subgroup of ℤ/992, we construct, on each member, a smooth background metric q and three effective circle actions with generators W1,W2,W3 such that the metric determined by g−1=q−1+∑a=13Wa⊗ Wa has positive sectional curvature. Each action is non-isometric for every partial metric, including its incoming metric and the final metric. We first describe the sphere by gauge transformations of the quaternionic Hopf bundle. We then construct compatible metrics on two disks and smooth their inverse metrics while preserving the action formula. A local conjugation makes the circle actions non-isometric. For each fixed member, we obtain an explicit positive lower bound 2−54(1+M0,k+M1,k)−28, where M0,k and M1,k are norms of the curvature and its first covariant derivative for its fixed connection. The bound may depend on the member of the family.
Related
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- Positive Sectional Curvature on All Smooth 7-spheres
- Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
- Positive sectional curvature on homotopy nine-spheres
- Positive sectional curvature on exotic spheres in infinitely many dimensions
BibTeX
@misc{deng2026positive,
title = {{Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres}},
author = {Shaoqiang Deng and Zhiguang Hu and Hui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32680},
eprint = {2609.32680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32680},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres
AU - Shaoqiang Deng
AU - Zhiguang Hu
AU - Hui Zhang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.32680
UR - https://arxiv.org/abs/2609.32680
AB - We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells--Kuiper invariants form the even subgroup of ℤ/992, we construct, on each member, a smooth background metric q and three effective circle actions with generators W_1,W_2,W_3 such that the metric determined by g^−1=q^−1+∑_a=1^3W_a⊗ W_a has positive sectional curvature. Each action is non-isometric for every partial metric, including its incoming metric and the final metric. We first describe the sphere by gauge transformations of the quaternionic Hopf bundle. We then construct compatible metrics on two disks and smooth their inverse metrics while preserving the action formula. A local conjugation makes the circle actions non-isometric. For each fixed member, we obtain an explicit positive lower bound 2^−54(1+M_0,k+M_1,k)^−28, where M_0,k and M_1,k are norms of the curvature and its first covariant derivative for its fixed connection. The bound may depend on the member of the family.
ER -