Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres
Year: 2026
Type: preprint
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 $\mathbb{Z}/992$, 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 $g^{-1}=q^{-1}+\sum_{a=1}^3W_a\otimes W_a$ 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.
Source: arXiv (Cornell University)
Authors: Deng, Shaoqiang, Hu, Zhiguang, Zhang, Hui
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2804188644, G6978731451, G8523238066
OpenAlex ID: W7214751247