发表机构
Nankai University; Tianjin Normal University; Southeast University（南开大学; 天津师范大学; 东南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过非等距圆周作用在具有偶数Eells--Kuiper不变量的同伦十一维球面上构造正截面曲率度量，并给出依赖于成员的显式正曲率下界。
AI 中文摘要
我们研究通过非等距圆周作用构造正截面曲率黎曼度量的问题。对于一族同伦十一维球面，其Eells--Kuiper不变量构成$\mathbb{Z}/992$的偶数子群，我们在每个成员上构造一个光滑背景度量$q$和三个有效的圆周作用，其生成元为$W_1,W_2,W_3$，使得由$g^{-1}=q^{-1}+\sum_{a=1}^3W_a\otimes W_a$确定的度量具有正截面曲率。每个作用对于每个部分度量（包括其输入度量和最终度量）都是非等距的。我们首先通过四元数Hopf丛的规范变换来描述该球面。然后我们在两个圆盘上构造相容的度量，并在保持作用公式的同时对其逆度量进行光滑化。一个局部共轭使得圆周作用成为非等距的。对于每个固定的成员，我们获得一个显式的正下界$2^{-54}(1+M_{0,k}+M_{1,k})^{-28}$，其中$M_{0,k}$和$M_{1,k}$是其固定联络的曲率及其一阶协变导数的范数。该下界可能依赖于族中的成员。
英文摘要
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.
