AI 中文总结
利用相容圆盘定理构造附着映射，在 Kervaire 九球面上获得正截面曲率，并通过 Dirac 指标排除部分类型的非负曲率。
AI 中文摘要
利用邓、胡和张的相容圆盘定理，我们在 Kervaire 九球面上获得了正截面曲率。我们构造了一个光滑的共轭等变附着映射，其扭转恰好是 Milnor--Munkres 换位子。其稳定化的拼接数据将所得球面识别为五球面上两个切丛管道的边界。我们描述了标记商度量，并给出了基于固定联络的曲率范数的正下界。实 Dirac 指标排除了八个光滑九球面类型中四个上的甚至非负截面曲率。该构造覆盖了标准型和 Kervaire 型；其余两个指标为零的类型具有正数量曲率，但其正截面曲率的存在性在此未作决定。
英文摘要
Using the compatible-disk theorem of Deng, Hu, and Zhang, we obtain positive sectional curvature on the Kervaire nine-sphere. We construct a smooth conjugation-equivariant attaching map whose twist is exactly a Milnor--Munkres commutator. Its stabilized clutching data identify the resulting sphere with the boundary of the plumbing of two tangent disk bundles over the five-sphere. We describe the marked quotient metrics and give a positive lower bound in terms of the curvature norms of a fixed connection. The real Dirac index excludes even nonnegative sectional curvature on four of the eight smooth nine-sphere types. The construction covers the standard and Kervaire types; the remaining two index-zero types admit positive scalar curvature, but their positive-sectional-curvature existence is not decided here.
