AI 中文总结
本文构造了微分同胚于球面的光滑非负弯曲度量，其零曲率点轨迹的豪斯多夫维数恰为1/2，方法基于凸图与正则化最大值嵌入。
AI 中文摘要
对于每个整数$m\geq 3$，我们在一个微分同胚于$S^m$的流形上构造一个光滑黎曼度量$g$，使得$\text{sec}_g\geq 0$，所有截面曲率在某个闭的无处稠密集之外严格为正，并且某些截面曲率消失的点轨迹的豪斯多夫维数恰好为$\frac{1}{2}$。该度量由$\mathbb{R}^{m+1}$中一个光滑凸体的边界诱导。关键的局部模型是一个凸图，其Hessian的核恰好在一坐标轴的Cantor子集上为一维；然后通过正则化最大值将该图嵌入到标准球面中，而不引入额外的退化点。本文主要内容由ChatGPT 5.6生成，并由作者验证。
英文摘要
For every integer $m\geq 3$, we construct a smooth Riemannian metric $g$ on a manifold diffeomorphic to $S^m$ such that $\text{sec}_g\geq 0$, every sectional curvature is strictly positive away from a closed nowhere-dense set, and the point locus on which some sectional curvature vanishes has Hausdorff dimension exactly $\frac{1}{2}$. The metric is induced on the boundary of a smooth convex body in $\mathbb{R}^{m+1}$. The essential local model is a convex graph whose Hessian has a one-dimensional kernel precisely on a Cantor subset of one coordinate axis; a regularized maximum then inserts this graph into a round sphere without introducing additional degenerate points. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.
Comments7 pages. AI generated and human verified. Comments are welcome
