标题： 低正则性黎曼度量的分布截面曲率界限 显示英文标题

标题： Distributional sectional curvature bounds for Riemannian metrics of low regularity

摘要： 截面曲率的界限在黎曼流形的研究中具有中心重要性，无论是在光滑微分几何中，还是在亚历山大洛夫空间的广义合成设置中。 黎曼度量以及具有有界截面曲率的度量空间享有各种往往刚性的几何性质。 本文的目的是引入并讨论一种新的截面曲率界限概念，适用于具有Geroch--Traschen正则性的连续黎曼度量的流形，即$H^1_{\mathrm{loc}} \cap C^0$，该概念基于经典公式的分布版本。 我们的主要结果指出，对于$g \in C^1$，这种新概念在亚历山大洛夫的意义下，能够恢复基于三角形比较的相应界限。 对于局部Lipschitz连续的度量，也证明了该陈述的一个较弱版本。 显示英文摘要

摘要： Sectional curvature bounds are of central importance in the study of Riemannian mani\-folds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of bounded sectional curvature enjoy a variety of, oftentimes rigid, geometric properties. The purpose of this article is to introduce and discuss a new notion of sectional curvature bounds for manifolds equipped with continuous Riemannian metrics of Geroch--Traschen regularity, i.e., $H^1_{\mathrm{loc}} \cap C^0$, based on a distributional version of the classical formula. Our main result states that for $g \in C^1$, this new notion recovers the corresponding bound based on triangle comparison in the sense of Alexandrov. A weaker version of this statement is also proven for locally Lipschitz continuous metrics.