Distributional Sectional Curvature Bounds for Riemannian Metrics of Low Regularity

TL;DR

This paper introduces a new way to define curvature bounds for manifolds with low-regularity metrics, extending classical geometric concepts.

Contribution

A novel distributional approach to sectional curvature bounds for continuous Riemannian metrics of Geroch–Traschen regularity.

Findings

The new curvature bound recovers Alexandrov triangle comparison for $C^1$ metrics.

A weaker version applies to locally Lipschitz continuous metrics.

Abstract

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, 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., \documentclass[12pt]{minimal}
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\begin{document}$$H^1_{\textrm{loc}} \cap C^0$$\end{document}Hloc1∩C0, based on a distributional version of the classical formula. Our main result states that for \documentclass[12pt]{minimal}
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\begin{document}$$g \in C^1$$\end{document}g∈C1, 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.