Distributional sectional curvature bounds for Riemannian metrics of low regularity

2025/03/21 by Darius Erös, Erös, Darius, Michael Kunzinger +5
Mathematics · #30L99 #46T30 #53B20 #53C21 #Advanced Harmonic Analysis Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2503.17337

openalex publication_date 2025/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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., H¹_loc ∩ C⁰, based on a distributional version of the classical formula. Our main result states that for 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.

Related

• On the equivalence of distributional and synthetic Ricci curvature lower bounds
• Curvature bounds for regularized riemannian metrics
• Smoothing Riemannian Metrics with Ricci Curvature Bounds
• Curvature bounds via Ricci smoothing
• Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics