Year: 2026
Type: article
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., $$H^1_{\textrm{loc}} \cap C^0$$ H loc 1 ∩ C 0 , based on a distributional version of the classical formula. Our main result states that for $$g \in C^1$$ g ∈ 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.
Source: Journal of Geometric Analysis
Institutions: University of Vienna, University of Toronto
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funders: Austrian Science Fund, Universität Wien
Award: G4727268999
OpenAlex ID: W7126194459