Distributional sectional curvature bounds for Riemannian metrics of low regularity
Darius Erös, Michael Kunzinger, Argam Ohanyan, Alessio Vardabasso
TL;DR
This work introduces distributional sectional curvature bounds for continuous Geroch--Traschen metrics and proves that, when the metric has $C^1$ regularity, these distributional bounds imply the corresponding Alexandrov curvature bounds (locally $\mathrm{CBB}(k)$ or locally $\mathrm{CAT}(k)$). The authors develop a robust regularization framework via manifold convolution to pass from distributional inequalities to smooth approximations, apply Toponogov-type arguments, and manage the sizes of comparison regions to obtain local curvature control. They also establish weaker, variable-parameter results for $C^{0,1}_{\mathrm{loc}}$ metrics and discuss classical Hartmann--Wintner examples to illustrate the limits of these bounds at low regularity. The work connects distributional and synthetic curvature concepts, highlighting future directions toward converse results, links with $RCD$-spaces, and possible Lorentzian extensions, thereby bridging low-regularity Riemannian geometry with Alexandrov geometry.
Abstract
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.