The infinitesimal structure of manifolds with non-continuous Riemannian metrics

TL;DR

This paper explores the geometric and analytic properties of manifolds equipped with possibly discontinuous Riemannian metrics, highlighting how such metrics affect the infinitesimal structure and related properties.

Contribution

It constructs examples of manifolds with discontinuous Riemannian metrics to analyze their failure to be infinitesimally Hilbertian or quasi-Riemannian.

Findings

Examples of manifolds with discontinuous metrics are constructed.

Discontinuous metrics can cause failure of infinitesimal Hilbertianity.

The regularity conditions of the metric influence the manifold's infinitesimal structure.

Abstract

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly discontinuous, with $g, g^{-1} \in L^\infty_{\mathrm{loc}} $ and $ g \in W^{1,p}_{\mathrm{loc}}$ for $ p \leq \mathrm{dim} M - 1 $.