The infinitesimal structure of manifolds with non-continuous Riemannian metrics

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 $.