The infinitesimal structure of manifolds with non-continuous Riemannian metrics

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 $g,g^{-1}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$ and $g\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$ for $p\le \mathrm{d}\mathrm{i}\mathrm{m}M-1$.