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 $g,g^{-1}\in L_{\mathrm{loc}}^{\infty }$ and $g\in W_{\mathrm{loc}}^{1,p}$ for $p\le \dim M-1$.