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 M endowed with a Riemannian metric g that is possibly discontinuous, with g, g⁻¹ ∈ L^∞ loc g , g - 1 ∈ L loc ∞ and g ∈ W1,ploc g ∈ W loc 1 , p for p ≤ M - 1 p ≤ dim M - 1 .