The space of rough Riemannian metrics

TL;DR

This paper explores the space of rough Riemannian metrics on manifolds, establishing its metric properties, connectivity, and the relationship to smooth and continuous metrics.

Contribution

It introduces a new extended metric space for rough Riemannian metrics and characterizes its structure and the closure of smooth metrics within it.

Findings

The space of rough Riemannian metrics is a complete length extended metric space.

Connectivity of the metric space depends on the compactness of the manifold.

The closure of smooth metrics corresponds to continuous metrics.

Abstract

On a smooth connected manifold, we consider all possible locally elliptic and locally bounded measurable coefficient Riemannian metrics called rough Riemannian metrics. We equip this set with an extended metric which is connected if and only if the manifold is compact. We prove that this is a complete length extended metric space and the components on which the distance is finite are path-connected. Moreover, we identify the closure of smooth metrics in this space to be continuous metrics.