Length structures on manifolds with continuous Riemannian metrics

TL;DR

This paper explores how to define and analyze length structures on manifolds equipped with low-regularity Riemannian metrics, showing that absolutely continuous curves induce the standard metric space structure despite limited smoothness.

Contribution

It extends low-regularity Riemannian geometry by proving that absolutely continuous curves induce the standard metric, even with continuous, non-smooth metrics.

Findings

Absolutely continuous curves induce the standard metric space structure.

Arc-length of absolutely continuous curves matches the metric length.

Low-regularity metrics can be analyzed using smooth approximations.

Abstract

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as a differentiable exponential map etc. are available nor (generalized) curvature bounds are imposed. In this paper we advance low-regularity Riemannian geometry by investigating general length structures on manifolds that are equipped with Riemannian metrics of low regularity. We generalize the length structure by proving that the class of absolutely continuous curves induces the standard metric space structure. The main result states that the arc-length of absolutely continuous curves is the same as the length induced by the metric. For the proof we use techniques from the analysis of metric spaces and employ specific smooth approximations of continuous Riemannian metrics. We thus show that when dealing with lengths of curves, the metric approach for low-regularity Riemannnian manifolds is still compatible with standard definitions and can successfully fill in for lack of differentiability.