 

Annegret Y. Burtscher
Length structures on manifolds with continuous Riemannian metrics view print


Published: 
April 27, 2015 
Keywords: 
length structures, curves, lowregularity Riemannian metrics, approximations of Riemannian metrics 
Subject: 
53C20, 53C23 


Abstract
It is wellknown 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 lengthminimizing 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 lowregularity 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 arclength 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 lowregularity Riemannnian manifolds is still compatible with standard definitions and can successfully fill in for lack of differentiability.


Acknowledgements
This research was supported by a "For Women in Science" fellowship by L'Oréal Austria, the Austrian commission of UNESCO and the Austrian Ministry of Science and Research, and by the Austrian Science Fund in the framework of project P23714.


Author information
Laboratoire JacquesLouis Lions, Université Pierre et Marie Curie (Paris 6), France, and Faculty of Mathematics, University of Vienna, Austria
Current Address: Mathematical Institute, University of Bonn, Endenicher Allee 60, 53111 Bonn, Germany.

