Can we detect Gaussian curvature by counting paths and measuring their length?

Abstract

The aim of this paper is to associate a measure for certain sets ofpaths in the Euclidean planeR2with fixed starting and ending points. Then,working on parameterized surfaces with a specific Riemannian metric, wedefine and calculate the integral of the length over the set ofpaths obtainedas the image of the initial paths inR2under the given parameterization.Moreover, we prove that this integral is given by the averageof the lengthsof the external paths times the measure of the set of paths if,and only if, thesurface has Gaussian curvature equal to zero.