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The aim of this paper is to associate a measure for certain sets of paths inthe Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then,working on parameterized surfaces with a specific Riemannian metric, we defineand calculate the integral of the length over the set of paths obtained as theimage of the initial paths in $\mathbb{R}^2$ under the given parameterization.Moreover, we prove that this integral is given by the average of the lengths ofthe external paths times the measure of the set of paths if and only if thesurface has Gaussian curvature equal to zero.

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