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

TL;DR

This paper investigates whether Gaussian curvature can be inferred by counting and measuring the lengths of paths on surfaces, establishing a connection between path integrals and curvature.

Contribution

It introduces a measure for path sets in the plane and relates the integral of path lengths on surfaces to Gaussian curvature, providing a new geometric characterization.

Findings

The integral of path lengths equals the average length times measure only if the surface is flat.

The method characterizes Gaussian curvature through path length measurements.

Provides a mathematical framework linking path counting to surface curvature.

Abstract

The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image 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 of the external paths times the measure of the set of paths if and only if the surface has Gaussian curvature equal to zero.