Tractors and Tractrices in Riemannian Manifolds

TL;DR

This paper extends the concept of bicycle tracks to tractor/tractrix systems in Riemannian manifolds, providing explicit formulas and estimates that depend on the manifold's curvature, and uses these to generate geodesics.

Contribution

It introduces a generalization of planar tractor/tractrix systems to Riemannian manifolds and derives explicit formulas and curvature-dependent estimates for their properties.

Findings

Explicit formulas for tractrix length and swept area.

Curvature-dependent estimates using Rauch's and Toponogov's theorems.

Length shortening property used to generate geodesics.

Abstract

We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. These expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we prove explicit estimates for them based on Rauch's and Toponogov's comparison theorems. Moreover, the general length shortening property of tractor/tractrix systems is used to generate geodesics in homotopy classes of curves in the ambient manifold.