Dynamical behavior of Darboux curves

TL;DR

This paper studies Darboux curves, special curves on surfaces preserved by Möbius transformations, by analyzing their properties through Lorentz geometry and providing explicit characterizations on simple surfaces.

Contribution

It characterizes Darboux curves using Lorentz geometry and generalizes classical results on these curves on simple surfaces.

Findings

Darboux curves are preserved by Möbius transformations.

Explicit descriptions of Darboux curves on simple surfaces.

Connections to classical results by Pell and Santalo.

Abstract

In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and prove some general properties and make them explicit them on simple surfaces, retrieving results of Pell (1900) and Santalo (1941).