Axial straight lines in the covering surface of a Finsler surface

Abstract

In the study of geodesics on surfaces, the subjects and methods are different for spheres, tori, other closed surfaces and non-compact surfaces. We make the method used to study geodesics on 2-tori applicable to surfaces with genus 2 or higher. Let $S$ be a torus with boundary embedded in an orientable geodesically complete Finsler surface $M$. We define a distance ${\delta }_S$ on $S$ in such a way that ${\delta }_S(p,q)$ is the minimum length of curves in $M$ from $p$ to $q$ homotopic to curves in $S$ with same endpoints $p,q\in S$. Those geodesics are minimal with respect to ${\delta }_S$ but not in $M$. We use this distance ${\delta }_S$ to study geodesics in $M$ homotopic to curves in $S$.