# Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic

## TL;DR

This paper characterizes when two real-analytic Finsler metrics on surfaces with negative Euler characteristic share the same unparametrized geodesics, showing they differ only by a scaling and a closed 1-form.

## Contribution

It provides a complete characterization of projectively equivalent Finsler metrics on such surfaces, extending understanding of their geometric structure.

## Key findings

- Two metrics have the same unparametrized geodesics iff they differ by a scaling and a closed 1-form.
- The result applies specifically to surfaces with negative Euler characteristic.
- The proof involves properties of real-analytic Finsler metrics and geodesic equivalence.

## Abstract

We proof that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.