# On Finsler surfaces of constant flag curvature with a Killing field

## TL;DR

This paper characterizes two-dimensional Finsler surfaces with constant flag curvature that admit a Killing field, providing a normal form depending on two functions and applying it to spherically symmetric cases including the Funk metric.

## Contribution

It introduces a normal form for such Finsler metrics with a Killing field, depending on two functions, and applies this to spherically symmetric surfaces.

## Key findings

- Normal form depending on two functions for Finsler surfaces with Killing fields.
- Method to compute these functions for spherically symmetric cases.
- Explicit normal form of the Funk metric on the unit disk.

## Abstract

We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to calculate these functions for spherically symmetric Finsler surfaces of constant flag curvature. In particular, we obtain the normal form of the Funk metric on the unit disk D^2.