A class of Finsler surfaces whose geodesics are circles

TL;DR

This paper classifies Finsler metrics of Randers type on the Euclidean plane with circular geodesics, revealing conditions on the Riemannian part and providing explicit examples including horocycles in the Poincare disk.

Contribution

It characterizes all Randers Finsler metrics with circular geodesics on subsets of the Euclidean plane, linking them to constant Gaussian curvature metrics and providing explicit constructions.

Findings

Riemannian part must have constant Gaussian curvature

Existence of Randers metrics with prescribed circular geodesics

Explicit example with horocycles in the Poincare disk

Abstract

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is a class of Randers metrics satisfying the condition, determined up to the addition of a total derivative, depending on a single parameter. As one of several applications we exhibit a Finsler metric whose geodesics are the oriented horocycles in the Poincare disk, in each of the two possible orientations.