The cut locus of a Randers rotational 2-sphere of revolution

TL;DR

This paper investigates the structure of the cut locus on Randers rotational 2-spheres of revolution, revealing how curvature monotonicity influences the cut locus's geometric configuration.

Contribution

It provides new characterizations of the cut locus structure on Randers spheres based on curvature properties, extending classical Riemannian results.

Findings

Cut locus is a point on a subarc of the opposite half bending meridian or antipodal parallel when curvature is monotone.

If curvature is not monotone but the cut locus of a point on the equator is a subarc of the same equator, then the cut locus of other points is a subarc of the antipodal parallel.

Examples illustrate differences between Randers and Riemannian cases.

Abstract

In the present paper we study the structure of the cut locus of a Randers rotational 2-sphere of revolution $(M, F = \alpha+\beta)$. We show that in the case when the Gaussian curvature of the Randers surface is monotone along a meridian, the cut locus of a point $q\in M$ is a point on a subarc of the opposite half bending meridian or of the antipodal parallel (Theorem 1.1). More generally, in the case when the Gaussian curvature is not monotone along the meridian, but the cut locus of a point $q$ on the equator is a subarc of the same equator, the cut locus of any point $\widetilde q\in M$ different from poles is a subarc of the antipodal parallel (Theorem 1.2). Some examples are also given at the last section and some differences with the Riemannian case are pointed out.