Randers and $(\alpha,\beta)$ equigeodesics for some compact homogeneous manifolds

TL;DR

This paper investigates Randers and $(eta,eta)$ equigeodesics on compact homogeneous manifolds, establishing their equivalence and providing classification criteria, especially on spheres with non-Riemannian homogeneous Randers metrics.

Contribution

It proves the equivalence of Randers and $(eta,eta)$ equigeodesics on compact homogeneous manifolds and offers a classification criterion, extending understanding of equigeodesics beyond Riemannian metrics.

Findings

Randers and $(eta,eta)$ equigeodesics are equivalent on compact homogeneous manifolds.

A criterion for classifying these equigeodesics is established.

Classification achieved for equigeodesics on certain homogeneous spheres.

Abstract

A smooth curve on $G/H$ is called a Riemannian equigeodesic if it is a homogeneous geodesic for all $G$-invariant Riemannian metrics on $G/H$. With the $G$-invariant Riemannian metric replaced by other classes of $G$-invariant metrics, we can similarly define Finsler equigeodesic, Randers equigeodesic, $(\alpha,\beta)$ equigeodesic, etc. In this paper, we study Randers and $(\alpha,\beta)$ equigeodesics. For a compact homogeneous manifold, we prove Randers and $(\alpha,\beta)$ equigeodesics are equivalent, and find a criterion for them. Using this criterion we can classify the equigeodesics on many compact homogeneous manifolds which permit non-Riemannian homogeneous Randers metrics, including four classes of homogeneous spheres.