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

A smooth curve on $G\mathrm{/}H$ is called a Riemannian equigeodesic if it is a homogeneous geodesic for all $G$-invariant Riemannian metrics on $G\mathrm{/}H$. 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.