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Randers and $(α,β)$ equigeodesics for some compact homogeneous manifolds

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, $(α,β)$ equigeodesic, etc. In this paper, we study Randers and $(α,β)$ equigeodesics. For a compact homogeneous manifold, we prove Randers and $(α,β)$ 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.