Homogeneous Einstein (α,β)-metrics on compact simple Lie groups and spheres

In this paper, we study homogeneous Einstein (α,β)-metrics on compact Lie groups and spheres. We first show that any left invariant Einstein (α,β)-metric on a connected compact simple Lie groups except SU(2) with vanishing S-curvature must be a Randers metric. Secondly, we prove that any Sp(n+1)-invariant Einstein (α,β)-metric on S4n+3(n∈N+) with vanishing S-curvature is either a Randers metric, or SU(2n+2)-invariant. Finally, we give a complete description of SU(n+1)-invariant Einstein Finsler metrics on S2n+1(n≥2).