On homogeneous Einstein (α,β)(α,β)-metrics

In this paper, we study homogeneous Einstein (α,β)(α,β)-metrics. First, we deduce a formula for Ricci curvature of a homogeneous (α,β)(α,β)-metric. Based on this formula, we obtain a sufficient and necessary condition for a compact homogeneous (α,β)(α,β)-metric to be Einstein and with vanishing S-curvature. Moreover, we prove that any homogeneous Ricci flat (α,β)(α,β) space with vanishing S-curvature must be a Minkowski space. Finally, we consider left invariant Einstein (α,β)(α,β)-metrics on Lie groups with negative Ricci constant. Under some appropriate conditions, we show that the underlying Lie groups must be two step solvable. We also present a more convenient sufficient and necessary condition for the metric to be Einstein in this special case.