On the flag curvature of invariant (α,β)-metrics

In this paper, we consider invariant (alpha,beta)-metrics which are induced by invariant Riemannian metrics (alpha) over tilde and invariant vector fields (X) over tilde on homogeneous spaces. We study the flag curvatures of invariant (alpha,beta)-metrics. We first give an explicit formula for the flag curvature of invariant (alpha,beta)-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.