On Matsumoto-type Finsler metrics

In this paper, we find a necessary and sufficient condition for the Matsumoto-type metric F=(α−β)qαq−1 to be projectively flat. Suppose that the Matsumoto-type metric F̄=(α−β)qαq−1 is constructed from a Randers metric F=α+βF=α+β on a manifold MM by a λλ-deformation, where λ∈C∞(M)λ∈C∞(M). We find a condition on λλ under which the corresponding λλ-deformation preserves the property of being projectively flat. We show that if F̄=λα2α−β is a λλ-deformation of a Randers metric F=α+βF=α+β, then FF is a Berwald metric if and only if F̄ is a Douglas metric if and only if FF is projectively related to F̄.