Projective vector fields on special (α,β)-metrics

In this paper, we study the projective vector fields on two special (α,β)-metrics, namely Kropina and Matsumoto metrics. First, we consider the Kropina metrics, and show that if a Kropina metric F = α^2/β admits a projective vector field, then this is a conformal vector field with respect to Riemannian metric a or F has vanishing S-curvature. Then we study the Matsumoto metric F = α^2/(α−β) and prove that if the Matsumoto metric F = α^2/β admits a projective vector field, then this is a conformal vector field with respect to Riemannian metric a or F has vanishing S-curvature.