Volume and projective equivalence between Riemannian manifolds

Let (M, g) and (M, (g) over bar) be two Riemannian metrics which are pointwise projectively equivalent, i.e. they have the same geodesics as point sets. We prove that the pointwise projective equivalence is trivial, if (M, g) is a noncompact complete manifold which has at most quadratic volume growth and nonnegative total scalar curvature, and (M, (g) over bar) has nonpositive Ricci curvature.