A Comparison Theorem on the Ricci Curvature in Projective Geometry

We show that if two Riemannian metrics and g are pointwiseprojectively equivalent and the Ricci curvatures satisfy Ric, then the projective equivalence is trivialprovided that g is complete. In this case, is parallel with respect to g and the Riemann curvatures of g and are equal.The Ricci curvature condition can be weakened when the manifold iscompact. This rigidity theorem actually holds for more general geometricstructures, such as Finsler metrics and sprays. In this paper, we willalso discuss several examples and show that the completeness of g cannot be dropped.