A Comparison Theorem on the Ricci Curvature in Projective Geometry

Abstract

We show that if two Riemannian metrics \({\tilde g}\)and g are pointwiseprojectively equivalent and the Ricci curvatures satisfy \(\widetilde{Ric}\)≤ Ric, then the projective equivalence is trivialprovided that g is complete. In this case, \({\tilde g}\)is parallel with respect to g and the Riemann curvatures of g and \({\tilde g}\) 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.