On positively curved Riemannian manifolds with bounded volume

It is very interesting and important to investigate the relations among curvatures, volumes and topological structures on Riemannian manifolds of positive curvature.The following theorems are well known.THEOREM A. (Bishop-Crittenden [5]) Let M be an n-dimensίonal complete Riemannian manifold with sectional curvature K ^ 1.Then we have vol M ^ vol S n , and equality holds only if M is isometric to a sphere S n with constant curvature 1, where we denote the volume of M by vol M. THEOREM B. (Heim [7]) Let M be an n-dίmensional complete Riemannian manifold with sectional curvature K^>1 and vol M > (1/2) vol S n .Then M is a homotopical sphere.In this paper we give a simple proof of Theorem B and prove the following theorem.THEOREM C. Let M be an n-dimensional complete Riemannian manifold with sectional curvature K^l and vol M ^ (1/2) vol S n .Then M is a homotopical sphere or isometric to the real protective space with constant curvature 1.