A Generalized Sphere Theorem

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.