On radius , systole , and positive Ricci curvature

A closed Riemannian n-manifold M with Ricci M > n 1 has diameter < 7r ([M]), and equality holds only if M is isometric to the unit sphere S ~ ([Cn]). Given these results, it is natural to ask whether M is diffeomorphic to S n if the diameter of M is almost 7r. This question was answered negatively by Anderson and Otsu, who showed that even the topology of such a space can be different from the topology of the sphere ([A], [O1]). Thus one must have stronger hypotheses to prove a differentiable sphere theorem. There are already results along these lines in [O2], [PSZ], [Wm], [Wu], and [Y1]. We will prove generalizations of all of these theorems and provide a corresponding characterization of real projective space. Recall that the radius of a compact metric space, (X, dist) is defined by