The Radius Rigidity Theorem for Manifolds of Positive Curvature

Recall that the radius of a compact metric space (X, dist) is given by rad X = minx∈X maxy∈X dist(x, y). In this paper we generalize Berger’s 1 4 -pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature ≥ 1 and radius ≥ π2 is either homeomorphic to the sphere or isometric to a compact rank one symmetric space. The classical sphere theorem states that a complete, simply connected Riemannian n-manifold with positive, strictly 1/4-pinched sectional curvature is homeomorphic to S ([Ber1], [K], and [Rch]). The weakly 1/4-pinched case is covered by Berger’s Rigidity Theorem Let M be a complete, simply connected Riemannian n-manifold with sectional curvature, 1 ≤ sec M ≤ 4. Then either (i) M is homeomorphic to S, or (ii) M is isometric to a compact rank one symmetric space.