Manifolds Close to the round Sphere

We prove that the manifold Mn of minimal radial curvature Kmin o ≥ 1 is homeomorphic to the sphere Sn if its radius or volume is larger than half the radius or volume of the round sphere of constant curvature 1. These results are optimal and give a complete generalization of the corresponding results for manifolds of sectional curvature bounded from below. 0. Introduction and results 0.1. A standard problem in Riemannian Geometry is to find conditions which guarantee that a given Riemannian manifold is topologically or metrically close to the round sphere S of constant curvature 1 (so-called sphere recognition theorems; see [GM1, GM2]). Two of the best known results of this type are contained in the following theorem. Theorem 1. Let M be an n-dimensional compact Riemannian manifold without boundary of sectional curvature K ≥ 1 and rad(M) > π − . Then we have: (1) (Grove-Shiohama [GS]) M is homeomorphic to S if rad(M) > π/2. (2) (Grove-Petersen [GP]) The Gromov-Hausdorff distance between M and S satisfies dGH(M, S) ≤ C( ) for some function C( )→ 0 as → 0. We consider the class of manifolds with minimal radial curvature bounded from below by 1. This class is substantially larger than the class of manifolds in Theorem 1 with sectional curvature bounded from below by 1. (Recall that a Riemannian manifold M has minimal radial curvature K o with a base point o bounded from below by k, K o ≥ k, if for an arbitrary point p and every minimal geodesic γ(t), 0 ≤ t ≤ r, connecting o and p the sectional Received February 15, 2000; received in final form August 22, 2000. 2000 Mathematics Subject Classification. 53C20, 53C21. Supported by CNPq. c ©2001 University of Illinois