Manifolds close to the round sphere

Source abstract

We prove that the manifold $M^n$ of minimal radial curvature $K^{\min}_o\geq 1$ is homeomorphic to the sphere $S^n$ 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.