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.
Citation
Valery Marenich. "Manifolds close to the round sphere." Illinois J. Math. 45 (2) 615 - 629, Summer 2001. https://doi.org/10.1215/ijm/1258138359
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