A sphere theorem for manifolds of positive Ricci curvature

Shiohama, Katsuhiro

doi:10.1090/S0002-9947-1983-0682734-1

A sphere theorem for manifolds of positive Ricci curvature
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Trans. Amer. Math. Soc. 275 (1983), 811-819 Request permission

Abstract:

Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that ${\text {Ri}}{{\text {c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}$ and the volume of $M \geqslant$ the volume of the $(\pi - \varepsilon )$-ball on the unit $m$-sphere, $m = {\text {dim }}M$. Then for a suitable choice of $\varepsilon = \varepsilon (m,k)$ every $M$ belonging to this class is homeomorphic to ${S^m}$.

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