A sphere theorem for manifolds of positive Ricci curvature

Abstract

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