2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75246We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close to $\frac{\Vol\sn}{#π_1(M)}$ is both Gromov-Hausdorff close and diffeomorphic to the space form $\frac{\sn}{π_1(M)}$. This extends results of T. Colding and T. Yamaguchi.To appear in Ann. Sci. Ec. Norm. supDifferential GeometryGeometric Topologytext