Pincements en courbure de Ricci positive

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$andits$n$-steigenvaluecloseto$n$ 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 $\text{Ric}\ge n-1$ and volume close to \frac{\Vol\sn}{#\pi_1(M)} is both Gromov-Hausdorff close and diffeomorphic to the space form $\frac{\text{sn}}{{\pi }_1(M)}$. This extends results of T. Colding and T. Yamaguchi.