Pincements en courbure de Ricci positive

Abstract

We 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 $\frac{\Vol\sn}{#\pi_1(M)}$ is both Gromov-Hausdorff close and diffeomorphic to the space form $\frac{\sn}{\pi_1(M)}$. This extends results of T. Colding and T. Yamaguchi.