Pincements en courbure de Ricci positive

Details Pincements en courbure de Ricci positive Pincements en courbure de Ricci positive

2005-05-19

arxiv.org/abs/math/0505408v1

Mathematics

Differential Geometry

To appear in Ann. Sci. Ec. Norm. sup

Scientific paper

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.

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