Pincement sur le spectre et le volume en courbure de Ricci positive

We shall show that a complete Riemannian manifold of dimension n with Ric⩾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 [Invent. Math. 138 (1999) 1] (as a by-product, we fill a gap stated in the erratum [Invent. Math. 155 (2004) 223]). We shall also show that a manifold with Ric⩾n−1 and volume close to VolSn#π1(M) is both Gromov-Hausdorff close and diffeomorphic to a space form Sn/π1(M). This extends results of T. Colding [Invent. Math. 124 (1996) 175] and T. Yamaguchi [Math. Ann. 284 (1989) 423].