An eigenvalue pinching theorem

In 1958 Lichnerowicz [7] showed that for a compact n-dimensional riemannian manifold M, whose Ricci curvature is bounded below by n 1 , the first non-zero eigenvalue, 21, of the laplacian satisfies 2 t>n . If, in fact, 21 =n Obata proved that M must be isometric to the standard sphere. A natural question is: Do there exist constants C(n)> 1, depending only on n such that if M is as above and C(n). n> 21 > n then M must be diffeomorphic to a sphere. Here, by combining the works of Gromov [3], Berard and Meyer [1], and Grove and Shiohama [4], we show