A compact Riemannian manifold with radial Ricci curvature bounded below, diameter l, and first nonzero eigenvalue at least that of a model ball is isometric to a spherically symmetric model manifold.
Chavel, Eigenvalues in Riemannian Geometry , Academic Press, New York (1984)
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A generalized sphere theorem and its applications
A compact Riemannian manifold with radial Ricci curvature bounded below, diameter l, and first nonzero eigenvalue at least that of a model ball is isometric to a spherically symmetric model manifold.