Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere Sⁿ(α) or else each nonzero eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following:.