Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature

Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥ n - 1. We prove that for 1 ≤ k ≤ n, the k^th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S^n-k / Γ of the unit (n - k)-sphere by a finite group Γ ⊂ O(n - k + 1) acting isometrically on S^{n - k} ⊂ ℝ^{n - k+1}.