Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature (Q1303874)

Let \(M\) be a compact \(n\) dimensional Riemannian orbifold. Assume that the Ricci curvature satisfies the bound Ric\(_M\geq n-1\). Assume the fixed point set of the local groups of the orbifold have codimension at least 2 so the singular set \(S_M\) is of codimension at least \(2\). Let \(\Delta\) be the Friedrichs extension of the Laplacian on \(C_0^\infty(M-S_M)\); this has a discrete spectrum given by \(0=\lambda_0<\lambda_1\leq\dots\uparrow\infty\). If \(M\) is smooth, then the Lichnerowicz-Obata theorem [\textit{A. Lichnérowicz}, `Géométrie des groupes de transformations' (1958; Zbl 0096.16001) and \textit{M. Obata}, J. Math. Soc. Japan 14, 333-340 (1962; Zbl 0115.39302)] shows that \(\lambda_1=n\) if and only if \(M\) is isometric to the sphere \(S^n\) with the standard metric. The author generalizes this result showing that the \(k\)th non-zero eigenvalue characterizes the \(k\)-times suspension structure of an orbifold: Theorem 1.1. Let \(M\) be as above. Then \(\lambda_1\geq n\). Furthermore one has that \(\lambda_k=n\) for \(1\leq k\leq n\) if and only if \(M\) is isometric to the \(k\) times spherical suspension of \(S^{n-k}/\Gamma\) where \(\Gamma\subset O(n-k)\) is a finite group acting isometrically on the sphere \(S^{n-k}\) with codimension 2 fixed point set.