Published online by Cambridge University Press: 01 May 2008
We observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls  is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics , non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls have the same first eigenvalue, the same volume and the distance spheres and, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,
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