Distance spheres and Myers-type theorems for manifolds with lower bounds on the Ricci curvature (Q1826142)

A new compactness criterion for complete Riemannian manifolds in terms of the lower bound on the Ricci curvature and the mean curvature of some distance sphere at a point is given. The result provides a simple and completely geometric proof of Calabi's theorem for the decay of radial Ricci curvature on complete noncompact manifolds of positive Ricci curvature.