OSCILLATION CRITERIA FOR THE COMPACTNESS OF RIEMANNIAN MANIFOLDS

In this paper, we prove some generalizations of the Ambrose (or Myers) theorem for complete Riemannian manifolds. We observe that the problem of finding the Ricci curvature conditions that guarantee the compactness of the manifold is reduced to the problem of finding the proper oscillation conditions of second-order linear differential equations. The proof of the theorems is based on the Riccati comparison theorem and some related oscillation conditions.