Compactness criteria for Riemannian manifolds

Abstract

Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers’ condition) which ensure the compactness of a complete Riemannian manifold. Using standard index form techniques we relate the problem of finding such Ricci curvature criteria to that of establishing the conjugacy of the scalar Jacobi equation. Using this relationship we obtain a Ricci curvature condition for compactness which is weaker than that of Ambrose and, in fact, which is best among a certain class of conditions.