Compactness criteria for Riemannian manifolds

Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers' condition) which ensure the compactness of a complete Riemamnan 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. One of the most well-known results relating the curvature and topology of a complete Riemannian manifold M is the classical theorem of Myers [8] which states that if the Ricci curvature with respect to unit vectors on M has a positive lower bound then M is compact. (Myers also gives a diameter estimate in terms of this bound.) In 1957 Ambrose [1] published an interesting generalization of Myers' theorem. He proved that if there is a point q M in such that along each geodesic y: [0, oo) -M emanating from q (and parameterized by arc length t) the Ricci curvature satisfies f Ric(dY dtY ) dt = +oo then M is compact. One of the important features of this result is that the Ricci curvature is not required to be everywhere nonnegative. The author, together with T. Frankel, has applied Ambrose's theorem to certain problems in general relativity (see [4]). In this paper we present a general technique for establishing compactness criteria for complete Riemannian manifolds. As an application of this technique we obtain a generalization of Ambrose's theorem, which, with respect to a certain class of curvature conditions, is best. This generalization can be used to improve some of the results in [4]. (See, especially, Theorem 5 and Corollary 6 of that paper.) We take a moment to introduce some notation and terminology. Throughout, let M denote a smooth complete Riemannian manifold of dimension n > 2. Let be the Riemannian metric on M and let V be the associated Levi-Civita connection. If t -y(t) is a curve in M, let D/dt be the covariant derivative operator on vector fields along y induced by the connection V. For vector fields X and Y let R(X, Y) be the Riemann curvature transformation, i.e. R(X, Y)Z = VxV yZ-V yVxZ -V[XyZ Received by the editors October 3, 1980. 1980 Mathematics Subject Classificatio. Primary 53C20. o 1982 American Mathematical Society 0002-9939/82/0000-0025/$02.25