On complete manifolds with nonnegative Ricci curvature

Complete open Riemannian manifolds (M n , g) with nonnegative sectional curvature are well understood.The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CGI].The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2].On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2].A different method to construct manifolds which carry a metric with Ric> 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB].This leads to the question (cf.also [YI]): Is there any finiteness result for complete Riemannian manifolds with Ric ~ O? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY].On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions.For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x Cp2 attached to it by surgery; cf. also [ShYI].It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions.The purpose of this paper is to establish the following main result.Theorem A. Let M n be a complete open Riemannian manifold with Ric ~ O. Suppose that M n has diameter growth of order o(rl/n).Then M n is homotopy equivalent to the interior of a compact manifold with boundary, provided the sectional curvature is bounded away from -00.The notion of diameter growth requires a precise definition.Roughly speaking, we would like to measure the diameters of the "essential components" of the distance spheres S(po' r) W.r.t. the intrinsic metric in Mn(po' , .r) , where !< , < I is a fixed number.Given any open set Q c M n , not necessarily connected, we shall write diam(L, Q) for the diameter of any connected subset L c Q measured W.r.t. the intrinsic distance function of the open submanifold