Abstract
Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CG1]. 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 [Y1]): Is there any finiteness result for complete Riemannian manifolds with Ric > 0 ? 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 CP 2 attached to it by surgery; cf. also [ShY 1]. 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.
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