Detlef Gromoll

On differentiable functions with isolated critical points

THE PURPOSE of this paper is to describe some quantitative aspects of a Morse theory for differentiable functions on a manifold which have only isolated but possibly degenerate critical points. We will show that around such points locally, a function splits into degenerate and non-degenerate parts, in terms of which the relative homology can be expressed in a certain way when passing a critical level. Our investigation originated in connection with a specific geometric problem, namely to prove the existence of infinitely many geometrically distinct periodic geodesics for a very large class of compact riemannian manifolds, see [3], but the results of this paper may be useful for other applications of Morse theory as well. We wish to thank Ralph Abraham and Alan Weinstein for helpful conversations.

On complete manifolds with nonnegative Ricci curvature

Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogovs 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.

Nonnegatively Curved Metrics on S2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M2 × ℝ2, where M2 is any compact 2-manifold.