This article presents an approach to the theory of open manifolds of nonnegative sectional curvature via the calculus of nonsmooth functions. This analytical approach makes possible a very compact development of the by now classical theory. The article also gives a summary of the historical development of the subject of open manifolds of nonnegative sectional curvature and of related topics. Some very recent results are also discussed, including results of the author jointly with P. Petersen and S. H. Zhu on curvature decay.

Introduction

At first sight, the study of noncompact manifolds seems necessarily more complicated than that of compact ones: Removal of a single point from a compact manifold gives a noncompact one, but not all noncompact manifolds arise in this way—in general, the topological one-point compactification of a noncompact C∞ manifold is not even a topological manifold. To look at the matter another way, more relevant to the subject of this article, a C∞compact manifold always admits a C∞ function with only nondegenerate critical points, and compactness implies that there are only a finite number of such critical points; thus, a compact manifold has the homotopy type of a finite CW-complex, or what we shall call loosely finite topology. A C∞ noncompact manifold (all manifolds will be C∞ from now on) of course also always admits a proper C∞ function with only nondegenerate critical points, but for some manifolds all such proper functions have infinitely many critical points. (Here proper means that the inverse of each set of the form (—∞, α] is compact: this is the natural context for Morse theory.)