Finiteness Theorems for Riemannian Manifolds

Abstract

1. The purpose of this paper is to show that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism classes of compact ndimensional manifolds admitting a metric for which these bounds are satisfied is finite. As an application, we show that the set of diffeomorphism classes of compact n-manifolds, (n , 4) for which some characteristic number is nonzero, and which admit an Einstein metric of nonnegative curvature, is finite. Our main tool is a theorem giving a lower bound for the injectivity radius of the exponential map. The results represent an improved version of a portion of the author's doctoral thesis and he again wishes to thank Professors S. Bochner and J. Simons for their advice and encouragement. A weaker version of part C) of Corollary 3. 3 is due independently to A.