Finiteness Theorems for Riemannian Manifolds

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.