Collapsing vs. Positive Pinching

Let M be a closed simply connected manifold and 0 < . Klingenberg and Sakai conjectured that there exists a constant such that the injectivity radius of any Riemannian metric g on M with can be estimated from below by i ₀. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the Klingenberg-Sakai conjecture: Given any metric d ₀ on M, there exists a constant , such that the injectivity radius of any -pinched d ₀-bounded Riemannian metric g on M (i.e., and can be estimated from below by i ₀. We also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on M with positively uniformly pinched curvature cannot converge to a metric space of strictly lower dimension. Submitted: October 1998, revised: December 1998, final version: May 1999.