Collapsing vs. Positive Pinching

Abstract

Let M be a closed simply connected manifold and 0 < $ \delta \le 1 $ . Klingenberg and Sakai conjectured that there exists a constant $ {i_0} = {i_0}(M,\delta) > 0 $ such that the injectivity radius of any Riemannian metric g on M with $ \delta \le {K_g} \le 1 $ 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 $ {i_0} = {i_0}(M,{d_0},\delta) > 0 $ , such that the injectivity radius of any $ \delta $ -pinched d 0-bounded Riemannian metric g on M (i.e., $ {\rm dist}_g \le d_0 $ and $ \delta \le K_g \le 1) $ 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.