SomeApplications of Collapsing with Bounded Curvature

In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 1. For any given numbers m, C and D, the class of m-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature |K| ≤ C and diameter ≤ D contains only finitely many diffeomorphism types. 2. Given any m and any δ > 0, there exists a positive constant i0 = i0(m, δ) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ricci curvature Ric ≥ δ, K ≤ 1, is bounded from below by i0(m, δ). I also intend to discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use and usefulness in the proof of these results. 2000 Mathematics Subject Classification: 53C. This note is about a couple of applications and variations of techniques developed in [CFG], which we found jointly with W. Tuschmann. Namely I will talk about injectivity radius estimates for positive pinching, a generalized notion of manifolds, and finiteness theorems for Riemannian manifolds with bounded curvature. The purpose of this note is to give an informal explanation of ideas in these proofs and for more details I refer the reader to [PT]. 1. Injectivity radius estimates and megafolds Is it true that positive pinching of the sectional curvatures of a simply connected manifold implies some lower positive bound for the injectivity radius, which does not depend on the manifold? For dimension = 3 this was proved by Burago and Toponogov [BT]. More generally, they proved the following: *Department of Mathematics, PSU, University Park, PA 16802, USA. E-mail: [email protected]