2003/04/18 by Anton Petrunin
Mathematics · #math.DG #msc:53C
published as Proceedings of the ICM, Beijing 2002, vol. 2, 315--322
arxiv created 2003/04/18 · arxiv updated 2009/11/30
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 \vert K \vert≤ 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.