## Abstract

Our main results can be stated as follows: 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 bounded in absolute value by |K| ≤ C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. 2. Given any m and any δ > 0, there exists a positive constant i_{0}= i_{0}(m,δ) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ric ≥ δ, K ≤ 1, is bounded from below by i_{0}(m, δ). In an appendix we discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use (and usefulness) in collapsing with bounded curvature.

Original language | English (US) |
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Pages (from-to) | 736-774 |

Number of pages | 39 |

Journal | Geometric and Functional Analysis |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 1999 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology