### Abstract

Let M be a closed simply connected manifold and 0 < δ ≤ 1. Klingenberg and Sakai conjectured that there exists a constant i_{o} = i_{o}(M, δ) > 0 such that the injectivity radius of any Riemannian metric g on M with δ ≤ K_{g} ≤ 1 can be estimated from below by i_{o}. 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_{o} on M, there exists a constant i_{o} = i_{o}(M, d_{o}, δ) > 0, such that the injectivity radius of any δ-pinched d_{o}-bounded Riemannian metric g on M (i.e., dist_{g} ≤ d_{o} and δ ≤ K_{g} ≤ 1) can be estimated from below by i_{o}. 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.

Original language | English (US) |
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Pages (from-to) | 699-735 |

Number of pages | 37 |

Journal | Geometric and Functional Analysis |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1999 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

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## Cite this

*Geometric and Functional Analysis*,

*9*(4), 699-735. https://doi.org/10.1007/s000390050100