## Abstract

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold M with a base point p ∈ M whose radial curvature at p is bounded from below by that of a non-compact model surface of revolution M which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point p ∈ M. Notice that our model M does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor's open conjecture on the fundamental group of complete open manifolds.

Original language | English |
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Pages (from-to) | 6293-6324 |

Number of pages | 32 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2010 |

Externally published | Yes |

## Keywords

- Cut locus
- Geodesic
- Radial curvature
- Toponogov's comparison theorem
- Total curvature

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics