### Abstract

In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold M has nonnegative Ricci curvature outside a compact set B and contains a line γ which does not intersect B, then the line splits in a maximal neighborhood that is contained in M \ B. We use this result to give a simplified proof that M has a bounded number of ends. We also prove that if M has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood U of a geodesic γ which is a line in U, then U splits along γ.

Original language | English (US) |
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Pages (from-to) | 1231-1239 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 120 |

Issue number | 4 |

DOIs | |

State | Published - 1994 |

### Keywords

- Asymptotic ray
- Line
- Ricci curvature

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Cai, M., Galloway, G. J., & Liu, Z. (1994). Local splitting theorems for riemannian manifolds.

*Proceedings of the American Mathematical Society*,*120*(4), 1231-1239. https://doi.org/10.1090/S0002-9939-1994-1186984-2