### 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) |
---|---|

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 |

### Fingerprint

### Keywords

- Asymptotic ray
- Line
- Ricci curvature

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*120*(4), 1231-1239. https://doi.org/10.1090/S0002-9939-1994-1186984-2

**Local splitting theorems for riemannian manifolds.** / Cai, MingLiang; Galloway, Gregory J; Liu, Z.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 120, no. 4, pp. 1231-1239. https://doi.org/10.1090/S0002-9939-1994-1186984-2

}

TY - JOUR

T1 - Local splitting theorems for riemannian manifolds

AU - Cai, MingLiang

AU - Galloway, Gregory J

AU - Liu, Z.

PY - 1994

Y1 - 1994

N2 - 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.

AB - 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.

KW - Asymptotic ray

KW - Line

KW - Ricci curvature

UR - http://www.scopus.com/inward/record.url?scp=84966237738&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966237738&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1994-1186984-2

DO - 10.1090/S0002-9939-1994-1186984-2

M3 - Article

AN - SCOPUS:84966237738

VL - 120

SP - 1231

EP - 1239

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -