### Abstract

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernández-López in the compact case.

Original language | English (US) |
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Pages (from-to) | 1803-1806 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 136 |

Issue number | 5 |

DOIs | |

State | Published - May 2008 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics