Complete shrinking ricci solitons have finite fundamental group

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63 Scopus citations


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 languageEnglish (US)
Pages (from-to)1803-1806
Number of pages4
JournalProceedings of the American Mathematical Society
Issue number5
StatePublished - May 2008
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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