Complete Shrinking Ricci Solitons have Finite Fundamental Group

2007/04/03 by William Wylie, Wylie, William
Mathematics · Physics and Astronomy · #53C20 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0704.0317

openalex publication_date 2007/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Fernandez-Lopez in the compact case.

Citations

• Some Geometry and Analysis on Ricci Solitons

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