The Measure Preserving Isometry Groups of Metric Measure Spaces - Guo, Yifan (Symmetry, Integrability and Geometry: Methods and Applications, 2020)
Bochner's theorem says that if is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso() is finite. In this article, we show that if (, , ) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure-preserving isometry group Iso(, , ) is finite. We also give an effective estimate on the order of the measure-preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Émery Ricci curvature, except for small portions.