The Measure Preserving Isometry Groups of Metric Measure Spaces

Abstract

Bochner's theorem says that if 𝑀 is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso(𝑀) is finite. In this article, we show that if (𝘟, 𝘥, 𝑚) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure-preserving isometry group Iso(𝘟, 𝘥, 𝑚) 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.