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 effective estimate on the order of measure preserving isometry group for compact weighted Riemannian manifold with an integral bound on Bakry-Emery Ricci curvature.