The isometry group of an ${\mathsf{RCD}}^{\ast }$-space is Lie

Abstract

We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD*-spaces and, under extra assumptions, by CD-spaces, CD*-spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups. More generally we show that spaces with good optimal transport properties meet as well the hypotheses.

V2:17 pages, 1 figure. Theorem 1.1 is improved now giving necessary and sufficient conditions. Little extra work is needed, which is done in the new Prop. 3.6. Previous Theorem 1.1's new name is Theorem 1.4. Additionally, slight changes in redaction to be coherent with changes. V1:16 pages, 2 figures