Theorems \[heb-vaug\] and \[main2\]: In the context of metric-mesure spaces arising from Riemannian manifolds, the group $H$ in Theorem \[main2\] is a subgroup of $\text{Diff}_{V_g}(X)$, the group of volume preserving diffeomorphisms of $(X,g)$; in Theorem \[heb-vaug\] the group $H$ is a compact subgroup of the smaller group $\text{Isom}(X,g)$. A classical result of S. Myers and N. Steenrod, see [@kobayashi], provides $\text{Isom}(X,g)$ with the structure of a finite dimensional Lie group, that is compact if $X$ is compact. In contrast, if $X$ is compact, H. Omori provided the larger group $\text{Diff}_{V_g}(X)$, and $\text{Diff}(X)$ as well, with the structure of an Inverse Limit of Hilbert manifolds, see [@ebin; @kobayashi]. The Lie algebras of both groups are *represented* by vector fields: those in the *formal* Lie algebra of $\text{Diff}_{V_g}(X)$ are free of divergence; those in the Lie algebra of $\text{Isom}(X,g)$ are Killing, a stronger condition. In every Riemannian manifold there are non-trivial vector fields free of divergence; on the other hand, the sign of the Ricci curvature imposes restrictions for Killing vector fields: if the Ricci tensor is non-positive and negative definite at some point, there are no non-trivial Killing vector fields, and the group $\text{Isom}(X,g)$ is finite [@Berard; @kobayashi].

2.  Concerning the 