# Isometries from Diffeomorphisms

Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?

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Fix any Riemann metric on $M$ and average it over the action of $G$. –  Ryan Budney Dec 4 '12 at 16:36
I need compactness for the Haar measure. But can't I do without connectedness? –  user48713 Dec 4 '12 at 17:03
Connectedness is irrelevant. A compact Lie group has a Haar measure. Just average any Riemann metric over $G$ wrt that measure. –  Mariano Suárez-Alvarez Dec 7 '12 at 21:10