# $\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth.

I found that as a sidenote somewhere. I really would like to see a prove of the above. So if somebody happens to know an online (free accessible) source, please tell me. I know $\operatorname{Isom}{(M)}$ is locally compact w.r.t. the compact-open topology. Is this also the topology of the Lie-Group?

-
Certainly this doesn't even come close to answering the question, but you should think about the fact that the Lie algebra will be $\Gamma(TM)$, the vector fields on $M$ (if you haven't already). –  Aaron Mazel-Gee May 1 '12 at 14:47