The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)

So, is every topological group isomorphic (in the category of topological groups) to the isometry group of a metric space?

And what about the differentiable version: Is every Liegroup isomorphic (in the category of Liegroups) to a isometry group of a Riemann manifold?