I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for (homogenous) transformation group $\mathrm{SE}(3)$. For the latter I have seen use of Riemannian metric. For both I am guessing (as this is not my field) that there are some kind of manifold structure involved, so I can't help to think that there are also good metrics (by good I mean metric that is considered "practical" when one considers rigid body motions).
Furthermore, is there a way to visualize the manifolds from $\mathrm{E}(3)$ and $\mathrm{SE}(3)$? Any good reference in this area?