metric on the Euclidean Group

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?

Solutions Collecting From Web of "metric on the Euclidean Group"

There are several natural metrics that you can use on transformation groups. For example, for any topological group you could ask for a left or a right or a bi-invariant metric. Another natural way to define a metric is via how it acts on a space. If you really want to think about the isometry group of $\mathbb R^3$, I suppose this is the direction you’d want to go. So let $f$ and $g$ be isometries of $\mathbb R^3$. Define

$$d(f,g) = \max \{ |f(x)-g(x)| : x \in \mathbb R^3, |x| \leq 1 \}$$

This has the advantage that it’s left-invariant already. IMO this is a pretty good metric. If $f$ and $g$ fix the origin they’re linear and this is the norm of $f-g$, which is considered a fairly standard metric in linear algebra.