Wiki says that the euclidean group: http://en.wikipedia.org/wiki/Euclidean_group is a topological group. Can you explain me what is the topology we take on it?
Thanks !
Wiki says that the euclidean group: http://en.wikipedia.org/wiki/Euclidean_group is a topological group. Can you explain me what is the topology we take on it?
Thanks !
Topologically the euclidean group is the product $O_n(\mathbb R)\times \mathbb R^n$ (but beware that the group structures are different: the euclidean group is a semi-direct product of $O_n(\mathbb R)$ and $\mathbb R^n$, not the direct product.)
Of course $ \mathbb R^n$ has its usual metric structure and $O_n(\mathbb R)$ has the induced topology from the inclusion $O_n(\mathbb R)\subset M_n(\mathbb R)\cong \mathbb R^{n^2}$.