I'm working on my thesis on the Hopf-Fibration. To discuss the $S^3\rightarrow S^2$ variant, I'm using Quaternions over matrices. During the last review, my advisor said I should probably justify why.
In particular, he said that
we would like to parameterize the $3$-d Manifold $SO(3)$ with $3$ reals, but this is not possible globally.
I on the one hand, $SO(3)$ is in fact $3$ dimensional, as (in matrix representation)
- the first column $p$ must be of unit length, i.e., from $S^2$
- the second column must be orthogonal to $p$, and of unit length, i.e., from $S^1$
- the last column is already determined fully, as the determinant must be $1$.
However, by looking at this argument, we can't find a global parameterization for the first two columns, as $S^2$ and $S^1$ are closed, but $\mathbb R^2$ and $\mathbb R^1$ are open.
Is there perhaps a more concise argument as to why $SO(3)$ is closed, which would show that the global parameterization with three reals is impossible?
$SO(3)$ is a closed subset of $\mathbb R^9 = (\mathbb R^3)^3$ because its defining equation $M M^{T} = I$ is a continuous function $\mathbb R^{9} \mapsto \mathbb R^9$ whose image is a closed subset, namely the one point subset $\{I\} \subset \mathbb R^9$.