Consider the space $\mathbb{R}^3\oplus \mathrm{SU}(2)$.
How do you construct a metric for it using the metrics for the subspaces?
I'm considering using something like: $d(x_1, x_2) = d_1\left(\vec{v}_1, \vec{v}_2\right) + d_2\left(\mathbf{q}_1, \mathbf{q}_2\right)$, where:
- $x_i \equiv \vec{v}_i\oplus\mathbf{q}_i$
$d_1\left(\vec{v}_1, \vec{v}_2\right) \equiv ||\vec{v}_1 - \vec{v}_2||$ in $\mathbb{R}^3$, and
$d_2\left(\mathbf{q}_1, \mathbf{q}_2\right) \equiv 1 - \mathbf{q}_1\cdot\mathbf{q}_2$
But I don't know how to go about proving that's a bad idea.
My second approach would be to try something along the lines of:
$d(x_1, x_2) = \left[d_1\left(\vec{v}_1, \vec{v}_2\right)^2 + d_2\left(\mathbf{q}_1, \mathbf{q}_2\right)^2\right]^{1/2}$.