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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:

$d(x_1, x_2) = \left[d_1\left(\vec{v}_1, \vec{v}_2\right)^2 + d_2\left(\mathbf{q}_1, \mathbf{q}_2\right)^2\right]^{1/2}$.

Alejandro Cámara
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    Either definition works, and they define the same "topology," which means that the idea of continuity from and to your space are equivalent for these two metrics. – Thomas Andrews Sep 12 '19 at 17:03
  • Umm, that is, assuming you use a better metric on $SU(2).$ Not sure if your $d_2$ works, since it doesn't return a real number. You should define $d_2(q_1,q_2)=\left|1-q_1\cdot q_2^{-1}\right|$ where $|\cdot|$ is some norm on the $2\times 2$ matrices. – Thomas Andrews Sep 12 '19 at 17:07
  • You can also use $$d(x_1,x_2)=\max\left(d_1\left(\vec{v}_1, \vec{v}_2\right),d_2\left(\mathbf{q}_1, \mathbf{q}_2\right)\right)$$ – Thomas Andrews Sep 12 '19 at 17:10
  • I'm abusing $\mathrm{SU}(2)$ as "quaternion-space", where the $\cdot$-product is the quaternion dot product. Would that not work as a metric? – Alejandro Cámara Sep 12 '19 at 17:21
  • Not sure, is $q\cdot q = 1$ for any $q\in SU(2)?$ – Thomas Andrews Sep 12 '19 at 18:13
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    https://en.wikipedia.org/wiki/Product_metric : your two definitions are the special cases of $p=1$ and $p=2$. See also the case of $p=\infty$. –  Sep 12 '19 at 18:14
  • $q\cdot q = 1$ is true for all unit quaternions. – Alejandro Cámara Sep 12 '19 at 19:05
  • @Stinking Bishop: my understanding is that the product metric is used for the Cartesian product of spaces, not the direct sum? – Alejandro Cámara Sep 12 '19 at 19:08
  • @AlejandroCámara I assumed you meant Cartesian product. Otherwise, what does a direct sum mean for a non-Abelian group such as $SU(2)$? Cf. https://math.stackexchange.com/questions/39895/the-direct-sum-oplus-versus-the-cartesian-product-times –  Sep 12 '19 at 19:29
  • Ah! I didn’t know the Cartesian product and sum were interchangeable for finite groups! Thanks for the link. Then you’re right, I can use the product metric right away :) – Alejandro Cámara Sep 12 '19 at 21:01

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