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This is a question coming from a physics background. Consider the surface of the unit sphere. For each point on this unit sphere I have an associated angle $\theta$. This would be familiar as the $SU(2)$ group. Now, there are points on this group which are "equivalent". These points are all the points on the equatorial arc and two orthogonal longitudinal arcs. And only when $\theta \in n\pi$ where $n \in \mathbf{Z} $.

I want to define a good metric in this space. The metric must be phase-agnostic, meaning $ dist(\mathbf{U_1},\mathbf{U_2}) = 0$, if $\mathbf{U_1} = e^{i\phi} \mathbf{U_2}$.

I would like to learn the details on this, any references and textbooks will work for me.

Thanks

Pratik Patnaik
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  • What are your equivalence classes, for instance, for a given $p$ on the sphere, what is ${q \in S | \theta(q) = \theta(p)}$? – Paul Tanenbaum Oct 15 '24 at 19:18
  • It's not clear to me what you're looking for. Are you looking for something like an $SU(2)$-invariant metric on the quotient $SU(2) / U(1)$ viewed as the $2$-sphere? – Travis Willse Oct 15 '24 at 19:21
  • @TravisWillse Yes, It would be an SU(2)-invariant metric on SU(2)/Center(SU(2)) – Pratik Patnaik Oct 15 '24 at 19:58
  • So the answer to @Travis's question suggests that we want the Fubini-Study metric on $\Bbb CP^1 (= S^2)$. You can find that discussed here, on wiki, and in numerous books. – Ted Shifrin Oct 15 '24 at 20:16
  • @PaulTanenbaum Sorry I misinterpreted your question. I am not sure if there are equivalence classes in SU(2) represented this way. – Pratik Patnaik Oct 15 '24 at 20:19
  • See https://math.stackexchange.com/questions/19712/the-action-of-su2-on-the-riemann-sphere – Travis Willse Oct 15 '24 at 20:23

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