Is it possible to write the Fubini-Study metric on $\mathbb{CP}^2$ in terms of spherical coordinates? How does such a metric look like?

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    $\begingroup$ What do you mean by spherical coordinates? $\endgroup$
    – Danu
    Oct 18, 2016 at 21:42
  • $\begingroup$ I mean the 4 real angular coordinates appearing in the round metric of $S^4$ $\endgroup$ Oct 20, 2016 at 14:16
  • $\begingroup$ That still doesn't make sense to me. Can you explain how you intend to use coordinates on $S^4$ for the complex projective plane? $\endgroup$
    – Danu
    Oct 20, 2016 at 17:03
  • $\begingroup$ Consider the second line of equation (4,4) here. arxiv.org/pdf/0801.1053.pdf This is claimed to be the Fubini-Study metric, written in terms of 4 angular coordinates $\sigma$, $\theta$, $\phi$, $\beta$. My question is what is exactly the change of coordinates which brings the the Fubini Study metric defined in en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric, and those 4 real angular coordinates. $\endgroup$ Oct 20, 2016 at 17:12
  • $\begingroup$ The change of coordinates is a bit involved, since you have to take into account the projective nature of your space. In particular, I'll do the example with $\mathbb{CP}^1$. The change of coordinates it's the stereographic projection, more details here: en.wikipedia.org/wiki/… $\endgroup$
    – Oscar
    Jan 19, 2017 at 12:04


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