# Fubini-Study in spherical coordinates.

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?

• What do you mean by spherical coordinates?
– Danu
Oct 18, 2016 at 21:42
• I mean the 4 real angular coordinates appearing in the round metric of $S^4$ Oct 20, 2016 at 14:16
• 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?
– Danu
Oct 20, 2016 at 17:03
• 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. Oct 20, 2016 at 17:12
• 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/… Jan 19, 2017 at 12:04