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There is a well-known result that pullback of Fubini-Study form on $\mathbb{C}P^1$ under stereo graphic map gives one quarter of the standard metric on $S^2$, for example see exercise 3.2 (Ⅴ) on page 3 of this note.

However, when I was trying to prove this I got exactly the same result as this question and I can't see where the process went wrong. This is not pointed out in the answer of that question. Can anybody tell me where I went wrong on this?

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Hint: Instead of using the stereographic projection on $\Bbb{C}P^1$ use it on $S^2$. So one has the following map $$\phi : \Bbb{R}^2 \to \Bbb{S}^2 \\ \phi(x_1,x_2) = (\frac{2x_1}{1+x_1^2 + x_2^2}, \frac{2x_2}{1+x_1^2 + x_2^2}, \frac{-1+x_1^2 + x_2^2}{1+x_1^2+x_2^2})$$ Now pullback the standard 2 form on the sphere to $\Bbb{R}^2$

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