I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration.
I tried to compute on homogenous coordinates $\varphi_0:U_0\rightarrow \mathbb{C}^{n}$, where $U_0=\lbrace[z_0:\dots:z_n] :z_0\neq0\rbrace$ and $\varphi_0[z_0:\dots:z_n] =(\frac{z_1}{z_0},\dots,\frac{z_n}{z_0})$.
For this i factored the map $\varphi_0^{-1}=\pi\circ\psi$, where $\pi$ is the projection $\pi:S^{2n+1}\rightarrow \mathbb{CP}^{n}$ and $\psi:\mathbb{C}^{n} \rightarrow S^{2n+1}$ given by $\psi(z)=\frac{(1,z)}{\sqrt{1+|z|^{2}}}$ (the image of $\psi$ is a submanifold N of $S^{2n+1}$).
Then $(\varphi_{0}^{-1})^{\ast}g_{FS}=(\psi^{\ast}\circ\pi^{\ast})g_{FS}$, now $\pi^{\ast}g_{FS}$ is the hermitian metric of $\mathbb{C}^{n+1}$ induced on the submanifold N.
So i just have to compute $\psi^{\ast}(dz_{a}\otimes d\bar{z}^{a})$, and the result i get is
\begin{equation} \sum_{a,b}\frac{(1+|z|^{2})\delta_{ab} -\frac{3}{4}\bar{z}_{i}z_{j}}{(1+|z|^{2})^{2}}dz_a\otimes d\bar{z}_b, \end{equation}
but the factor $\frac{3}{4}$ should be $1$. So i ask what am i doing wrong?
The Fubini Study metric on $C\mathbb P^n$ in normal coordinates $(t, θ)$, $t$ positive real and $θ ∈ S^{2n−1}(1) ⊂ T_pC\mathbb P^n$ is given by
$$ds^2=dt^2+\frac{1}{4}\sin^2(2t)d\theta^2|_{F}+\sin^2(t)d\theta^2|_{F^\perp}$$ where where $F$ is tangent to the Hopf fibration $S^{2n−1}(1) → C\mathbb P^{n−1}$ in the tangent space and $F^⊥$ orthogonal to it
For proof see