The [Fubini-Study metric](https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric#In_local_affine_coordinates) $g:=g\_{FS}$ is the unique $U(n+1)$-invariant
Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space
which by $U(n+1)$-invariance can be wlog definined on
tangent bundle over affine chart
$ U\_0 :=\{Z \in \mathbb{CP}^n \ \vert
z\_0 \neq 0 \} \subset \mathbb{CP}^n $ and then it extends uniquely by $U(n+1)$-action
to all affine pieces $U\_i$.


So we can restict our considerations to tangent space $T\_Z \mathbb{CP}^n$ over a point


$$Z =[1:z\_1:....z\_n] \in U\_0 :=\{Z \in \mathbb{CP}^n \ \vert
z\_0 \neq 0 \} \subset \mathbb{CP}^n$$


Let $\{\partial \_{1},\ldots ,\partial \_{n}\}$ the be frame of tangent space  

$T\_Z \mathbb{CP}^n$ at $Z$, ie the canonical $\mathbb{C}$-basis of
$T\_Z \mathbb{CP}^n$. Then the Fubini Study metric is defined by


$$\tag{FS} g\_{i{\bar {j}}}=h(\partial \_{i},{\bar {\partial }}\_{j})=
{\frac {(1+|\mathbf {z} |^{2})\delta \_{i{\bar {j}}}-
{\bar {z}}\_{i}z\_{j}}{(1+|\mathbf {z} |^{2})^{2}}} $$


At first glance one can think that $g$ somehow "has hallen from heaven"
but it is well knows that it's not.
One way to derive it is using Kähler potentials and canonical
choice of a distinguished connection: Chern connection: See [Here](https://math.stackexchange.com/questions/3748449/the-first-chern-class-of-o1-is-positive)
the excellent answer by Arctic Char.


The concern of *this* question is to find out if it also possible to derive $g$ on
$\mathbb{CP}^{n}$ naturally by recognizing $(\mathbb{CP}^{n},g)$ as
*Riemannian submersion* of $(\mathbb{C}^{n+1} \backslash \{0\},h)$
where $h$ is to restriction of standard Hermitian metric on
$\mathbb{C}^{n+1}$ to $\mathbb{C}^{n+1} \backslash \{0\}$.


What do I mean by 'naturally"?


Well, $\mathbb{CP}^{n}$ can be considered as submersion of complex manifold
$\mathbb{C}^{n+1} \backslash \{0\}$ via canonical projection


$$\pi: \mathbb{C}^{n+1} \backslash \{0\} \to \mathbb{CP}^n$$


The complex manifold is moreover Hermitian since
$ \mathbb{C}^{n+1} \backslash \{0\}$ inherits naturally the standard
Hermitian metric $h$ on $ \mathbb{C}^{n+1} $.
Futhermore by $\pi$ the $\mathbb{CP}^{n}$ can also be considered as
quotient $ \mathbb{C}^{n+1} \backslash \{0\} / \mathbb{C}^\*$.


By definition a Riemannian submersion $F: (M,g\_M) \to (N, g\_N)$
is a
submersion from one Riemannian manifold $M$ with Riemannian
metric $g^M$ to another Riemannian manifold $N$ with Riemannian
metric $g^N$
that respects the metrics, meaning that it is an
orthogonal projection on tangent spaces.


Namely that means that for every $x \in M$ and $y=F(X)$
the restriction of the differential map $dF\_x$ to
$ker(dF\_x)^{\perp} \subset T\_x M $ gives
an isometry $d\_x F: ker(dF\_x)^{\perp} \to T\_y N$
such that the metric $g^N$ is induced by $g^M$ is following sense:


Let $A, B \in T\_y N$ and denote by $\overline{A}, \overline{B}
 \in ker(dF\_x)^{\perp}$ their unique preimages with respect the
isometry $d\_x F: ker(dF\_x)^{\perp} \to T\_y N$.


Then


$$\tag{RieSub} g^N \_y(A,B)= g^M \_x(\overline{A}, \overline{B})$$


**QUESTION:**


How can be proved that the Fubini-Study metric $g$ arises exactly on this way? ie that
the canonical projection


$\pi: \mathbb{C}^{n+1} \backslash \{0\} \to \mathbb{CP}^n$


extends naturally to Riemannian submersion
$\pi: (\mathbb{C}^{n+1} \backslash \{0\},h) \to 
(\mathbb{CP}^n, g)$


is the sense above, where $h$ is the restriction of the
stadard Hermitian metric on $\mathbb{C}^{n+1}$ and
$g$ is the Fubini- Study metric defined above in (FS).


In other words we need to verify that the metrics $h$ and $g$ are related
to each other as in (RieSub).


Take as $y= Z =[1:z\_1:....z\_n]$ and $x =(1,z\_1,..., z\_n) \in 
\mathbb{C}^{n+1}$ and $\partial \_{i}, \partial \_{j} \in 
T\_y \mathbb{CP}^n$ element of the canonical basis of the tangent space
at $y$. Firstly we need an explicit identification between the tangent space
$T \mathbb{CP}^n$ and $ker(d \pi)^{\perp}$. We need this isometry explicitly
since in order to prove (FS) we have to work with unique preimages of elements
in $T\_y \mathbb{CP}^n$ living in $ker(d \pi\_x)^{\perp}$.


But I don't see how can I show that indeed


$$g(\partial \_{i},{\bar {\partial }}\_{j})= 
{\frac {(1+|\mathbf {z} |^{2})\delta \_{i{\bar {j}}}-
{\bar {z}}\_{i}z\_{j}}{(1+|\mathbf {z} |^{2})^{2}}} $$


holds. On the left side we have already
identified $\partial \_{i}, \partial \_{j} \in 
T\_y \mathbb{CP}^n$ with their unique preimages in
$ker(d\_x \pi)^{\perp}$.



Fubini-Study metric induced by submersion
Let me start from scratch. Note that everything below uses only the definition of complex projective space and the natural Hermitian inner product on $\mathbb{C}^{n+1}$. Also, the construction is coordinate-independent in the sense that everything below can be done with an abstract complex vector space with a Hermitian inner product without using any basis.


Recall that canonical projection map is defined to be
\begin{align\*}
\mathbb{C}^{n+1}\backslash\{0\} &\rightarrow \mathbb{C}P^n\\
z &\mapsto [z],
\end{align\*}
where $[z] = \{ tz\ :\ t \in \mathbb{C} \}$. For each $z \ne 0$, the tangent space of $\mathbb{C}^{n+1}$ at $z$ splits naturally
$$
T\_z\mathbb{C}^{n+1} = [z] \oplus z^\perp,
$$
where $z^\perp = \{ w \in \mathbb{C}^{n+1}\ :\ \langle w,z\rangle = 0 \}$. For each $z$, the pushforward map $\pi\_\*: T\_z\mathbb{C}^{n+1} \rightarrow T\_{[z]}\mathbb{C}P^n$ is an isomorphism if restricted to $z^\perp$.


We want to use the Hermitian inner product to define a Kähler metric $g$ on $\mathbb{C}P^n$. Observe that, given such a metric, $\pi^\*g$ is an degenerate Hermitian $2$-tensor $h$ on $T\_z\mathbb{C}$ that is scale invariant in the sense that if $R\_t(z) = tz$, then $(R\_t)^\*h = h$. Conversely, a Hermitian metric on $\mathbb{C}^{n+1}$ defines one on $\mathbb{C}P^n$ only if its restriction to each $z^\perp$ is scale invariant.


If we want to define the metric using only the standard Hermitian inner product pointwise on $\mathbb{C}^{n+1}$, then the only possibility is one of the form
$$
g = f(|z|^2)|dz|^2
$$
restricted to $z^\perp$, for each $z \in \mathbb{C}^{n+1}\backslash\{0\}$. On the other hand, if you work out all the definitions carefully, you find that
$$
R\_t^\*g = f(|t|^2|z|^2)|t|^2|dz|^2,
$$
which equals $g$ if and only if there exists a real constant $c$ such that $f(|z|^2) = c|z|^{-2}$. By the observations above, this metric can be pushed down to a metric on $\mathbb{C}P^n$, and it can be verified that it is, up to a scale factor, the Fubini-Study metric.


Note that the explanation above is just the definition of a Riemannian submersion but for the specific situation here.


