The projective space $\mathbb{C}P^n$ has a special metric, the Fubini-Study metric, in which projectivities are isometries. It is described by a Kähler potential in each affine chart:

where every other coefficients of the metric follows:

To compute the Christoffel symbols and other curvature coefficients may be very difficult, so we'll try an indirect approach. Instead of deriving directly, we'll compute the determinant of the metric.

This way,

where $P_B(\lambda)=\det(B-\lambda I)$ is the characteristic polynomial of $B$. But how is this characteristic polynomial? Since $B$ has rank 1, it has the eigenvalue 0 with multiplicity $n-1$, so in fact

(since the trace comes with the term of degree $n-1$), and in the end

Having found the determinant, we have the Ricci tensor and form

and in the same way

That is, the Fubini-Study metric in $\mathbb{C}P^n$ is Einstein with constant $n+1$, and in particular has constant scalar curvature