I am trying to solve an exercise of the course of Riemannian Geometry and I have to deal with isometries of $\mathbb{P}^n\mathbb{C}$ and $\mathbb{H}^n\mathbb{C}$ (the first with the Fubini-Studi metric which I denote $ell_c$ and the second thought as the quotient: $\{ (z_0, \dots, z_n) \in \mathbb{C}^{n+1} : \sum_{i=1}^{n} |z_i|^2 - |z_0|^2 = -1 \}$). I already observed that we have: $$ \begin{array}{c} U(n+1) \subseteq Isom(\mathbb{P}^n\mathbb{C})\\ U(n,1) \subseteq Isom(\mathbb{H}^n\mathbb{C}) \end{array} $$ Now I am tempted (along the way one find the isometries of $\mathbb{S}^n$) to say those are equalities, but I am not sure. A tentatively proof I made is: given an isometry $f : \mathbb{P}^n \mathbb{C} \rightarrow \mathbb{P}^n \mathbb{C}$ let $N = [1: 0 :\dots : 0]$ and $P = f(N)$. There exists $A \in U(n+1)$ such that $A \cdot P = N$, then $f_A \circ f$ is an isometry that fixes $N$. For an isometry $g$ of $\mathbb{P}^n \mathbb{C}$ that fixes $N$ we have: $$ d_N g \in O(T_N \mathbb{P}^n \mathbb{C}, ell_c) = O(N^{\bot_{Herm}}, sph_{(1,0 \dots, 0)}) = O(\mathbb{C}^n, eucl) = U(n)$$
where the second equality comes from the fact that the Fubini-Studi metric in a point $z$ is the pushforward of the metric on the sphere $\mathbb{S}^{2n+1}$ via the differential of the projection map restricted to the subspace $z^{\bot_{Herm}} \subset T_{z} \mathbb{S}^{2n+1}$ (actually restricted to this subspace the differential of the projection is an isomorphism of vector spaces) and that the metric on the sphere is the euclidean metric restricted to the sphere. Now using that an isometry of a connected riemannian manifold is determined uniquely by the image of a point and by its differential in that point we obtain that: $$ g = f_C \quad C = \left( \begin{array}{cc} 1 & 0\\ 0 & d_Ng \end{array} \right) $$ where $f_C$ is the map associated to $C \in U(n+1)$. Coming back to $f_A \circ f_B$ we then obtain that $f_a \circ f = f_C$ where we took $g = f_A \circ f$, so $f = f_{A^{-1}C}$. Can someone tell whether this proof is correct and if it isn't which are the isometry groups of those spaces?
Thank you.