This is again a request for references. I'd appreciate a pointer to any published proof of the following:



> 
> **Proposition.** Given $n \in \mathbb{N}^+$, let
>  $\Phi$ be a function $\mathbb{C}^n
> > \to \mathbb{C}^n$. Then $\Phi$ is an
>  isometry of
>  $(\mathbb{C}^n,\|\cdot\|)$ into itself
>  if and only if there exist a unitary $U \in
> > \mathbb{C}^{n,n}$ and an orthogonal
>  $O \in \mathbb{R}^{n,n}$ such that
>  $\Phi(z) - \Phi(0) = U\; (\Re(z) + i\; O \;
> > \Im(z))$ for all $z \in
> > \mathbb{C}^n$.
> 
> 
> 


Here $\|\cdot\|$ stands for the usual norm $\mathbb{C}^n \times \mathbb{C}^n \to \mathbb{R}: (z\_1, \ldots, z\_n) \mapsto \left(\sum\_{k=1}^n |z\_k|^2\right)$, and an isometry of $(\mathbb{C}^n,\|\cdot\|)$ into itself is a function $f: \mathbb{C}^n \to \mathbb{C}^n$ such that $\|f(z)-f(w)\| = \|z-w\|$ for all $z,w \in \mathbb{C}^n$.


**Edit.** I almost forgot! Thanks in advance for any feedback.



All the isometries of $\mathbb{C}^n$ into itself are made like these
The proposition of the posting can't be true in general, I'm afraid: e.g., the isometry group of $\mathbb{C}^2$ has (real) dimension $4+6=10$, while $\dim U(2)+\dim O(2)+4=9$. However, the isometries of $\mathbb{R}^n$ (and in particular, of $\mathbb{C}^n$) are easy to describe.


Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a map such that $d(x,y)=d(f(x,f(y))$ for all $x,y\in\mathbb{R}^n$ where $d$ is the metric induced by a positive definite quadratic form $(\cdot,\cdot)$.


Suppose $f$ preserves the origin. Then for all $x,y\in\mathbb{R}^n$ of unit length such that $(x,y)=0$ we have $(f(x),f(y))=0$, since $f(x)$ and $f(y))$ are of unit length and the distance between them is $\sqrt{2}$.


Now choose an orthogonal frame $e\_1,\ldots,e\_n$ of $\mathbb{R}^n$ and take a composition $h=g\circ f$ such that $g$ is of the form $x\mapsto Ax+b$ for some $A\in O(n),b\in\mathbb{R}^n$ and $h$ preserves the origin and any $e\_i$. If $x=\sum a\_i x\_i\in\mathbb{R}^n$ and we know $d=d(x,0)$ and all $d\_i=d(x,e\_i)$, then we can recover $a\_i$ as $a\_i=\frac{1}{2}(d^2-d\_i^2+1)$. So the map $h$ is in fact the identity, which means that $f$ is a composition of a translation and an orthogonal map.


The question of the posting is the case when $\mathbb{R}^n=\mathbb{C}^{n/2}$ and $(\cdot,\cdot)$ is the real part of the standard hermitian form.


