I met a problem: Let $R,\;S$ be two Riemann surfaces, with universal cover $\mathbb{C}$. Let $f:R\rightarrow S$ be a holomorphic function. Then it preserves the constant curvature metric up to a constant scaling factor.
However, if I take $R=S=\mathbb{C}$, then consider $f(z)=z^2$. Then the distance between 0,1 becomes 1, and distance between 0,2 becomes 4=4-0. Then it is not a scaling. Do I understand anything wrong?