I am unable to understand difference between local co-ordinate and coordinate charts for a Riemann surface. And unable to prove the following:
Let $f:X\to Y$ be a non constant holomorphic map between two compact Riemann surfaces. Take any local co-ordinate $z$ near $p$ and $w$ near $f(p)$ ; say $p$ corresponds to $z_0$ and $f(p)$ corresponds to $w_0$ . In terms of these coordinate the map $f$ may be written as $w=h(z)$ where $h$ is holomorphic and $h(z_0)=w_0$ . How is this happening?
Someone please explain this precisely. Thanks in advance.