Let $X,Y$ be Riemann surfaces and $\pi:X\longrightarrow Y$ be an $n$-sheeted covering map. Suppose that $\pi$ is not given explicitly. Under what conditions can it be deduced that $\pi$ is a local biholomorphism?
Alternatively, let $U\subset X$ be a single sheet of $\pi$. Under what conditions is $\pi\mid_U$ holomorphic?