Analyticity of Covering Maps

by Lucas Schuman   Last Updated May 16, 2019 04:20 AM

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?

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