# 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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