# Why is this map between Riemann surfaces a covering map?

In Donaldson's book

http://wwwf.imperial.ac.uk/~skdona/RSPREF.PDF

Theorem 3 of Chapter 6 asserts that, given a compact Riemann surface $$X$$ with a holomorphic 1-form with no zeros $$\omega$$, there is a lattice $$\Lambda \subset \mathbb{C}$$ and isomorphism $$\mathbb{C}/\Lambda \cong X$$ identifying $$\omega$$ with $$dz$$.

The proof starts by considering the universal covering space $$p:\tilde{X} \to X$$. We then note that there exists a holomorphic function $$F: \tilde{X} \to \mathbb{C}$$ such that $$dF = p^*\omega$$. This last equation implies in particular that $$F$$ is a local homeomorphism. The next claim is that $$F$$ is actually a covering map.

I am confused because the proof given in the book doesn't seem to show that $$F$$ is surjective. Is there a simple way of seeing that?

## 1 Answer

The number of "sheets" is locally constant for covering maps. Since $$\mathbf C$$ is connected, it follows that $$\#(F^{-1}(z))$$ is the same for all $$z\in \mathbf C$$. Hence $$F$$ is surjective.

Note that the definition of the book of covering map doesn't require the map to be surjective.