This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of One Complex Variable. I've picked up Donaldson's book on Riemann surfaces, and the following exercise has me a bit confused. It says:

show that the set of points $(w,z) \in \mathbb C^2$ satisfying the equation $w^2= \sin (z) $ is a Riemann surface.

I know the definition of a Riemann surface, and I recognize the similarity of this definition to smooth manifolds from my experience with geometry (in fact I have already proven that every Riemann surface is an orientable smooth manifold).

I guess my problem here is that I don't understand what's being asked of me, because I don't see how to come up with charts on this space. One of my first inclinations was to think of it as a function which takes $z$ as inputs and gives two values (the square roots of the $w$'s) as outputs.

I can see how this defines some kind of manifold then, it looks like maybe it has two sheets locally homeomorphic to the plane. I guess I could take the disks in the plane to be the chart domains, but then I don't see how to come up with overlap maps. Is this the right line of thinking at all?

I'm sorry if this is a really crappy question. I've been working with manifolds for about a month now and I should probably have a handle on this. Just something about this question is really giving me a hard time.

  • $\begingroup$ Perhaps you want a holomorphic version of the implicit function theorem. You can derive it from the usual smooth version and an an application of $\partial/\partial\bar z_j$. $\endgroup$ – Ted Shifrin Jul 14 '15 at 1:40
  • $\begingroup$ @TedShifrin That's a really good idea! I didn't even think of that - but now that you say it it feels natural. I'll work on this approach for a little bit and update the question probably tomorrow. $\endgroup$ – Alfred Yerger Jul 14 '15 at 2:54

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