I am looking for conditions on $f$ such that $f:S^2 \to \mathbb C$ is such that the image, $f(S^2)$ is a Riemann surface. Must $f$ be analytic, or something stronger? Or are there no simple conditions that we can impose such that this is true?
I am an undergraduate student in maths and have just been introduced to manifolds etc. and don't know anything about homology yet. If there is a simple proof of the answer above, it would be very useful if I could see it. Otherwise, if you know a place where I can find a proof of it, that would be useful also.
This question came up when I was trying to find the Riemann surface of $\sqrt z$ and I needed to prove that the result (two spheres identified along a meridian) was indeed a Riemann surface. I'm now looking at more complicated examples, where I don't know if I can prove that they're Riemann surfaces, without using a result like the one I'm asking for.