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I don't quite understand some work I'm doing creating the normal Riemann surface for the function ##f(z)=\frac{A}{\sqrt{(1-z^2)(k^2-z^2)}}##. I can use Schwarz-Christoffel transforms to map the function to a rectangular polygon in the zeta-plane then map this rectangle onto a torus. But I don't understand what becomes the atlas to make the surface of the torus a manifold. I was wondering if someone could help me understand this.
Thanks
Thanks