Riemann Surfaces

Abstract

A Riemann surface X is a connected surface with a special collection of coordinate charts φ_α: U_α→X. As before, U_α is a subset of ℝ² but now we identify ℝ² with the complex numbers ℂ. The requirement to be a Riemann surface is that the change of coordinate mappings φ _βα from U_αβ ⊂ U_α to U_βα ⊂ U_β are not just C^∞, but they must also be analytic, or holomorphic. Recall (see §9d) that an analytic function f on an open set in ℂ is a complex-valued function that is locally expandable in a power series, i.e., at each point z₀ in the open set, there is a power series $\Sigma _{{n = 0{\kern 1pt} }}^{\infty }{{a}_{n}}{{(z - {{z}_{0}})}^{n}}$ that converges to f(z) for all z in some neighborhood of z₀. As before, another atlas of charts is compatible with a given one (and defines the same Riemann surface) if the changes of coordinates from charts in one to charts in the other are all analytic.