Riemann Surfaces

  • William Fulton
Part of the Graduate Texts in Mathematics book series (GTM, volume 153)


A Riemann surface X is a connected surface with a special collection of coordinate charts φα: UαX. As before, Uα 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 UαβUα to UβαUβ 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.


Riemann Surface Fundamental Group Disjoint Union Half Plane Compact Riemann Surface 
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Copyright information

© Springer Science+Business Media, Inc. 1995

Authors and Affiliations

  • William Fulton
    • 1
  1. 1.Mathematics DepartmentUniversity of ChicagoChicagoUSA

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