Riemann Surfaces

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

Abstract

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.

Keywords

Posite Dinate 

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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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