# On Poicarés Uniformization Theorem

The study of Riemann surfaces begun in the 19th century when Riemann introduced them in his doctoral dissertation Foundations for a general theory of functions of a complex variable. An important perspective in the study of Riemann surfaces is the concept of uniformization, which was developed by Poincar´e, Klein and others.The theory states that every closed orientable Riemann surface admits a Riemann metric of constant curvature. The main geometries are the sphere, the Euclidian plane and the hyperbolic plane with curvature 1, 0 and −1 respectively.

Contents

1 Introduction
1.1 Background
1.2 Chapter outline
2 General Setting
2.1 The Extended Complex Plane
2.1.1 The Riemann Sphere
2.1.2 Mobius Transformations
2.2 Surfaces
2.2.1 Riemann Surfaces
2.2.2 Automorphisms
2.3 The Hyperbolic Plane
2.4 Fuchsian groups
3 Covering Maps
3.1 Fundamental Groups
3.2 Group Actions on Surfaces
3.3 Universal Coverings
4 Riemann Surfaces as Orbifolds
4.1 2-Orbifolds
4.2 Branched Coverings
4.2.1 Universal Branched Coverings
5 Poincar´es Theorem
5.1 Fundamental Domains
5.2 The Quotient Space H/Γ
5.3 Poincar´es Theorem

Author: Bartolini, Gabriel

Source: Linköping University

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