The Geometry of Möbius Transformations and an Introduction to Riemann Surfaces

We explore the geometric properties of Möbius transformations as mappings of the extended complex plane and use this viewpoint to introduce the idea of a Riemann surface. We begin by discussing the key properties of Möbius transformations and emphasize the geometric viewpoint. In many places, we will avoid standard analytical proofs in a favor of geometric and constructive proofs. Then, we provide an introduction to the concept of a Riemann surface and show how subgroups of the general group of Möbius transformations generate Riemann surfaces via quotient spaces. The final goal is to explain the idea of the Uniformization Theorem and provide examples of various Riemann surfaces including compact surfaces of genus one.