Abstract Riemann Surfaces

This chapter presents abstract Riemann surfaces. Examples of Riemann surfaces are provided. The authors discuss the genus of a compact Riemann surface. They treat triangulation of a Riemann surface, and consider rational functions on a Riemann surface. The authors give the argument principle on a Riemann surface. The chapter also presents the Riemann-Hurwitz formula, and considers holomorphic and meromorphic differential forms. It looks at harmonic differential forms, and the authors show that every Riemann surface admits a non-constant meromorphic function.