Introducción a superficies de Riemann: fórmula de Riemann-Hurwitz

Abstract

Riemann surfaces are complex manifolds of dimension 1. An analytical approach to the study of Riemann surfaces is motivated by the identification of these geometrical objects with the complex plane, in such a way that the classical results of complex analysis apply in this context. However, an algebraic approach becomes more prominent when we talk about describing and classifying these objects, and is primarily motivated by algebraic geometry. In this sense, we study Riemann surfaces in order to understand a first approach of this concept. We make a considerable number of classic examples explicit together with the respective results. We also study the algebraic properties of these objects, how they are related to each other, and some very important invariants.