An introduction to Riemann surfaces

Abstract

In this document we give a first view to Riemann surface theory. Starting from definition and examples in chapter I, in the next chapter one sees the relation between the well known oriented smooth surfaces and this new object, with the result that any oriented smooth surfaces is equivalent to a Riemann surface. In order to prove this result, almost-complex structures and isothermal coordinates (between others) are explained, and the key point is the existence of these isothermal coordinates for a smooth surface as we will see. Finally in Chapter III we stablish the relation between Riemann surfaces and algebraic curves. First we construct a Riemann surface from a polynomial, which is relatively easy, and then we give and prove the Main Theorem for Riemann surfaces using Hilbert space techniques and some tools like the Riesz Representation Theorem. This Main Theorem is the key to prove the existence of meromorphic functions on a Riemann surface and the fact that any compact Riemann surface arises from a polynomial.