Introduction

Summary

Riemann surfaces have played a central role in mathematics ever since their introduction by Riemann in his dissertation in 1851; for a biography of Riemann, see Riemann, topology and physics, Birkhäuser, 1987 by M. Monastyrsky. Following Riemann, we first consider a Riemann surface to be the natural maximal domain of some analytic function under analytic continuation, and this point of view enables one to put the theory of ‘manyvalued functions’ on a firm foundation. However, one soon realises that Riemann surfaces are the natural spaces on which one can study complex analysis and then an alternative definition presents itself, namely that a Riemann surface is a one dimensional complex manifold. This is the point of view developed by Weyl in his classic text (The concept of a Riemann surface, Addison-Wesley, 1964) and this idea leads eventually on to the general theory of manifolds. These two points of view raise interesting questions. If we start with a Riemann surface as an abstract manifold, how do we know that it supports analytic functions? On the other hand, if we develop Riemann surfaces from the point of view of analytic continuation, how do we know that in this way we get all complex manifolds of one (complex) dimension? Fortunately, it turns out that these two different views of a Riemann surface are indeed identical.