The graph Sis a very simple example of a (concrete, non-singular) Riemann surface. Thus, the basic idea of Riemann surface theory is to replace the domain of a multi-valued function, e.g. a function de ned by a polynomial equation P(z;w) = wn+ p n 1(z)wn 1 + + p 1(z)w+ p 0(z) by its graph S= f(z;w) 2C2 jP(z;w) = 0g;

A Riemann surface is the datum of the topological space X together with the atlas (U i, f i) i2I. In particular, it is possible to have two riemann surfaces with the same underlying topological spaces but different atlases. Exercise 1.6. Show that Cˆ, the Riemann sphere (the one point com-pactification of C, whose underlying set is C[f¥gand ...

2025年1月31日 · A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99).

Riemann Surfaces provide a nice way to visualize multiple valued functions in the complex plane. In fact: on its Riemann Surface a multiple valued function is single valued, it is only when the Riemann Surface is projected on the Complex Plane that the multiple values arise.

In this chapter we will define (complex) algebraic curves (rep-resented by “C”),1 complex 1-manifolds (represented by “M”), and Riemann surfaces, and start to consider under what additional hy-potheses they are equivalent concepts. 2.1. Algebraic curves. 2.1.1. DEFINITION.

At this point we introduce the three most important Riemann surfaces, and illustrate the primary method by which one can construct other more complicated Riemann surfaces. It is immediately obvious from the de nition that the complex plane C is a Riemann surface; the atlas containing the single chart, the identity map, is analytic.

A Riemann surface is a topological surface with an atlas of charts to C, for which the transition functions are holomorphic. Constructions.

2018年2月9日 · The term “Riemann surface” is often used in a narrower sense. A Riemann surface in the narrower sense is a branched covering of the complex plane. That means that we have a one-dimensional complex manifold together with a projection map from a dense open subset of that manifold to the complex plane.

2022年1月6日 · A Riemann surface (with boundary) is a triangulable and orientable manifold with a countable base and, hence, it is separable and metrizable.

We present three aproaches to the uniformization of Riemann surfaces. We first prove the uniformization theorem via the construction of a harmonic function by the Dirichlet principle. We then give an alter-nate proof by triangulating the surface …