# The Type of a Riemann Surface

Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)

## Abstract

We restrict ourselves at first to the class considered in XI, § 2, of simply connected Riemann surfaces F q whose branch points project onto a finite number q of image points w 1, .... w q . These surfaces can be represented concretely by the line complexes defined in the place just mentioned. To each half sheet of F q there corresponds a node point in the line complex. By means of the line segments connecting the node points, the plane is decomposed into elementary regions which correspond one-to-one to the branch points of F in such a way that one elementary region with 2m sides is associated with a branch point of order m — 1. Corresponding in the complex to the sheets which are unramified over the base points are two-sided figures (double 1ine segments).

## References

1. 1.
L. Ahlfors [9]. I am indebted to aletter from Mr. Ahlfors for the presentation given in the text.Google Scholar
2. 1.
3. 1.
This result is obvious for a = 0. If a ≠ 0 one has to transform w by the rotation of the sphere that takes a into 0.Google Scholar
4. 1.
This visualization of Kobayashi’s theorem I likewise owe to a communication from Mr. Ahlfors.Google Scholar
5. 1.
This theorem was first proved by the author [8] in a way whose natural extension the technique of Kobayashi can be considered.Google Scholar
6. 2.
For surfaces with nothing but algebraic branch points there is a closely related theorem of Ahlfors [4]. In this regard also cf. A. Speiser [2], [3], E. Ullrich [3].Google Scholar
7. 3.
P. J. Myrberg [2]. For newer results concerning the type problem cf. H.Wittich [7], [8], O. Teichmüller [5], F. E. Ullrich [1], R. Nevanlinna [15], Le-Van [2] (in the last article one finds an extensive bibliography on the type problem), C. Blanc [1], [2], Z. Kobayashi [2], P. Laasonen [1].Google Scholar