The Type of a Riemann Surface
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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).
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- 1.L. Ahlfors . I am indebted to aletter from Mr. Ahlfors for the presentation given in the text.Google Scholar
- 1.Cf. L. Ahlfors .Google Scholar
- 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
- 1.This visualization of Kobayashi’s theorem I likewise owe to a communication from Mr. Ahlfors.Google Scholar
- 1.This theorem was first proved by the author  in a way whose natural extension the technique of Kobayashi can be considered.Google Scholar
- 2.For surfaces with nothing but algebraic branch points there is a closely related theorem of Ahlfors . In this regard also cf. A. Speiser , , E. Ullrich .Google Scholar
- 3.P. J. Myrberg . For newer results concerning the type problem cf. H.Wittich , , O. Teichmüller , F. E. Ullrich , R. Nevanlinna , Le-Van  (in the last article one finds an extensive bibliography on the type problem), C. Blanc , , Z. Kobayashi , P. Laasonen .Google Scholar