Type: Article
Publication Date: 1956-01-01
Citations: 10
DOI: https://doi.org/10.1090/s0002-9947-1956-0094450-1
This paper contains results of a somewhat varied nature, all obtained from a detailed examination of a special class of surfaces. These results include a method for the isometric embedding of Riemann surfaces in space, an application of this method for the construction of a hyperbolic surface in space, and several criteria in the problem of type. The first three sections are devoted to developing the necessary preliminary material. In ?1 we give an intuitive picture of the embedding process, and also the exact definitions to be used throughout the paper. ?2 contains a brief discussion of the surfaces to be considered, simply-connected covering surfaces of the plane having only algebraic singularities over finite points, which are denoted as surfaces of class A. A method is given for subdividing these surfaces into sheets, this subdivision being the main tool for the embedding. In ?3 we define and study the Euclidean metric on the surface. The embedding itself is carried out in ?4, where we show how a large class of Riemann covering surfaces can be represented isometrically as nonselfintersecting surfaces in 3-space. The interest in the embedding method is partly theoretical, since the existence of such an embedding is somewhat surprising in itself, and partly practical, in the sense tha,t by viewing a surface in the imbedded form one often obtains a much better intuitive picture of its internal structure. The author has indicated at the 1954 International Congress [8] how this embedding method could be used to settle a question on the existence of certain hyperbolic surfaces in 3-space. The details are carried out in ?5 with slight refinements so that the surface obtained is everywhere infinitely differentiable. The method used here for removing the singular points may be used in conjunction with the author's paper [9] to provide a different way of settling the original question. Finally, we turn to the problem of type for surfaces of class A. These surfaces were originally considered by Ahlfors [1 ] who introduced the function n(t) as a measure of the branching and found a sufficient condition for parabolic type. ?6 contains a number of new results on the use of the function n(t) in the determination of type.