The Profiles of Riemann Surfaces

Abstract

Riemann surfaces which are set by algebraic, algebroid and inverse functions are considered. A method for describing these Riemann surfaces by graphs is proposed. Each such Riemann surface is assigned to a special type of graph - profile. In terms of graph theory, the necessary and sufficient conditions of profile existence are clarified. The conditions of a one-to-one correspondence between Riemann surfaces and profiles are formulated. In the graph theory terminology the criterion of profiles existence is formulated and proved. The resulting criterion can be used as a criterion of existence of Riemann surfaces with a set signature. The proposed method of describing Riemann surfaces by profiles corresponds to the intuitive notion of a Riemann surface as a covering surface over a Riemann sphere. Examples of the Riemann surface profile of the algebraic function and the Riemann surface profile of the inverse function of algebroid are given.