The Profiles of Riemann Surfaces

TL;DR

This paper introduces a graph-based method to describe Riemann surfaces, establishing conditions for their existence and correspondence with profiles, which enhances understanding of their structure and classification.

Contribution

It proposes a novel graph-theoretic approach to characterize Riemann surfaces via profiles, providing necessary and sufficient conditions for their existence and correspondence.

Findings

Established criteria for profile existence on Riemann surfaces

Proved one-to-one correspondence between Riemann surfaces and profiles

Applied method to algebraic and inverse functions' 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.