Spherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere

TL;DR

This paper demonstrates that various types of arc-lengths serve as global conformal parameters for analytic curves across different geometries, including Euclidean, hyperbolic, and Riemann sphere, with generalizations to higher dimensions.

Contribution

It establishes the universality of arc-lengths as global conformal parameters for analytic curves in multiple geometries and extends these results to higher-dimensional spaces.

Findings

Euclidean and spherical arc-lengths are global conformal parameters in the complex plane.

Hyperbolic arc-length is a global parameter for curves in the hyperbolic plane.

Results are generalized to Euclidean and complex n-space, and discussed for the Riemann sphere.

Abstract

We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euclidean n-space and complex n-space and we discuss the situation of curves in the Riemann sphere.