Spherical arc-length as a global holomorphic parameter for analytic curves in the Riemann sphere
Résumé
The subject of this paper is the interaction between real analytic curves and complex geometry. We prove that for every analytic curve in the complex plane $\mathbb{C}$, Euclidean and spherical arc-lengths are global holomorphic 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 $\mathbb{R}^n$ and $\mathbb{C}^n$ and we discuss the situation of curves in the Riemann sphere $\mathbb{C}\cup\{\infty\}$.