On the geometry of curves and conformal geodesics in the Mobius space

TL;DR

This paper explores the properties of immersed curves in conformal spheres, focusing on conformal geodesics, their equations, and a surprising reduction to 4-spheres, extending previous work with explicit solutions.

Contribution

It proves that all conformal geodesics in s are contained in a 4-sphere and provides explicit solutions for their curvature equations, extending prior research.

Findings

Conformal geodesics in s lie in a 4-sphere.

Explicit solutions for the curvature equations of conformal geodesics.

Extension of previous work with new explicit expressions.

Abstract

This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the M\"obius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of the paper is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any $n$, we prove an interesting codimension reduction, namely that every conformal geodesic in $\mathds{Q}_n$ lies, in fact, in a totally umbilical 4-sphere $\mathds{Q}_4$. We then extend and complete the work in (Musso, "The conformal arclength functional", Math Nachr.) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.