Conformal trajectories in 3-dimensional space form

TL;DR

This paper introduces conformal trajectories in three-dimensional space forms, characterizing their geometric properties and providing explicit descriptions in Euclidean, spherical, and hyperbolic spaces.

Contribution

It defines conformal trajectories in 3D space forms and characterizes their curvature and torsion properties for various conformal vector fields.

Findings

Conformal trajectories in ${ m S}^3$ and ${ m H}^3$ have constant curvature.

Torsion of these trajectories is a linear combination of trigonometric or hyperbolic functions.

In ${ m R}^3$, all conformal trajectories for the radial vector field are characterized.

Abstract

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in\mathfrak{X}(M^3)$, a conformal trajectory of $V$ is a regular curve $\gamma$ in $M^3$ satisfying $\nabla_{\gamma'}\gamma'=q\, V\times\gamma'$, for some fixed non-zero constant $q\in {\mathbb{R}}$. In this paper, we study conformal trajectories in the space forms ${\mathbb{R}}^3$, ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$. For (non-Killing) conformal vector fields in ${\mathbb{S}}^3$ (respectively in ${\mathbb{H}}^3$), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ${\mathbb{R}}^3$, we obtain the same result for the radial vector field and characterising all conformal trajectories.