Conformal trajectories in 3-dimensional space form

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.