Conformal trajectories in three-dimensional space forms

Abstract

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), 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 [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.