Conformal trajectories in 3-dimensional space form

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds M³. Given a conformal vector field V(M³), a conformal trajectory of V is a regular curve γ in M³ satisfying ∇γ'γ'=q\, V×γ', for some fixed non-zero constant q∈ R. In this paper, we study conformal trajectories in the space forms R³, S³ and H³. For (non-Killing) conformal vector fields in S³ (respectively in H³), 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 R³, we obtain the same result for the radial vector field and characterising all conformal trajectories.