A Characterization of curves that lie on a totally umbilical surface of a space form

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms \({\mathbb {H}}^3\) ${\mathbb{H}}^3$ and \({\mathbb {S}}^3\) ${\mathbb{S}}^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the value of the curvature of the curve. Also numerical pictures of these curves are shown.