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 ${\text{h}}^3$ and ${\text{s}}^3$ 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.