A Characterization of Curves that Lie on a Totally Umbilical Surface of a Space Form

TL;DR

This paper characterizes the curvature and torsion conditions for regular curves in hyperbolic and spherical space forms to lie on totally umbilical surfaces, providing explicit curvature values for constant torsion cases and visual illustrations.

Contribution

It establishes necessary and sufficient conditions for such curves to be on totally umbilical surfaces in space forms, including explicit curvature values for constant torsion curves.

Findings

Derived conditions for curves to lie on totally umbilical surfaces.

Calculated curvature values for curves with constant torsion.

Provided visual representations of the characterized curves.

Abstract

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ 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.