Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

Abstract

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

Keywords

• Bertrand curves
• Mannheim curves
• spherical regular curves
• spherical framed curves
• singularity