Bertrand-B Curves in the Three Dimensional Sphere

Abstract

We dene a Bertrand-B curve in Riemannian manifold M such that thereexists an isometry \phi of M, that is, \left( \phi \circ \beta \right) (s)=X\left( s,t(s)\right) and the binormal vector of another curve \beta is the paralel vector of binormal vector of \alpha at corresponding points. We obtain the conditions of existence of a Bertrand-B curve in the event E^3, S^3 and H^3 of M. The rst of our main results is that the curve \alpha in E^3 is a Bertrand-B curve if and only if it is planar. Second one, we prove that the curve \alpha with the curvatures \epsilon _{1},\epsilon _{2} in S^3 is a Bertrand-B curve if and only if it is satises \epsilon _{1}^{2}+\epsilon _{2}^{2}=1. Finally, we state that there not exists a Bertrand-B curve in H^3.

Bertrand partner P-trajectories in the Euclidean 3-space $E^3$

“…By following the steps similar to those of Bertrand, this topic was expanded to different moving frames. For example, the studies [21], [14] and [7] expanded this topic to the type-2 Bishop frame, Darboux frame and qframe, respectively. Also, many mathematicians presented various studies about the concept of Bertrand curve couple with different perspectives.…”