On a variational problem due to the B‐Darboux frame in Euclidean 3‐space

Abstract

In this paper, we introduce a new frame which is called the B-Darboux frame on a surface in Euclidean 3-space. We know that the parallel transport frame is derived from the Frenet frame along a space curve on a surface. Analogously, we derive the B-Darboux frame from the Darboux frame along a space curve on a surface in Euclidean 3-space. Then, we obtain the intrinsic equations due to the B-Darboux frame for a generalized relaxed elastic line on an oriented surface and give some applications of this obtained results in Euclidean 3-space.

A New Approach to Hyperbolic Spinor B-Darboux Equations

“…Here ( [12]), the geodesic curvature , the normal curvature , and the relative torsion are given with the following equations:…”

“…Here, the hyperbolic angle is between the timelike vector and the new timelike normal plane vector . Thus ( [12]) the rotation matrix between B-Darboux and Darboux frames, is indicated with the equation…”

“…Additionally, the Darboux frame features curves in surfaces [10]. Recently, an original frame has been offered along curves lying in surfaces, designated as the Bishop Darboux frame in Euclidean and Lorentzian spaces [11][12][13].…”

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

“…Note that PAF is a generic adapted moving frame just like Bishop frame [2], Darboux frame [6], B-Darboux frame [8] etc. Generic adapted moving frames are obtained from Serret-Frenet frame by a rotation (see [5] for more details on generic adapted moving frame).…”

Spinor Description of B-Darboux Equations

“…Recently a new frame was introduced along the curve on the surface, called as the Bishop Darboux frame, which is formed of three orthonormal vectors { }. The Bdarboux frame which is developed as the Bishop version of the Darboux frame, has a tangent parallel to the tangent of the Darboux frame, but the vector fields and are obtained by the parallel transport method unlike the Darboux frame [20,21].…”