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.