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.
In this study, we examine the condition of the conchoidal surface to be a Bonnet surface in Euclidean 3-space. Especially, we consider the Bonnet conchoidal surfaces which admit an infnite number of isometries. In addition, we study the necessary conditions which have to be fulflled by the surface of revolution with the rotating curve <em>c</em>(<em>t</em>) and its conchoid curve <em>c<sub>d</sub></em>(<em>t</em>) to be the Bonnet surface in Euclidean 3-space.
A basic goal of this paper is to calculate, Weyl curvature of R-flat (Ricci-flat) spray of isotropic curvature and a locally projectively R-flat (Ricci-flat) spray, which is a projective invariance. Besides, the equivalents of E ̅-curvature and H-curvature that are closely related to the mean Berwald curvature have been found for a locally projectively R-flat spray of isotropic curvature.
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