Curves Lying on Non-lightlike Surface: Differential Equation for Position Vector

Abstract

The main purpose of this study is to examine curves lying on a given non-lightlike surface with the help of its position vectors. For this purpose, the darboux frame is used and the position vector of the curve is expressed as a linear combination of the darboux frame with differentiable functions. Then, nonhomogeneous systems of differential equations revealed by the position vector of the curve are obtained for timelike and spacelike surfaces, respectively. For both timelike and spacelike surfaces, the solutions of nonhomogeneous systems of differential equations are obtained depending on the character of the curves and the values kg, kn and tr. The general solutions of the systems of differential equations are obtained separately for each case. Moreover, by considering only the particular solution of the systems of differential equations, new results regarding the differential geometric structure of the curves on the surface are presented with the help of the position vector