Darboux Vector-based Non-linear Differential Equations

Abstract

The definition of slant helices in Euclidean space were given by İzumiya and Takeuchi.  It is well known that there exist three kinds of curves, i.e. space-like curve, time-like curve and light-like curve in Minkowski 4- space. This paper mainly focuses on some notions of the space-like curve and the curvature in Minkowski 4-space. In this study the Darboux frame is given for a space-like curve in Minkowski 4-space R⁴₁. By using k-type slant helices of space-like curve in Minkowski 4-space playing the role of the Darboux vectors, we obtain non-linear differential equations. Also, the k-type slant helices in terms of their curvature are characterized.