On non-null relatively normal-slant helices in Minkowski 3-space

Abstract

By using the Darboux frame |?, ?, ?| of a non-null curve lying on a timelike surface in Minkowski 3-space, where ? is the unit tangent vector of the curve, ? is the unit spacelike normal vector field restricted to the curve and ? = ?? ? ?, we define relatively normal-slant helices as the curves satisfying the condition that the scalar product of the fixed vector spanning their axis and the non-constant vector field ? is constant. We give the necessary and sufficient conditions for non-null curves lying on a timelike surface to be relatively normal-slant helices. We consider the special cases when non-null relatively-normal slant helices are geodesic curves, asymptotic curves, or lines of the principal curvature. We show that an asymptotic spacelike hyperbolic helix lying on the principal normal surface over the helix and a geodesic spacelike general helix lying on the timelike cylindrical ruled surface, are some examples of non-null relatively normal-slant helices in E31.