On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space

Abstract

In this paper, we derive the Bishop frames of the pseudo null curve and show that its normal Bishop vectors N1 and N2 can be obtained by applying the hyperbolic rotation and the composition of three rotations about two lightlike and one spacelike axis to the Frenet vectors N and B respectively. We also derive Bishop frame of the null Cartan curve and show that among all null Cartan curves in E13, only the null Cartan cubic has two Bishop frames, one of which coincides with its Cartan frame. As an application, we obtain some solutions of the Da Rios vortex filament equation in terms of the Bishop frames of the pseudo null curves and null Cartan cubic.