On Bishop frame of a null Cartan curve in Minkowski space-time

Abstract

In this paper, we define the Bishop frame of a null Cartan curve in Minkowski space-time E-1(4). We obtain the Bishop's frame equations of a null Cartan curve which lies in the timelike hyperplane of E-1(4). We show that a null Cartan cubic lying in the timelike hyperplane of E-1(4) has two Bishop frames, one of which coincides with its Cartan frame. We also derive the Bishop's frame equation of the null Cartan curve which has the third Cartan curvature K-3(s) not equal 0. As an application, we find a solution of the null Betchov-Da Rios vortex filament equation in terms of a null Cartan curve and its Bishop frame, which generates a timelike Hasimoto surface.