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

Abstract

© 2018 World Scientific Publishing Company. In this paper, we define the Bishop frame of a null Cartan curve in Minkowski space-time 14. We obtain the Bishop's frame equations of a null Cartan curve which lies in the timelike hyperplane of 14. We show that a null Cartan cubic lying in the timelike hyperplane of 14 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 κ3(s)≠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.