On Bishop frame of a partially null curve in Minkowski space-time E^4_1

Abstract

In this paper, we introduce Bishop frame of a partially null curve α in Minkowski space-time E41. We prove that such curve has three Bishop frames determined by the particular solutions of the third order non-linear differential equation. We show that the Frenet frame of α can be obtained by rotating the Bishop frame that corresponds to the particular solution θ(s) =R κ1(s) ds of the mentioned differential equation. As an application, we obtain the parametrization of a lightlike hypersurface and focal surface with base curve α in terms of its Bishop frame. Finally, we prove that a lightlike focal surface along a partially null helix has no critical value set.