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

Jelena Djordjević,Emilija Nešović

doi:10.3906/mat-1910-11

Abstract

In this paper, we introduce the Bishop frame of a pseudo null curve $\alpha$ in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using space-time geometric algebra, we derive the Darboux bivectors $D$ and $\tilde{D}$ of the Frenet and the Bishop frame of $\alpha$, respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of $\alpha$ in terms of areas of the projections of the corresponding Darboux bivectors onto the planes spanned by the frame vector's fields.

Highlights

The Bishop frame {T, N1, N2} of a regular curve in Euclidean space E3 can be obtained by applying rotation of the Frenet frame {T, N, B} about the tangent
We introduce the Bishop frame of a pseudo null curve α in Minkowski space-time
In Minkowski space-time E41, the Frenet frame of a nonnull curve with a nonnull principal normal has Darboux bivector D that can be regarded as generalization of the Darboux vector ([11,12,13])

Summary

Introduction

The Bishop frame {T, N1, N2} (relatively parallel adapted frame, rotation-minimizing frame) of a regular curve in Euclidean space E3 can be obtained by applying rotation of the Frenet frame {T, N, B} about the tangent. In Minkowski space-time E41 the Bishop frame {T1, N1, N2, N3} of a null Cartan curve contains the tangent vector field T1 of the curve and three vector fields whose derivatives N1′ , N2′ , and N3′ with respect to pseudo-arc are collinear with N2 [7]. In Minkowski space-time E41 , the Frenet frame of a nonnull curve with a nonnull principal normal has Darboux bivector D that can be regarded as generalization of the Darboux vector ([11,12,13]) It is show in [11] that the Frenet frame of a timelike curve α in E41 has Darboux bivector which satisfies the Darboux equations in terms of the inner product of spacetime geometric algebra. For further properties of spacetime geometric algebra, we refer to [12, 13]

The Bishop frame of a pseudo null curve in E41

The Frenet frame’s and the Bishop frame’s Darboux bivectors