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

Authors: JELENA DJORDJEVIC, EMILIJA NESOVIC

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.

Keywords: Bishop frame, Darboux bivector, pseudo null curve, space-time geometric algebra, Minkowski space-time

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