$(1,2)$-Null Bertrand Curves in Minkowski Spacetime

TL;DR

This paper investigates null Bertrand curves in Minkowski spacetime R_{1}^{4} with Cartan frames, establishing conditions under which such curves are or are not Bertrand curves based on the linear independence of their derivative vectors.

Contribution

It introduces conditions for null Cartan curves in R_{1}^{4} to be Bertrand curves, clarifying when they occur based on derivative vector linear dependence.

Findings

Null Cartan curves with linearly independent derivatives are not Bertrand curves.

Null Bertrand curves occur only when derivative vectors are linearly dependent.

Conditions for null Cartan curves to be Bertrand curves are explicitly characterized.

Abstract

In this paper we study null Bertrand curves in $R_{1}^{4}$ under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in $R_{1}^{4}$ is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R_{1}^{4}$ occurs only if the derivative vectors of the curve is linearly dependent. We will introduce an idea of Bertrand curves and abiding by this idea we bring to light under which conditions a null Cartan curve in $R_{1}^{4}$ is a Bertrand curve.