Timelike Bertrand Curves in Semi-Euclidean Space

TL;DR

This paper investigates the properties of timelike Bertrand curves in semi-Euclidean spaces, proving their non-existence in certain dimensions and introducing a generalized concept called timelike (1,3)-Bertrand curves with their characterization.

Contribution

It establishes the non-existence of special timelike Frenet Bertrand curves in specific semi-Euclidean spaces and introduces and characterizes the generalized timelike (1,3)-Bertrand curves.

Findings

No special timelike Frenet curve is a Bertrand curve in _2^4.

Bertrand curves are only well-defined in _1^2 and _1^3.

Introduction and characterization of timelike (1,3)-Bertrand curves in _2^4.

Abstract

In this paper, it is proved that, no special timelike Frenet curve is a Bertrand curve in $\mathbb{E}_2^4$ and also, in $\mathbb{E}_\nu^{n+1}$ $ ({n \ge 3})$, such that the notion of Bertrand curve is definite only in $\mathbb{E}_1^2$ and $\mathbb{E}_1^3$. Therefore, a generalization of timelike Bertrand curve is defined and called as timelike (1,3)-Bertrand curve in $\mathbb{E}_2^4$. Moreover, the characterization of timelike (1,3)-Bertrand curve is given in $\mathbb{E}_2^4$.