Timelike $B_2$-slant helices in Minkowski space $E_1^4$

TL;DR

This paper investigates timelike $B_2$-slant helices in Minkowski 4-space, characterizing curves where the binormal vector $B_2$ maintains a constant scalar product with a fixed direction, expanding understanding of generalized helices in this setting.

Contribution

It introduces and characterizes a new class of generalized helices in Minkowski 4-space where the $B_2$ binormal vector has a constant scalar product with a fixed direction.

Findings

Characterizations of timelike $B_2$-slant helices.

Conditions for the constancy of $<B_2,U>$.

New geometric properties of these helices.

Abstract

We consider a unit speed timelike curve $\alpha$ in Minkowski 4-space $E_1^4$ and denote the Frenet frame of $\alpha$ by $\{T,N,B_1,B_2\}$. We say that $\alpha$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $E_1^4$. In this work we study those helices where the function $<B_2,U>$ is constant and we give different characterizations of such curves.