Time-like Salkowski and anti-Salkowski curves in Minkowski space $\e_1^3$

Abstract

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to time-like curves in Minkowski 3-space $\e_1^3$. Thereafter, we introduce an explicit parametrization of a time-like Salkowski curves and a time-like Anti-Salkowski curves in Minkowski space $\e_1^3$. Also, we characterize them as space curve with constant curvature or constant torsion and whose normal vector makes a constant angle with a fixed line.