On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space

TL;DR

This paper explores the geometric properties of space-like and time-like Bertrand curves in Minkowski 3-space, establishing their relations with constant slope surfaces, Darboux images, and evolutes, with new definitions and explicit examples.

Contribution

It introduces Lorentzian Sabban frames and de Sitter evolutes for space-like curves, and constructs Bertrand curves from curves on de Sitter and hyperbolic spaces, revealing new geometric relations.

Findings

Relations between Bertrand curves and helices established.

Pseudo-spherical Darboux images of Bertrand curves are identified.

Connections between Bertrand curves and space-like constant slope surfaces demonstrated.

Abstract

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and hyperbolic space $\mathbb{H}^{2}$, respectively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evolutes in Minkowski 3-space $\mathbb{R}^{3}_{1}$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in $\mathbb{R}^{3}_{1}$. Finally, we give some examples to illustrate our main results.