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

TL;DR

This paper investigates the geometric properties of time-like curves on de Sitter 2-space, explores their invariants, and establishes relationships between space-like Bertrand curves, helices, and constant slope surfaces in Minkowski 3-space.

Contribution

It introduces Lorentzian Sabban frames, studies invariants and evolutes of time-like curves, and links space-like Bertrand curves with helices and constant slope surfaces in Minkowski space.

Findings

Invariants of time-like curves on de Sitter 2-space are characterized.

De Sitter evolutes of these curves are studied and related to Darboux images.

Relations between space-like Bertrand curves and constant slope surfaces are established.

Abstract

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Sitter evolutes of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ are studied. A relation between space-like Bertrand curves and helices is obtained. De Sitter Darboux images of space-like Bertrand curves are equal to de Sitter evolutes. The relations between time-like constant slope surfaces lying in the space-like cone and space-like Bertrand curves in Minkowski 3-space $\mathbb{R}^{3}_{1}$ are obtained.