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

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 ${\mathbb{S}}_1^2$. 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 ${\mathbb{S}}_1^2$ 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 ${\mathbb{R}}_1^3$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in ${\mathbb{R}}_1^3$. Finally, we give some examples to illustrate our main results.

arXiv comment16 pages, 4 figures. Accepted for publication in the Scientific Annals of "Al.I. Cuza" University of Iasi