Helix surfaces and slant helices in the three-dimensional anti-De Sitter space

Abstract

In this paper, we define slant helices in the three-dimensional anti-De Sitter space \(\mathbb {H}^{3}_{1}\) and give characterizations in terms of the curvatures of the curve. We also introduce helix surfaces in \(\mathbb {H}^{3}_{1}\) and, after proving that every helix surface is a flat surface, we give characterizations of helix surfaces based on the geometric model of pseudo-quaternions for \(\mathbb {H}^{3}_{1}\). These characterizations, with the support of Hopf fibrations, allow us to build helix surfaces in \(\mathbb {H}^{3}_{1}\) from two curves in the base space (\(\mathbb {H}^{2}\) or \(\mathbb {S}^{2}_{1}\)), one of them being a general helix. Finally, we obtain a geometric integration of slant helices by proving that these curves can be viewed as geodesics (or other special curves) in certain flat surfaces of \(\mathbb {H}^{3}_{1}\). This gives us a method to geometrically construct all the slant helices in \(\mathbb {H}^{3}_{1}\).