[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.j160508

SLANT HELICES IN THE THREE-DIMENSIONAL SPHERE

Lucas, Pascual (Departamento de Matematicas Universidad de Murcia Campus de Espinardo)
Ortega-Yagues, Jose Antonio (Departamento de Matematicas Universidad de Murcia Campus de Espinardo)

Publication Information

Journal of the Korean Mathematical Society / v.54, no.4, 2017 , pp. 1331-1343 More about this Journal

Abstract

A curve ${\gamma}$ immersed in the three-dimensional sphere ${\mathbb{S}}^3$ is said to be a slant helix if there exists a Killing vector field V(s) with constant length along ${\gamma}$ and such that the angle between V and the principal normal is constant along ${\gamma}$ . In this paper we characterize slant helices in ${\mathbb{S}}^3$ by means of a differential equation in the curvature ${\kappa}$ and the torsion ${\tau}$ of the curve. We define a helix surface in ${\mathbb{S}}^3$ and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in ${\mathbb{S}}^3$ . Finally, we obtain a geometric approach to the problem of solving natural equations for slant helices in the three-dimensional sphere. We prove that the slant helices in ${\mathbb{S}}^3$ are exactly the geodesics of helix surfaces.

Keywords

slant helix; 3-sphere; helix surface; Killing field; Hopf field;

Citations & Related Records

Reference

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