SLANT HELICES IN THE THREE-DIMENSIONAL SPHERE

A curve gamma immersed in the three-dimensional sphere 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 s(3) by means of a differential equation in the curvature and the torsion tau of the curve. We define a helix surface in s(3) and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in s(3). Finally, we obtain a geometric approach to the problem of solving natural equations for slant helices in the threedimensional sphere. We prove that the slant helices in s(3) are exactly the geodesics of helix surfaces.