Journal of the
Korean Mathematical Society JKMS

A curve $\gamma$ immersed in the three-dimensional sphere $\S3$ 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 $\S3$ by means of a differential equation in the curvature $\kappa$ and the torsion $\tau$ of the curve. We define a helix surface in $\S3$ and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in $\S3$. 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 $\S3$ are exactly the geodesics of helix surfaces.

Keywords: slant helix, 3-sphere, helix surface, Killing field, Hopf field

MSC numbers: 53B25, 53B20