SLANT HELICES IN THE THREE-DIMENSIONAL SPHERE

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 h...

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Bibliographic Details
Published inJournal of the Korean Mathematical Society pp. 1331 - 1343
Main Authors Pascual Lucas, Jose Antonio Ortega-Yagues
Format Journal Article
LanguageEnglish
Published 대한수학회 01.07.2017
Subjects
수학
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Summary: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. KCI Citation Count: 4
ISSN:0304-9914
2234-3008
DOI:10.4134/JKMS.j160508