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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.
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참고문헌 (17)
참고문헌 신청
M. Barros / 1997 / General helices and a theorem of Lancret / Proc. Amer. Math. Soc 125(5):1503~1509
V. N. Berestovskii / 2008 / Killing vector fields of constant length on Riemannian manifolds / Sib. Math. J 49(3):395~407
F. Dillen / 2007 / Constant angle surfaces in S2 × R / Monatsh. Math 152(2):89~96
A. J. Di Scala / 2009 / Helix submanifolds of euclidean spaces / Monasth. Math 157(3):205~215
J. M. Espinar / 2013 / Locally convex surfaces immersed in a Killing submersion / Bull. Braz. Math. Soc. (N.S.) 44(1):155~171
V. N. Berestovskii / 2008 / Killing vector fields of constant length on Riemannian manifolds / Sib. Math. J 49(3):395~407
F. Dillen / 2007 / Constant angle surfaces in S2 × R / Monatsh. Math 152(2):89~96
A. J. Di Scala / 2009 / Helix submanifolds of euclidean spaces / Monasth. Math 157(3):205~215
J. M. Espinar / 2013 / Locally convex surfaces immersed in a Killing submersion / Bull. Braz. Math. Soc. (N.S.) 44(1):155~171
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