Some Characterizations of Slant Helices in the Euclidean Space En

Year 2010, Volume: 39 Issue: 3, 327 - 336, 01.03.2010

Ahmad T. Ali Melih Turgut

Abstract

In this work, the notion of a slant helix is extended to the space En First, we introduce the type-2 harmonic curvatures of a regular curve.
Thereafter, by using this, we present some necessary and sufficient conditions for a curve to be a slant helix in Euclidean n-space. We also express some integral characterizations of such curves in terms of the curvature functions. Finally, we give some characterizations of slant helices in terms of type-2 harmonic curvatures.

Keywords

Euclidean n-space, Frenet equations, Slant helices, Type-2 Harmonic Curvatures, 2000 AMS Classification: 53 A 04

SOME CHARACTERIZATIONS OF SLANT HELICES IN THE EUCLIDEAN SPACE En

Abstract

References

• Ali, A. Inclined curves in the Euclidean 5-space E5, J. Advanced Research in Pure Math. (1), 15–22, 2009.
• Ali, A. and L´opez, R. Slant helices in Minkowski space E1, preprint, 2008: arXiv: 1464v1 [math.DG].
• Ali, A. and L´opez, R. Timelike B2-slant helices in Minkowski space E4, Archivum Math. (1), 39–46, 2010.
• Ali, A. Position vetors of slant helices in Euclidean 3-space, preprint, 2009: arXiv: 0750v1 [math.DG].
• Barros, M. General helices and a theorem of Lancert, Proc. Amer. Math. Soc. 125, 1503– , 1997.
• Camcı, C¸ ., ˙Ilarslan, K., Kula, L. and Hacısaliho˘glu, H. H. Harmonic curvatures and gener- alized helices in En, Chaos, Solitons and Fractals 40, 2590–2596, 2009.
• Ekmek¸ci, N., Hacisaliho˘glu, H. H. and ˙Ilarslan, K. Harmonic curvatures in Lorentzian space, Bull. Malaysian Math. Soc. (Second Series) 23 (2), 173–179, 2000.
• Erdo˘gan, M. and Yılmaz, G. Null generalized and slant helices in 4-dimensional Lorentz- Minkowski space, Int. J. Contemp. Math. Sci. 3 (23), 1113–1120, 2008.
• Gluck, H. Higher curvatures of curves in Euclidean space, Amer. Math. Monthly 73, 699– , 1966.
• Izumiya, S. and Takeuchi, N. New special curves and developable surfaces, Turk. J. Math. (2), 531–537, 2004.
• Kula, L. and Yayli, Y. On slant helix and its spherical indicatrix, Appl. Math. Comput. (1), 600ˆu-607, 2005.
• Kula, L., Ekmek¸ci, N., Yayli Y. and ˙Ilarslan, K. Characterizations of slant helices in Eu- clidean 3-space, Turk. J. Math. 169 (1), 600ˆu-607, 2009.
• Milman, R. S. and Parker, G. D. Elements of Differential Geometry (Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977).
• Monterde, J. Curves with constant curvature ratios, Bull. Mexican Math. Soc. Ser. 3A (1), 177–186, 2007.
• Monterde, J. Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. 26, 271–278, 2009.
• ¨Onder, M., Kazaz, M., Kocayi˘git, H. and Kili¸c, O. B2-slant helix in Euclidean 4-space E4, Int. J. Contemp. Math. Sci. 3 (29), 1433–1440, 2008.
• ¨Ozdamar, E. and Hacisaliho˘glu, H. H. A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci. Univ. Ankara, Ser. A1 24, 15–23, 1975.
• ¨Ozt¨urk, G., Arslan, K. and Hacisalihoglu, H. H. A characterization of ccr-curves in Rm, Proc. Estonian Acad. Sci. 57 (4), 217–224, 2008.
• Petrovic-Torgasev, M. and Sucurovic, E. W-curves in Minkowski spacetime, Novi. Sad. J. Math. 32 (2), 55–65, 2002.
• Scofield, P. D. Curves of constant precession, Amer. Math. Monthly 102, 531–537, 1995.
• Turgut, M. and Yilmaz, S. Characterizations of some special helices in E4, Sci. Magna. (1), 51–55, 2008.
• Turgut, M. and Yilmaz, S. Some characterizations of type-3 slant helices in Minkowski space-time, Involve J. Math.2 (1), 115–120, 2009.