Slant Helices that Constructed from Hyperspherical Curves in the n-dimensional Euclidean Space

In this work, we study slant helices in the n-dimensional Euclidean space. We give methods to determine the position vectors of slant helices from arclength parameterized curves that lie on the unit hypersphere. By means of these methods, first we characterize slant helices and Salkowski curves which lie on 2n-dimensional hyperboloid. After that, we characterize rectifying slant helices which are geodesics of 2n-dimensional cone.

Keywords:

Slant helix, Salkowski curve, rectifying curve, hyperspherical curve geodesic of a hypersurface,

PDF

___

[1] Ahmad TA, Turgut M. Some characterizations of slant helices in the Euclidean space En. Hac J Math Sta 2010; 39: 327-336.
[2] Arslan K, Celik Y, Deszcz C, Ozgur C. Submanifolds all of whose normal sections are W-curves. Far East J Math Sci 1997; 5: 537-544.
[3] Altunkaya B, Aksoyak FK, Kula L, Aytekin C. On rectifying slant helices in Euclidean 3-space. Kon J Math 2016; 4: 17-24.
[4] Altunkaya B, Kula L. General helices that lie on the sphere S2n in Euclidean space E2n+1. Uni J Math App 2018; 1: 166-170.
[5] Camci C, Ilarslan K, Kula L, Hacisalihoglu HH. Harmonic cuvature and general helices, Chaos Solitons Fractals 2009; 40: 2590-2596.
[6] Cambie S, Goemans W, Van Den Bussche I. Rectifying curves in the n-dimensional Euclidean space, Turkish J Math 2016; 40: 210-223.
[7] Chen BY. When does the position vector of a space curve always lie in its rectifying plane?, Amer Math Monthly 2003; 110: 147-152.
[8] Chen BY, Dillen F. Rectifying curves as centrodes and extremal curves. Bull Inst Math Aca Sinica 2005; 33: 77-90.
[9] Chen BY. Differential geometry of rectifying submanifolds. Int Elec J Geo 2016; 9: 1-8.
[10] Chen BY. Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang J Math 2017; 48: 209-214.
[11] Deshmukh S, Chen BY, Alshammari, SH. On rectifying curves in Euclidean 3-space. Turkish J Math 2017; 42: 609-620.
[12] Gluck H. Higher curvatures of curves in Euclidean space, Amer Math Monthly 1966; 73: 699-704.
[13] Ilarslan K, Nesovic E. Some characterizations of rectifying curves in Euclidean space E4, Turkish J Math 2008; 32:21-30.
[14] Izumiya S, Takeuchi N. New special curves and developable surfaces. Turkish J Math 2004; 28: 153-163.
[15] Izumiya S, Takeuchi N. Generic properties of helices and Bertrand curves. J Geom 2002; 74: 97-109.
[16] Kula L, Yaylı Y. On slant helix and its spherical indicatrix. App Math Comp 2005; 169: 600-607.
[17] Kula L, Ekmekci N, Yaylı Y, Ilarslan K. Characterizations of slant helices in Euclidean 3-space. Turkish J Math 2010; 34: 261-273.
[18] Lucas P, Ortega-Yagues JA. Rectifying curves in the three-dimensional sphere. J Math Anal Appl 2015; 421: 1855–1868.
[19] Monterde J. Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion. Com Aided Geo Design2009; 26(3): 271-278.
[20] O’Neill B. Elementary Differential Geometry. London, UK: Academic Press Inc, 2006.
[21] Salkowski, E. Zur transformation von raumkurven. Mathematische Annalen 1909; 66(4); 517-557.
[22] Yayli Y, Ziplar E. On slant helices and general helices in Euclidean n-space. Mathematica Aeterna 2011; 1: 599-610.
[23] Yayli Y, Gok I, Hacisalihoglu HH. Extended rectifying curves as new kind of modified Darboux vectors. TWMS J Pure Appl Math 2018;9: 18-31.

International Electronic Journal of Geometry-Cover

Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2008
Yayıncı: Prof.Dr. H.Hilmi Hacısalioğlu

Arşiv