SLANT HELICES IN DUAL LORENTZIAN SPACE D3 1

Abstract

In this paper, we consider a unit speed dual Lorentzian curve a in dual Lorentzian space D31 and denote by {T ,N, B } the dual Frenet frame of a. We say that a is a slant helix if there exists a non-zero dual constant vector …eld U in D 3 1 such that the dual function <N;U> is a dual constant.
Moreover, we give some characterizations of slant helice in terms of their dual
curvatures. Finally, we show that dual tangent indicatrices and dual binormal
indicatrices of slant helices are dual helices.

Keywords

• Dual Lorentzian space; Dual Frenet equations; Dual slant helices; Dual curves

Slant helices in dual Lorentzian Space

Öz

References

1. Ali, A.T. and Lopez, R., Slant helices in Minkowski space E₁³, J. Korean Math. Soc. 48. 2011; no.1: 159-167.
2. Guggenheimer, H., W., Differential Geometry, McGraw-Hill, New York, 1963.
3. Özkaldı, S., İlarslan K. and Yaylı, Y., On mannheim partner curves in dual Lorentzian space, Hacettepe Journal of Mathematics and Statistics. 40. 2011; 649-661.
4. Kula L. and Yayli, Y., On slant helix and its spherical indicatrix, Appl. Math. Comput. 169. 2005; no.1: 600-607999.
5. O'Neill, B., Semi Riemannian geometry with applications to relativity, London: Academic Press. 1983.
6. Önder, M., Kazaz, M., Kocayiğit, H. and Kilic, O., B₂ slant helix in Euclidian 4-space E⁴, Int. J. Contemp. Math. Sci. 3. 2008; no.29-32: 1433-1440.
7. Önder, M. and Uğurlu, H., Normal and spherical curves in dual space D³, Mediterr. J. Math. 10. 2013; 1527-1537.
8. Özbey, E. and Oral, M., A study on rectifying curves in the dual Lorentzian space, Bull. Korean Math. Soc 46. 2009; no 5: 967-978.

9. Sağlam, D. and Kalkan, Ö., Some characterizations of slant helices in the Minkowski space E_{υ}ⁿ, Comptes rendus de l'Académie bulgare des Sciences.Tome 64. 2011; No 2: 173-184.
10. Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electronic J. of Geometry . 2014; Vol 7, No.1: 44-107.
11. Uğurlu, H., H. and Çalışkan, A., The study mapping for directed spacelike and timelike lines in Minkowski 3-Space R₁³, Mathematical and Computational Applications, 1. 1996; no 2: 142-148.
12. Veldkamp, G., R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mechanism and Machine Theory, 11. 1976; no 2: 141-156.
13. Lee, J.W., Choi, J.H. and Jin, D.H., Slant dual Mannheim partner curves in the dual space, Int, J. Contemp. Math. Sciences, 6(31). 2011; 1535-1544.
14. Şahiner, B.and Önder, M., Slant helices, Darboux helices and similar curves in dual space D³, Mathematica Moravica (In press).