Characterizations of slant and spherical helices due to pseudo-Sabban frame

Abstract

In this paper, we investigate that under which conditions of the geodesic curvature of unit speed curve $\gamma$ that lies on $S_1^2$ or $H^2,$ the curve $\alpha$ which is obtained by using $\gamma$, is a spherical helix or slant helix in Minkowski 3-space.

Keywords

• Geodesic curvature
• Sabban frame
• Minkowski space
• Pseudosphere
• Hyperbolic plane
• Helix

References

1. [1] Ali Ahmad, T. and Lopez, R., Slant helices in Minkowski space E3 1 , J. Korean Math. Soc. 48, no. 1, 159–167. 2011.
2. [2] Altunkaya, B. and Kula, L., Some characterizations of slant and spherical helices due to sabban frame, Mathematical Sciences and Applications E-Notes, Vol 3, No. 2, 64-73, 2015.
3. [3] Babaarslan, M. and Yaylı, Y., On spacelike constant slope surfaces and Bertrand curves in Minkowski 3-space, Annals of the Alexandru Ioan Cuza University -Mathematics, 2015. doi:10.1515/aicu-2015-0009.
4. [4] Encheva, R. and Georgiev, G., Shapes of space curves, Journal for Geometry and Graphics, Vol 7, No. 2, 145-155, 2003.
5. [5] Güner, G. and Ekmekci, N., On the spherical curves and Bertrand curves in Minkowski 3-space, J. Math. Comput. Sci., No. 4, 898-906, 2012.
6. [6] Izuyama, S., Pei, D.H., Sano, T., and Torii, E., Evolutes of hyperbolic plane curves, Acta Mathematica Sinica, Vol.20, no.3, pp. 543-550, 2004.
7. [7] Izumiya, S. and Takeuchi, N., New special curves and developable surfaces, Turk. J. Math. 28, 153-163, 2004.
8. [8] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, arXiv:0810.3351v1 [math.DG], 2008.

9. [9] Pekmen, U¨ . and Pas¸alı, S., Some characterizations of Lorentzian spherical spacelike curves, Mathematica Moravica 3, 31-37, 1999.
10. [10] Petrovic-Torgasev, M. and Sucurovic, E., Some characterizations of Lorentzian spherical spacelike curves with the timelike and the null principal normal, Math. Moravica, 4, 83-92, 2000.
11. [11] Petrovic-Torgasev, M. and Sucurovic, E., Some characterizations of Lorentzian spherical timelike and null curves, Matematicki Vesnik, 53, 21-27, 2001.
12. [12] Petrovic-Torgasev, M. and Sucurovic, E., Some characterizations of the spacelike, the timelike and the on the pseudohyperbolic space H2 0 in E3 1 , Krugajevac J. Math., 22, 71-82, 2000.