ON SPACELIKE BERTRAND CURVES IN MINKOWSKI 3-SPACE

Abstract

In this paper, we study spacelike Bertrand curves in Minkowski 3-space. According to the casual character of the principal normal vector, we show that the Bertrand mate curve can be spacelike, timelike or null curve. Thus, we give the necessary and sucient conditions for spacelike curves to be Bertrand curves and we also give the related examples.

Keywords

• Bertrand curve,Minkowski 3-space,Cartan null curve,non-null curve

References

1. [1] H. Balgetir, M. Bektaş and M. Ergüt, Bertrand curves for non-null curves in 3-dimensional Lorentzian space, Hadronic J. 27 (2004), no. 2, 229-236.
2. [2] H. Balgetir, M. Bektaş and J. Inoguchi, Null Bertrand curves in Minkowski 3-space and their characterizations, Note Mat. 23 (2004/05), no. 1, 7-13.
3. [3] J. M. Bertrand, Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 36, (1850).
4. [4] W. B. Bonnor, Null curves in a Minkowski space-time, Tensor 20 (1969), 229-242.
5. [5] W. B. Bonnor, Curves with null normals in Minkowski space-time. A random walk in relativity and cosmology, Wiley Easten Limited (1985), 33-47.
6. [6] N. Ekmekci and K. İlarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.
7. [7] D. H. Jin, Null Bertrand curves in a Lorentz manifold, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 15 (2008), no. 3, 209-215.
8. [8] S. Ozkaldı Karakuş, K. _Ilarslan and Y. Yaylı, A new approach for characterization of curve couples in Euclidean 3-space, Honam Mathematical J. 36(2014), no.1, 113-129.

9. [9] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, Braunschweig, Wiesbaden, (1999).
10. [10] H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 88 (2008), 120-126.
11. [11] H. Matsuda and S. Yorozu, Notes on Bertrand curves, Yokohama Math. J. 50 (2003), no. 1-2, 41-58.
12. [12] L. R. Pears, Bertrand curves in Riemannian space, J. London Math. Soc. Volumes 1-10, Number 2, 180-183 , July 1935.
13. [13] A. Uçum, K. İlarslan and S. Ozkaldı Karakuş, Curve couples and spacelike Frenet planes in Minkowski 3-space, Honam Mathematical J. 36 (2014), No. 3, 475-492.
14. [14] A. Uçum, K. _Ilarslan and S. Ozkaldı Karakuş, On curve couples with joint lightlike Frenet planes in Minkowski 3-space, Acta Univ. Apulensis Math. Inform. 41 (2015), 111-129.
15. [15] A. Uçum, K. _Ilarslan and S. Ozkaldı Karakuş, On curve couples with joint timelike Frenet planes in Minkowski 3-space, Acta Univ. Apulensis Math. Inform. 44 (2015), 87-108.
16. [16] A. Uçum, E. Nesovic and K. _Ilarslan, On generalized timelike Mannheim curves in Minkowski space-time, Journal of Dynamical Systems & Geomatric Theories, 13(1) (2015), 71-94.
17. [17] A. Uçum, E. Nesovic and K. _Ilarslan, On generalized spacelike Mannheim curves in Minkowski space-time, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, Volume 86, Issue 2 (2016), pp 249-258.
18. [18] A. Uçum, K. İlarslan and M. Sakaki, On (1; 3)-Cartan Null Bertrand curves in Semi-Euclidean 4-Space with index 2, J. Geom., Volume 107, Issue 3 (2015), pp 579591.
19. [19] A. Uçum, O. Kecilioğlu and K. _Ilarslan, Generalized Pseudo Null Bertrand curves in Semi- Euclidean 4-Space with index 2, Rendiconti del Circolo Matematico di Palermo, 65, Issue 3 (2016), pp 459-472.
20. [20] A. Uçum, O. Kecilioğlu and K. _Ilarslan, Generalized Bertrand curves with spacelike (1; 3)- normal plane in Minkowski space-time, Turk. J. Math. 40 (2016), 487-505.
21. [21] A. Uçum, O. Kecilioğlu and K. _Ilarslan, Generalized Bertrand curves with timelike (1; 3)- normal plane in Minkowski space-time, Kuwait J. Sci. 42 (2015), 10-27.
22. [22] A. Uçum and K. _Ilarslan, On timelike Bertrand Curves in Minkowski 3-space, Honam Mathematical Journal, Vol.38, No.3 (2016), pp.467-477.
23. [23] B. Saint Venant, Memoire sur les lignes courbes non planes, Journal de l'Ecole Polytechnique, vol. 18, pp.1-76, 1845.