Bertrand Curves and B-Lift Curves in Lorentzian 3-Space

Abstract

In this article, based on Thorpe's definition, we define a new curve called the B-lift curve in Lorentzian 3-space and examine the Frenet vectors of the B-lift curve. Furthermore, we examine the relationship between the Frenet vectors of the Bertrand curve and the Frenet vectors of the natural lift curve. Finally, we give an example on these results.

Keywords

• B-lift curve
• Bertrand curve
• Lorentzian space

References

1. Ekmekci, N., İlarslan, K., On Bertrand curves and their characterization, Differential Geometry Dynamical Systems, 3(2)(2001), 17-24.
2. Ergün, E., Bayram, E., Surface family with a common natural geodesic lift, International J.Math. Combin., 1(2016), 34-41.
3. Ergün, E., Çalışkan, M., The natural lift curve of the spherical indicatrix of a non-null curve in Minkowski 3-Space, International Mathematical Forum, 7(2012), 707-717.
4. Ergün, E., Çalışkan, M., On natural lift of a curve, Pure Mathematical Sciences, 2(2012), 81-85.
5. Ergün, E., Çalışkan, M., On the natural lift curve and the Bertrand mate, Journal of Science and Arts, 42(1)(2018), 75-94.
6. Karaca, E., Çalışkan, M., Ruled surfaces and tangent bundle of unit 2-sphere of natural lift curves, Gazi University Journal of Science, 33(2020), 751-759.
7. O'Neill, B., Semi Riemann Geometry, Academic Press, New York and London, 1983.
8. Önder, M., Uğurlu, H.H., Frenet frames and invariants of timelike ruled surfaces, Ain Shams Engineering Journal, 4(2013), 507-513.

9. Öztekin, H.B., Bektaş, M., On representation formulae for Bertrand curves in the Minkowski 3-space, Scientia Magna, 6(11)(2010), 89-96.
10. Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, Inc., 1994.
11. Thorpe, J.A., Elementary Topics in Differential Geometry, Springer-Verlag, New York, Heidelberg-Berlin, 1979.
12. Walrave, J., Curves and Surfaces in Minkowski Space, K. U. Leuven Faculteit, Der Wetenschappen, 1995.