Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Annals of the Alexandru Ioan Cuza University - Mathematics

The Journal of "Alexandru Ioan Cuza" University from Iasi

Editor-in-Chief: Oniciuc, Cezar

2 Issues per year


CiteScore 2016: 0.34

SCImago Journal Rank (SJR) 2016: 0.231
Source Normalized Impact per Paper (SNIP) 2016: 0.488

Mathematical Citation Quotient (MCQ) 2015: 0.10

Open Access
Online
ISSN
1221-8421
See all formats and pricing
More options …

Generalized Null Bertrand Curves In Minkowski Space-Time

Ferdag Kahraman Aksoyak / Ismail Gok / Kazim Ilarslan
Published Online: 2014-11-24 | DOI: https://doi.org/10.2478/aicu-2013-0031

Abstract

Çöken and ÇIFTCI proved that a null Cartan curve in Minkowski space-time E41 is a null Bertrand curve if and only if k2 is nonzero constant and k3 is zero. That is, the null curve with non-zero curvature k2 is not a Bertrand curve in Minkowski space-time E41.

So, in this paper we defined a new type of Bertrand curve in Minkowski space-time E41 for a null curve with non-zero curvature k3 by using the similar idea of generalized Bertrand curve given by Matsuda and Yorozu and we called it a null (1, 3)-Bertrand curve. Also, we proved that if a null curve with non-zero curvatures in Minkowski space-time E41 is a null (1, 3)-Bertrand curve then it is a null helix. We give an example of such curves.

Keywords: Minkowski space-time; null curve; Frenet vectors; Bertrand curves

MSC: 53C50; 53B30

References

  • 1. Balgetir, H.; Bektaş, M.; Inoguchi, J. - Null Bertrand curves in Minkowski 3-space and their characterizations, Note Mat., 23 (2004/05), 7-13.Google Scholar

  • 2. Balgetir, H.; Bektąs, M.; Ergüt, M. - Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic J., 27 (2004), 229-236.Google Scholar

  • 3. Bertrand, J.M. - Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 36 (1850).Google Scholar

  • 4. Bioche, Ch. - Sur les courbes de M. Bertrand, Bull. Soc. Math. France, 17 (1889), 109-112.Google Scholar

  • 5. Cöken, A.C.; Ciftçi, U. - On the Cartan curvatures of a null curve in Minkowski spacetime, Geom. Dedicata 114 (2005), 71-78.Google Scholar

  • 6. Burke, J.F. - Bertrand curves associated with a pair of curves, Math. Mag., 34 (1960), 60-62.CrossrefGoogle Scholar

  • 7. Ekmekci, N.; Ilarslan, K. - On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 3 (2001), 17-24 (electronic).Google Scholar

  • 8. Göçmen, M.; Keleş, S. - Notes on null Bertrand curves in Minkowski spacetime, Int. J. Contemp. Math. Sci., 6 (2011), 2105-2120.Google Scholar

  • 9. Ilarslan, K. - Some special curves on non-Euclidean manifolds, Doctoral thesis, Ankara University, Graduate School of Natural and Applied Sciences, 2002.Google Scholar

  • 10. Jin, D.H. - Null Bertrand curves in a Lorentz manifold, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 15 (2008), 209-215.Google Scholar

  • 11. Kühnel, W. - Differential Geometry: Curves-Surfaces-Manifolds, Braunschweig, Wiesbaden, 1999.Google Scholar

  • 12. Matsuda, H.; Yorozu, S. - Notes on Bertrand curves, Yokohama Math. J., 50 (2003), 41-58.Google Scholar

  • 13. O'Neill, B. - Semi-Riemannian Geometry, With applications to relativity, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.Google Scholar

  • 14. Pears, L.R. - Bertrand curves in Riemannian space, J. London Math. Soc, s1-10, 2 (1935), 180-183.Google Scholar

  • 15. Saint Venant, B. - Memoire planes, Journal de l'Ecole Polytechnique, 18 (1845), 1-76.Google Scholar

  • 16. Walrave, J. - Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium), 1995.Google Scholar

About the article

Received: 2011-11-24

Revised: 2012-10-23

Accepted: 2012-12-13

Published Online: 2014-11-24


Citation Information: Annals of the Alexandru Ioan Cuza University - Mathematics, ISSN (Online) 1221-8421, DOI: https://doi.org/10.2478/aicu-2013-0031.

Export Citation

© 2014 Ferdag Kahraman Aksoyak et. al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in