• Lucas, Pascual ;
  • Ortega-Yagues, Jose Antonio
  • Received : 2012.04.24
  • Published : 2013.07.31


Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.


Bertrand curve;general helix;null curve;non-null curve


  1. Y. A. Aminov, Differential Geometry and Topology of Curves, Gordon and Breach Science Publishers, Singapore, 2000.
  2. M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1503-1509.
  3. M. Barros, A. Ferrandez, P. Lucas, and M. A. Merono, Solutions of the Betchov-Da Rios soliton equation: A Lorentzian approach, J. Geom. Phys. 31 (1999), no. 2-3, 217-228.
  4. M. Barros, A. Ferrandez, P. Lucas, and M. A. Merono, General helices in the 3-dimensional Lorentzian space forms, Rocky Mountain J. Math. 31 (2001), no. 2, 373-388.
  5. J. Bertrand, Memoire sur la theorie des courbes e double courbure, Comptes Rendus 36 (1850); Journal de Mathematiques Pures et Appliquees 15 (1850), 332-350.
  6. G. S. Birman and K. Nomizu, Trigonometry in Lorentzian geometry, Amer. Math. Monthly 91 (1984), no. 9, 543-549.
  7. G. S. Birman and K. Nomizu, Gauss-Bonnet theorem for 2-dimensional spacetimes, Michigan Math. J. 31 (1984), no. 1, 77-81.
  8. Y.-M. Cheng and C.-C. Lin, On the generalized Bertrand curves in Euclidean N-spaces, Note Mat. 29 (2009), no. 2, 33-39.
  9. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, 1996.
  10. N. Ekmekci and K. Ilarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.
  11. S. Ersoy and M. Tosun, Timelike Bertrand Curves in Semi-Euclidean Space, arXiv:1003.1220v1[math.DG]
  12. A. Ferrandez, A. Gimenez, and P. Lucas, Null helices in Lorentzian space forms, Internat. J. Modern Phys. A 16 (2001), no. 30, 4845-4863.
  13. A. Gorgulu and E. Ozdamar, A generalization of the Bertrand curves as general inclined curves in En, Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist. 35 (1986), no. 1-2, 53-60.
  14. A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapter 3.5, Clothoids, pages 64-66. CRC Press, Boca Raton, FL, 2nd edition, 1997.
  15. G. Harary and A. Tal, 3D Euler spirals for 3D curve completion, In Proceedings of the 2010 Annual Symposium on Computational Geometry SoCG'10 (2010), 393-402. ISBN:978-1-4503-0016-2.
  16. M. Kulahc and M. Ergut, Bertrand curves of AW(k)-type in Lorentzian space, Nonlinear Anal. 70 (2009), no. 4, 1725-1731.
  17. H. F. Lai, Weakened Bertrand curves, Tohoku Math. J. 19 (1967), 141-155.
  18. H. Matsuda and S. Yorozu, Notes on Bertrand curves, Yokohama Math. J. 50 (2003), no. 1-2, 41-58.
  19. E. Nessovic, M. Petrovic-Torgasev, and L. Verstraelen, Curves in Lorentzian spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005), no. 3, 685-696.
  20. H. B. Oztekin, Weakened Bertrand curves in the Galilean space $G_3$, J. Adv. Math. Stud. 2 (2009), no. 2, 69-76.
  21. L. R. Pears, Bertrand curves in Riemannian space, J. London Math. Soc. (1935) s1-10 (3), 180-183.
  22. J. C. Saint-Venant, Memoire sur les lignes courbes non planes, Journal d'Ecole Polytechnique 30 (1845), 1-76.
  23. M. Y. Yilmaz and M. Bektas, General properties of Bertrand curves in Riemann-Otsuki space, Nonlinear Anal. 69 (2008), no. 10, 3225-3231.

Cited by

  1. Curves in Three Dimensional Riemannian Space Forms vol.66, pp.3-4, 2014,
  2. Mannheim Curves in Nonflat 3-Dimensional Space Forms vol.2015, 2015,