Journal of Geometry

, 110:8 | Cite as

Bi-null curves with constant curvatures in \({\mathbb {R}}_{2}^{5}\)

  • Ali Uçum
  • Makoto Sakaki
  • Kazım İlarslanEmail author


In the present paper, we classify bi-null curves with constant curvatures in semi-Euclidean 5-space \({\mathbb {R}}_{2}^{5}\) with index 2.


Bi-null curves semi-Euclidean space curvatures Frenet equations 

Mathematics Subject Classification

Primary 53B30 53A04 



The authors express thank to the referees and Prof. Dr. Hans Havlicek (Editor-in-Chief) for their valuable suggestions and remarks.


  1. 1.
    Chen, B.Y., Deprez, J., Verheyen, P.: Immersions with geodesics of 2-type. In: Dillen, F. (ed.) Geometry and Topology of Submanifolds, IV, Proceedings of the Conference on Differential Geometry and Vision, Leuven 27–29 June 1991, pp. 87–110. World Scientific, Singapore (1992)Google Scholar
  2. 2.
    Ferrandez, A., Gimenez, A., Lucas, P.: Null helices in Lorentzian space forms. Int. J. Mod. Phys. A 16(30), 4845–4863 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ferrandez, A., Gimenez, A., Lucas, P.: Degenerate curves in pseudo-Euclidean spaces of index two. In: Geometry, Integrability and Quantization (Varna, pp. 209–223, p. 2002. Coral Press Scientific Publication, Sofia) (2001)Google Scholar
  4. 4.
    İlarslan, K., Nesovic, E.: On Bishop frame of a null Cartan curve in Minkowski space-time. Int. J. Geom. Methods Mod. Phys 15(8), 16 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Klein, F., Lie, S.: Über diejenigen ebenen Curven welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen. Math. Ann. 4(1), 50–84 (1871)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lopez, R.: Differential geometry of curves and surfaces in Lorentz–Minkowski space. Int. Electron. J. Geom. 7(1), 44–107 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Low, R.J.: Framing curves in Euclidean and Minkowski space. J. Geom. Symmetry Phys. 27, 83–91 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Petrovic-Torgasev, M., Sucurovic, E.: W-curves in Minkowski space-time. Novi Sad J. Math 32, 55–65 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Petrovic-Torgasev, M., İlarslan, K., Nesovic, E.: On partially null and pseudo null curves in the semi-Euclidean space \({{\mathbb{R}}_{2}^{4}}\). J. Geom. 84, 106–116 (2005)Google Scholar
  10. 10.
    Sakaki, M.: Bi-null Cartan curves in semi-Euclidean spaces of index 2. Beitr. Algebra Geom. 53(2), 421–436 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Synge, J.L.: Timelike helices in flat space-time. Proc. R. Ir. Acad. A65, 27–42 (1967)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Uçum, A., Sakaki, M., İlarslan, K.: On bi-null curves in semi-Euclidean 6-space with index 3. J. Geom. 107, 719–728 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Walrave, J.: Curves and surfaces in Minkowski space. Doctoral thesis, K.U. Leuven, Faculty of Science, Leuven (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and ArtsKırıkkale UniversityKırıkkaleTurkey
  2. 2.Graduate School of Science and TechnologyHirosaki UniversityHirosakiJapan

Personalised recommendations