AIMS Mathematics, 2019, 4(3): 384-396. doi: 10.3934/math.2019.3.384.

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Non-null slant ruled surfaces

Delibekirli Village, Tepe Street, No. 63, 31440 Kırıkhan, Hatay, Turkey

In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space $E_{1}^{3} $. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in $E_{1}^{3} $. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line.
  Article Metrics

Keywords non-null ruled surface; Frenet frame; slant ruled surface

Citation: Mehmet Önder. Non-null slant ruled surfaces. AIMS Mathematics, 2019, 4(3): 384-396. doi: 10.3934/math.2019.3.384


  • 1.R. A. Abdel-Baky, Slant ruled surface in the Euclidean 3-space, Sci. Magna, 9 (2013), 107-112.
  • 2.A. T. Ali and R. Lopez, Slant helices in Minkowski space $E_{1}^{3}$, J. Korean Math. Soc., 48 (2011) 159-167.
  • 3.A. T. Ali and M. Turgut, Position vector of a time-like slant helix in Minkowski 3-space, J. Math. Anal. Appl., 365 (2010) 559-569.
  • 4. A. T. Ali, Position vectors of slant helices in Euclidean 3-space, J. Egypt. Math. Soc., 20 (2012) 1-6.
  • 5.M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125 (1997) 1503-1509.
  • 6.J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, New York: Marcel Dekker, 1981.
  • 7.N. Ekmekçi and H. H. Hacı}salihoğlu, On helices of a Lorentzian manifold, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 45 (1996) 45-50.
  • 8.A. Ferrandez, A. Gimenez and P. Lucas, Null helices in Lorentzian space forms, Int. J. Mod. Phys. A, 16 (2001) 4845-4863.
  • 9.S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2004) 153-163.
  • 10.O. Kaya and M. Önder, Position vector of a developable $h$-slant ruled surface, TWMS J. App. Eng. Math., 7 (2017) 322-331.
  • 11.O. Kaya and M. Önder, Position vector of a developable $q$-slant ruled surface, Korean J. Math., 26 (2018) 545-559.
  • 12.E. Kasap and N. Kuruoğlu, The Bertrand offsets of ruled surfaces in $IR_{1}^{3}$, Acta Math. Vietnam., 31 (2006) 39-48.
  • 13.H. Kocayiğit and M. Önder, Timelike curves of constant slope in Minkowski space $E_{1}^{4}$, J. Sci. Techn. Beykent Univ., 1 (2007) 311-318.
  • 14.L. Kula and Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169 (2005) 600-607.
  • 15.A. Küçük, On the developable timelike trajectory ruled surfaces in Lorentz 3-space $IR_{1}^{3}$, Appl. Math. Comput., 157 (2004) 483-489.
  • 16.B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, London: Academic Press, 1983.
  • 17.M. Önder, Similar ruled surfaces with variable transformations in Minkowski 3-space, TWMS J. App. Eng. Math., 5}(2) (2015) 219-230.
  • 18.M. Önder, H. Kocayiğit and M. Kazaz, Spacelike helices in Minkowski 4-space $E_{1}^{4}$, Ann. Univ. Ferrara, 56 (2010) 335-343.
  • 19.M. Önder and H. H. Uğurlu, Frenet frames and invariants of timelike ruled surfaces, Ain Shams Eng. J., 4 (2013) 507-513.
  • 20.M. Önder and H. H. Uğurlu, On the developable Mannheim offsets of timelike ruled surfaces, Proc. Natl. Acad. Sci., India, Sect. A, 84 (2014) 541-548.
  • 21.M. Önder and H. H. Uğurlu, Frenet frames and Frenet invariants of spacelike ruled surfaces, Dokuz Eylul Univ. Fac. Eng. J. Sci. Eng., 19 (2017) 712-722.
  • 22.M. Önder and O. Kaya, Slant null scrolls in Minkowski 3-space $E_{1}^{3}$, Kuwait J. Sci., 43 (2016) 31-47.
  • 23.M. Önder and O. Kaya, Characterizations of slant ruled surfaces in the Euclidean 3-space, Caspian J. Math. Sci., 6 (2017) 31-46.
  • 24.M. Önder, Slant ruled surfaces, Trans. J. Pure Appl. Math., 1 (2018) 63-82.
  • 25.D. J. Struik, Lectures on Classical Differential Geometry, 2 Eds., Dover: Addison Wesley, 1988.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved