AIMS Mathematics, 2020, 5(4): 3169-3181. doi: 10.3934/math.2020204

Research article

Export file:


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


  • Citation Only
  • Citation and Abstract

Darboux helices in three dimensional Lie groups

Department of Mathematics, Faculty of Science, University of Çankırı Karatekin, 18100 Çankırı, Turkey

In this paper, we introduce Darboux helices in a three dimensional Lie group G with a bi-invariant metric and give some characterizations of Darboux helices. Besides, we give some relations between some special curves (general helices and slant helices) and Darboux helices. Moreover, we prove that all Darboux helices are not a slant helix if G is commutative.
  Article Metrics


1. D. J. Struik, Lectures on classical differential geometry, Reading, MA: Addison, 1988.

2. S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2004), 153-163.

3. A. Menninger, Characterization of the slant helix as successor curve of the general helix, International Electronic Journal of Geometry, 7 (2014), 84-91.

4. M. Barros, A. Ferrández, P. Lucas, et al. General helices in the three-dimensional Lorentzian space forms, Rocky Mt. J. Math., 31 (2001), 373-388.    

5. P. Lucas, J. A. Ortega-Yagües, Slant helices in the Euclidean 3-space revisited, B. Belg. Math. Soc-Sim., 23 (2016), 133-150.

6. E. Ziplar, A. Senol, Y. Yayli, On Darboux helices in euclidean 3-space, Global Journal of Science Frontier Research Mathematics and Decision Sciences, 12 (2012), 73-80.

7. N. Macit, M. Düldül, Relatively normal-slant helices lying on a surface and their characterizations, Hacet. J. Math. Stat., 46 (2017), 397-408.

8. A. Şenol, E. Ziplar, Y. Yayli, et al. A new approach on helices in Euclidean n-space, Math. Commun., 18 (2013), 241-256.

9. I. Gök, C. Camci, H. H. Hacisalihoğlu, Vn-slant helices in Euclidean n-space En, Math. Commun., 14 (2009), 317-329.

10.L. Kula, N. Ekmekci, Y. Yaylı, et al. Characterizations of slant helices in Euclidean 3-space, Turk. J. Math., 34 (2010), 261-273.

11. L. Kula, Y. Yayli, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169 (2005), 600-607.

12. E. Özdamar, H. H. Hacisalihoğlu, A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci. Univ. Ankara Sér. A1 Math., 24 (1975), 15-23.

13. G. Öztürk, B. Bulca, B. Bayram, et al. Focal representation of k-slant helices in $\Bbb E^{m+1}$, Acta Universitatis Sapientiae, Mathematica, 7 (2015), 200-209.

14. B. Uzunoğlu, I. Gök, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput., 275 (2016), 317-323.

15. U. Çiftçi, A generalization of Lancret's theorem, J. Geom. Phys., 59 (2009), 1597-1603.    

16.O. Zeki Okuyucu, I. Gök, Y. Yayli, et al. Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013), 672-683.

17. A. Yampolsky, A. Opariy, Generalized helices in three-dimensional Lie groups, Turk. J. Math., 43 (2019), 1447-1455.    

18. N. do Espírito-Santo, S. Fornari, K. Frensel, et al. Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math., 111 (2003), 459-470.    

© 2020 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

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved