Natural and conjugate mates of Frenet curves in three-dimensional Lie group

Abstract

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Keywords

• Natural mate
• conjugate mate
• helix
• slant helix
• spherical curve
• rectifying curve
• Salkowski curve
• anti-Salkowski curve

References

1. Bozkurt, Z., Gok, I., Okuyucu, O. Z., Ekmekci, F. N., Characterizations of rectifying, normal and osculating curves in three dimensional compact lie groups, Life Science Journal, 10 (3) (2013), 819-823.
2. Çiftçi, Ü., A generalization of lancret's theorem, Journal of Geometry and Physics, 59 (12) (2009), 1597-1603.
3. Chen, B.-Y., When does the position vector of a space curve always lie in its rectifying plane?, The American mathematical monthly, 110 (2) (2003), 147-152.
4. Choi, J. H., Kim, Y. H., Associated curves of a Frenet curve and their applications, Appl.Math. Comput., 218 (18) (2012), 9116-9124, https://dx.doi.org/10.1016/j.amc.2012.02.064
5. Deshmukh, S., Chen, B.-Y., Alghanemi, A., Natural mates of Frenet curves in Euclidean 3-space, Turkish J. Math., 42 (5) (2018), 2826-2840.
6. Do Carmo, M. P., Differential geometry of curves and surfaces: revised and updated second edition, Courier Dover Publications, 2016.
7. do Esprito-Santo, N., Fornari, S., Frensel, K., Ripoll, J., Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math., 111 (4) (2003), 459-470. https://dx.doi.org/10.1007/s00229-003-0357-5
8. Fokas, A. S., Gelfand, I. M., Surfaces on Lie groups, on Lie algebras, and their integrability, Comm. Math. Phys., 177 (1) (1996), 203-220.

9. Gök, I., Okuyucu, O. Z., Ekmekci, N., Yayl , Y., On Mannheim partner curves in three dimensional Lie groups, Miskolc Math. Notes, 15 (2) (2014), 467-479, https://dx.doi.org/10.18514/mmn.2014.682
10. Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2) (2004), 153-164.
11. Kızıltug, S., Önder, M., Associated curves of Frenet curves in three dimensional compact Lie group, Miskolc Math. Notes, 16 (2) (2015), 953-964, https://dx.doi.org/10.18514/MMN.2015.1324
12. Lancret, M. A., Memoire sur less courbes a double courbure, Memoires presentes a Institut, 1 (1806), 416-454.
13. Monterde, J., Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design, 26 (3) (2009), 271-278.
14. Okuyucu, O. Z., Gök, I., Yayli, Y., Ekmekci, N., Slant helices in three dimensional lie groups, Applied Mathematics and Computation, 221 (2013), 672-683.
15. Okuyucu, O. Z., Gok, I., Yayli, Y., Ekmekci, N., Bertrand curves in three dimensional lie groups, Miskolc Mathematical Notes, 17 (2) (2016), 999-1010.
16. Oztürk, U., Alkan, Z. B., Darboux helices in three dimensional lie groups, AIMS Mathematics, 5 (4) (2020), 3169.
17. Salkowski, E., Zur transformation von raumkurven, Mathematische Annalen, 66 (4) (1909), 517-557.
18. Struik, D. J., Lectures on classical di erential geometry, Courier Corporation, 1961.
19. Yoon, D. W., General helices of AW(k)-type in the Lie group, J. Appl. Math. (2012), Art.ID 535123, 10. https://dx.doi.org/10.1155/2012/535123