On special singular curve couples of framed curves in 3D Lie groups

Bahar Doğan Yazıcı; Osman Zeki Okuyucu; Murat Tosun

doi:10.31801/cfsuasmas.1197154

Research Article

On special singular curve couples of framed curves in 3D Lie groups

Year 2023, Volume: 72 Issue: 3, 710 - 720, 30.09.2023

Bahar Doğan Yazıcı Osman Zeki Okuyucu Murat Tosun

https://doi.org/10.31801/cfsuasmas.1197154

Abstract

In this paper, we introduce Bertrand and Mannheim curves of framed curves, which are a special singular curve in 3D Lie groups. We explain the conditions for framed curves to be Bertrand curves and Mannheim curves in 3D Lie groups. We give relationships between framed curvatures and Lie curvatures of Bertrand and Mannheim curves of framed curves. In addition, we obtain the characterization of Bertrand and Mannheim curves according to the various frames of framed curves in 3D Lie groups.

Keywords

Bertrand curves, Mannheim curves, framed curves, Lie groups

References

Aminov, Y., Differential Geometry and the Topology of Curves, Translated from the Russian by V. Gorkavy, Amsterdam, the Netherlands: Gordon and Breach Science Publishers, 2000.
Bertrand, J., Memoire sur la theorie des courbes a double courbure, J. de Methematiques Pures et Appliquees, 15 (1850), 332–350 (in French).
Çiftçi, Ü., A generalization of Lancert’s theorem, J. Geom. Phys., 59 (2009), 1597–1603. https://doi.org/10.1016/j.geomphys.2009.07.016
Crouch, P., Silva Leite, F., The dynamic interpolation problem: On Riemannian manifolds, Lie groups and symmetric spaces, J. Dyn. Control Syst., 1(2) (1995), 177–202. https://doi.org/10.1007/BF02254638
do Espirito-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://doi.org/10.1007/s00229-003-0357-5
Doğan Yazıcı, B., Karakuş, S. Ö., Tosun, M., On the classification of framed rectifying curves in Euclidean space, Math. Methods Appl. Sci., 45(18) (2022), 12089–12098. http://dx.doi.org/10.1002/mma.7561
Doğan Yazıcı, B., Okuyucu, O. Z., Tosun, M., Framed curves in three-dimensional Lie groups and a Berry phase model, J. Geom. Phys., 182 (2022), 104682. https://doi.org/10.1016/j.geomphys.2022.104682
Fukunaga, T., Takahashi, M., Existence conditions of framed curves for smooth curves, J. Geo., 108 (2017), 763–774. https://doi.org/10.1007/s00022-017-0371-5
Gök, İ., Okuyucu, O. Z., Ekmekci, N., Yaylı, Y., On Mannheim partner curves in three dimensional Lie groups, Miskolc Math. Notes, 15(2) (2014), 467–479. https://doi.org/10.18514/mmn.2014.682
Honda, S., Takahashi, M., Framed curves in the Euclidean space, Adv. Geo., 16 (2016), 265–276. https://doi.org/10.1515/advgeom-2015-0035
Honda, S., Takahashi, M., Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turk. J. Math., 44 (2020), 883–899. https://doi.org/10.3906/mat-1905-63
Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, J. Geo., 74 (2002), 97–109. https://doi.org/10.1007/PL00012543
Liu, H., Wang, F., Mannheim partner curves in 3-space, J. Geo., 88 (2008), 120–126. https://doi.org/10.1007/s00022-007-1949-0
Okuyucu, 0. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013), 672–683. https://doi.org/10.1016/j.amc.2013.07.008
Okuyucu, 0. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Bertrand curves in three dimensional Lie groups, Miskolc Math. Notes, 17(2) (2017), 999–1010. https://doi.org/10.18514/MMN.2017
Wang, Y., Pei, D., Gao, R., Generic properties of framed rectifying curves, Mathematics, 7(1) (2019), 37. https://doi.org/10.3390/math7010037

Year 2023, Volume: 72 Issue: 3, 710 - 720, 30.09.2023

Bahar Doğan Yazıcı Osman Zeki Okuyucu Murat Tosun

https://doi.org/10.31801/cfsuasmas.1197154

Abstract

References

Aminov, Y., Differential Geometry and the Topology of Curves, Translated from the Russian by V. Gorkavy, Amsterdam, the Netherlands: Gordon and Breach Science Publishers, 2000.
Bertrand, J., Memoire sur la theorie des courbes a double courbure, J. de Methematiques Pures et Appliquees, 15 (1850), 332–350 (in French).
Çiftçi, Ü., A generalization of Lancert’s theorem, J. Geom. Phys., 59 (2009), 1597–1603. https://doi.org/10.1016/j.geomphys.2009.07.016
Crouch, P., Silva Leite, F., The dynamic interpolation problem: On Riemannian manifolds, Lie groups and symmetric spaces, J. Dyn. Control Syst., 1(2) (1995), 177–202. https://doi.org/10.1007/BF02254638
do Espirito-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://doi.org/10.1007/s00229-003-0357-5
Doğan Yazıcı, B., Karakuş, S. Ö., Tosun, M., On the classification of framed rectifying curves in Euclidean space, Math. Methods Appl. Sci., 45(18) (2022), 12089–12098. http://dx.doi.org/10.1002/mma.7561
Doğan Yazıcı, B., Okuyucu, O. Z., Tosun, M., Framed curves in three-dimensional Lie groups and a Berry phase model, J. Geom. Phys., 182 (2022), 104682. https://doi.org/10.1016/j.geomphys.2022.104682
Fukunaga, T., Takahashi, M., Existence conditions of framed curves for smooth curves, J. Geo., 108 (2017), 763–774. https://doi.org/10.1007/s00022-017-0371-5
Gök, İ., Okuyucu, O. Z., Ekmekci, N., Yaylı, Y., On Mannheim partner curves in three dimensional Lie groups, Miskolc Math. Notes, 15(2) (2014), 467–479. https://doi.org/10.18514/mmn.2014.682
Honda, S., Takahashi, M., Framed curves in the Euclidean space, Adv. Geo., 16 (2016), 265–276. https://doi.org/10.1515/advgeom-2015-0035
Honda, S., Takahashi, M., Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turk. J. Math., 44 (2020), 883–899. https://doi.org/10.3906/mat-1905-63
Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, J. Geo., 74 (2002), 97–109. https://doi.org/10.1007/PL00012543
Liu, H., Wang, F., Mannheim partner curves in 3-space, J. Geo., 88 (2008), 120–126. https://doi.org/10.1007/s00022-007-1949-0
Okuyucu, 0. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013), 672–683. https://doi.org/10.1016/j.amc.2013.07.008
Okuyucu, 0. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Bertrand curves in three dimensional Lie groups, Miskolc Math. Notes, 17(2) (2017), 999–1010. https://doi.org/10.18514/MMN.2017
Wang, Y., Pei, D., Gao, R., Generic properties of framed rectifying curves, Mathematics, 7(1) (2019), 37. https://doi.org/10.3390/math7010037

There are 16 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Articles
Authors	Bahar Doğan Yazıcı 0000-0001-5690-4840 Osman Zeki Okuyucu 0000-0003-4379-0252 Murat Tosun 0000-0002-4888-1412
Publication Date	September 30, 2023
Submission Date	October 31, 2022
Acceptance Date	February 6, 2023
Published in Issue	Year 2023 Volume: 72 Issue: 3

Cite

APA	Doğan Yazıcı, B., Okuyucu, O. Z., & Tosun, M. (2023). On special singular curve couples of framed curves in 3D Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(3), 710-720. https://doi.org/10.31801/cfsuasmas.1197154
AMA	Doğan Yazıcı B, Okuyucu OZ, Tosun M. On special singular curve couples of framed curves in 3D Lie groups. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2023;72(3):710-720. doi:10.31801/cfsuasmas.1197154
Chicago	Doğan Yazıcı, Bahar, Osman Zeki Okuyucu, and Murat Tosun. “On Special Singular Curve Couples of Framed Curves in 3D Lie Groups”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 3 (September 2023): 710-20. https://doi.org/10.31801/cfsuasmas.1197154.
EndNote	Doğan Yazıcı B, Okuyucu OZ, Tosun M (September 1, 2023) On special singular curve couples of framed curves in 3D Lie groups. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 3 710–720.
IEEE	B. Doğan Yazıcı, O. Z. Okuyucu, and M. Tosun, “On special singular curve couples of framed curves in 3D Lie groups”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 3, pp. 710–720, 2023, doi: 10.31801/cfsuasmas.1197154.
ISNAD	Doğan Yazıcı, Bahar et al. “On Special Singular Curve Couples of Framed Curves in 3D Lie Groups”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/3 (September 2023), 710-720. https://doi.org/10.31801/cfsuasmas.1197154.
JAMA	Doğan Yazıcı B, Okuyucu OZ, Tosun M. On special singular curve couples of framed curves in 3D Lie groups. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:710–720.
MLA	Doğan Yazıcı, Bahar et al. “On Special Singular Curve Couples of Framed Curves in 3D Lie Groups”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 3, 2023, pp. 710-2, doi:10.31801/cfsuasmas.1197154.
Vancouver	Doğan Yazıcı B, Okuyucu OZ, Tosun M. On special singular curve couples of framed curves in 3D Lie groups. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(3):710-2.