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

Year 2021, Volume: 70 Issue: 1, 522 - 540, 30.06.2021
https://doi.org/10.31801/cfsuasmas.785489

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.

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References

  • 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.
  • Çiftçi, Ü., A generalization of lancret's theorem, Journal of Geometry and Physics, 59 (12) (2009), 1597-1603.
  • 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.
  • 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
  • Deshmukh, S., Chen, B.-Y., Alghanemi, A., Natural mates of Frenet curves in Euclidean 3-space, Turkish J. Math., 42 (5) (2018), 2826-2840.
  • Do Carmo, M. P., Differential geometry of curves and surfaces: revised and updated second edition, Courier Dover Publications, 2016.
  • 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
  • Fokas, A. S., Gelfand, I. M., Surfaces on Lie groups, on Lie algebras, and their integrability, Comm. Math. Phys., 177 (1) (1996), 203-220.
  • 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
  • Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2) (2004), 153-164.
  • 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
  • Lancret, M. A., Memoire sur less courbes a double courbure, Memoires presentes a Institut, 1 (1806), 416-454.
  • 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.
  • Okuyucu, O. Z., Gök, I., Yayli, Y., Ekmekci, N., Slant helices in three dimensional lie groups, Applied Mathematics and Computation, 221 (2013), 672-683.
  • Okuyucu, O. Z., Gok, I., Yayli, Y., Ekmekci, N., Bertrand curves in three dimensional lie groups, Miskolc Mathematical Notes, 17 (2) (2016), 999-1010.
  • Oztürk, U., Alkan, Z. B., Darboux helices in three dimensional lie groups, AIMS Mathematics, 5 (4) (2020), 3169.
  • Salkowski, E., Zur transformation von raumkurven, Mathematische Annalen, 66 (4) (1909), 517-557.
  • Struik, D. J., Lectures on classical di erential geometry, Courier Corporation, 1961.
  • 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
Year 2021, Volume: 70 Issue: 1, 522 - 540, 30.06.2021
https://doi.org/10.31801/cfsuasmas.785489

Abstract

Project Number

---

References

  • 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.
  • Çiftçi, Ü., A generalization of lancret's theorem, Journal of Geometry and Physics, 59 (12) (2009), 1597-1603.
  • 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.
  • 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
  • Deshmukh, S., Chen, B.-Y., Alghanemi, A., Natural mates of Frenet curves in Euclidean 3-space, Turkish J. Math., 42 (5) (2018), 2826-2840.
  • Do Carmo, M. P., Differential geometry of curves and surfaces: revised and updated second edition, Courier Dover Publications, 2016.
  • 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
  • Fokas, A. S., Gelfand, I. M., Surfaces on Lie groups, on Lie algebras, and their integrability, Comm. Math. Phys., 177 (1) (1996), 203-220.
  • 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
  • Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2) (2004), 153-164.
  • 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
  • Lancret, M. A., Memoire sur less courbes a double courbure, Memoires presentes a Institut, 1 (1806), 416-454.
  • 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.
  • Okuyucu, O. Z., Gök, I., Yayli, Y., Ekmekci, N., Slant helices in three dimensional lie groups, Applied Mathematics and Computation, 221 (2013), 672-683.
  • Okuyucu, O. Z., Gok, I., Yayli, Y., Ekmekci, N., Bertrand curves in three dimensional lie groups, Miskolc Mathematical Notes, 17 (2) (2016), 999-1010.
  • Oztürk, U., Alkan, Z. B., Darboux helices in three dimensional lie groups, AIMS Mathematics, 5 (4) (2020), 3169.
  • Salkowski, E., Zur transformation von raumkurven, Mathematische Annalen, 66 (4) (1909), 517-557.
  • Struik, D. J., Lectures on classical di erential geometry, Courier Corporation, 1961.
  • 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
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Mahmut Mak 0000-0003-3558-9161

Project Number ---
Publication Date June 30, 2021
Submission Date August 25, 2020
Acceptance Date February 1, 2021
Published in Issue Year 2021 Volume: 70 Issue: 1

Cite

APA Mak, M. (2021). Natural and conjugate mates of Frenet curves in three-dimensional Lie group. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 522-540. https://doi.org/10.31801/cfsuasmas.785489
AMA Mak M. Natural and conjugate mates of Frenet curves in three-dimensional Lie group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2021;70(1):522-540. doi:10.31801/cfsuasmas.785489
Chicago Mak, Mahmut. “Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 1 (June 2021): 522-40. https://doi.org/10.31801/cfsuasmas.785489.
EndNote Mak M (June 1, 2021) Natural and conjugate mates of Frenet curves in three-dimensional Lie group. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 1 522–540.
IEEE M. Mak, “Natural and conjugate mates of Frenet curves in three-dimensional Lie group”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 1, pp. 522–540, 2021, doi: 10.31801/cfsuasmas.785489.
ISNAD Mak, Mahmut. “Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/1 (June 2021), 522-540. https://doi.org/10.31801/cfsuasmas.785489.
JAMA Mak M. Natural and conjugate mates of Frenet curves in three-dimensional Lie group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:522–540.
MLA Mak, Mahmut. “Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 1, 2021, pp. 522-40, doi:10.31801/cfsuasmas.785489.
Vancouver Mak M. Natural and conjugate mates of Frenet curves in three-dimensional Lie group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(1):522-40.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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