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Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$

Year 2021, Volume: 9 Issue: 2, 346 - 355, 15.10.2021

Abstract

If there exists a quaternionic Bertrand curve in $\mathbb{E}^{4}$, then its torsion or
bitorsion vanishes. So we can say that there is no quaternionic Bertrand
curves whose torsion and bitorsion are non-zero. Hence by using the
method which is given by Matsuda and Yorozu [13], we give the denition
of quaternionic $(1,3)-$Bertrand curve according to Type 2-Quaternionic
Frame and obtain some results about these curves.

References

  • [1] Bertrand J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15, 332-350, 1850.
  • [2] Bharathi K., Nagaraj M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 18 (6) 507-511.
  • [3] Çetin M. Kocayiğit H., On the quaternionic Smarandache curves in Euclidean 3-space, Int.J. Contemp Math Sci 8(3), 139-150, 2013.
  • [4] Ersoy S., Tosun M., Timelike Bertrand curves in semi-Euclidean space, Int. J. Math. Stat., 14(2), 78-89, 2013.
  • [5] Gök İ., Okuyucu O.Z., Kahraman F., Hacısalihoğlu H. H., On the quaternionic B2-slant helices in the Euclidean space E4: Adv. Appl. Cli ord Algebr., 21, 707-719, 2011.
  • [6] Gök İ., Kaya Nurkan S., İlarslan K., On pseudo null Bertrand curves in Minkowski space-time, Kyungpook Math. J. 54(4), 685-697, 2014.
  • [7] Güngör M. A. and Tosun M., Some characterizations of quaternionic rectifying curves, Di er. Geom. Dyn. Syst. 13, 89-100, 2011.
  • [8] Irmak Y., Bertrand Curves and Geometric Applications in Four Dimensional Euclidean Space, MSc thesis, Ankara University, Institute of Science, 2018.
  • [9] Kahraman Aksoyak F., Gök İ.., İlarslan K., Generalized null Bertrand curves in Minkowski space-time, An. Ştiint. Univ. Al. I. Cuza, Iasi, Mat. (N.S.) 60 (2), 489-502, 2014.
  • [10] Kahraman Aksoyak F., A new type of quaternionic Frame in R4; Int. J. Geom. Methods Mod. Phys., 16 (6), 1950084 (11 pages), 2019.
  • [11] Karadağ M., Sivridağ A., Quaternion valued functions of a single real variable and inclined curves, Erciyes Univ. J. Inst. Sci. Technol 13, 23-36,1997.
  • [12] Keçilioğlu O., İlarslan K. , Quaternionic Bertrand curves in Euclidean 4- space. Bull. Math. Anal. Appl. 5 (3), 27{38, 2013.
  • [13] Matsuda H. and Yorozu S., Notes on Bertrand curves. Yokohama Math. J. 50 (1-2), 41-58, 2003.
  • [14]  Önder M., Quaternionic Salkowski curves and quaternionic similar curves, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci., 90 (3), 447-456, 2020.
  • [15]  Öztürk G., Kişi İ., Büyükkütük S. , Constant ratio quaternionic curves in Euclidean spaces. Adv. Appl. Cli ord Algebr. 27 (2), 1659-1673, 2017.
  • [16] Pears L. R., Bertrand curves in Riemannian space, J. London Math. Soc. 1-10 (2), 180-183, 1935.
  • [17] Şenyurt S., Cevahir C., Altun Y., On spatial quaternionic involute curve a new view. Adv. Appl. Cli ord Algebr. 27 (2), 1815-1824, 2017.
  • [18] Uçum A., İlarslan K., Sasaki M., On (1,3)-Cartan null Bertrand curves in semi-Euclidean 4-space with index 2, J. Geom., 107 (3), 579-591, 2016.
  • [19] Uçum A., Keçilioğlu O., İlarslan K., Generalized Bertrand curves with spacelike (1,3)-normal plane in Minkowski space-time, Turkish J. Math., 40 (3), 487-505, 2016. [20] Uçum A., Keçilioğlu O., İlarslan K., Generalized Bertrand curves with timelike (1,3)-normal plane in Minkowski space-time, Kuwait J. Sci., 42 (3), 10-27, 2015.
  • [21] Yıldız Ö.G., İçer  O., A note on evolution of quaternionic curves in the Euclidean space R4; Konuralp J. Math., 7(2), 462-469, 2019.
  • [22] Yoon D.W. , On the quaternionic general helices in Euclidean 4-space, Honam Mathematical J. 34(3), 381-390, 2012.
  • [23] Yoon D.W., Dae Won, Y. Tuncer, Yilmaz, M.K. Karacan, Generalized Mannheim quaternionic curves in Euclidean 4-space. Appl. Math. Sci. (Ruse) 7, 6583-6592, 2013.
Year 2021, Volume: 9 Issue: 2, 346 - 355, 15.10.2021

Abstract

References

  • [1] Bertrand J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15, 332-350, 1850.
  • [2] Bharathi K., Nagaraj M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 18 (6) 507-511.
  • [3] Çetin M. Kocayiğit H., On the quaternionic Smarandache curves in Euclidean 3-space, Int.J. Contemp Math Sci 8(3), 139-150, 2013.
  • [4] Ersoy S., Tosun M., Timelike Bertrand curves in semi-Euclidean space, Int. J. Math. Stat., 14(2), 78-89, 2013.
  • [5] Gök İ., Okuyucu O.Z., Kahraman F., Hacısalihoğlu H. H., On the quaternionic B2-slant helices in the Euclidean space E4: Adv. Appl. Cli ord Algebr., 21, 707-719, 2011.
  • [6] Gök İ., Kaya Nurkan S., İlarslan K., On pseudo null Bertrand curves in Minkowski space-time, Kyungpook Math. J. 54(4), 685-697, 2014.
  • [7] Güngör M. A. and Tosun M., Some characterizations of quaternionic rectifying curves, Di er. Geom. Dyn. Syst. 13, 89-100, 2011.
  • [8] Irmak Y., Bertrand Curves and Geometric Applications in Four Dimensional Euclidean Space, MSc thesis, Ankara University, Institute of Science, 2018.
  • [9] Kahraman Aksoyak F., Gök İ.., İlarslan K., Generalized null Bertrand curves in Minkowski space-time, An. Ştiint. Univ. Al. I. Cuza, Iasi, Mat. (N.S.) 60 (2), 489-502, 2014.
  • [10] Kahraman Aksoyak F., A new type of quaternionic Frame in R4; Int. J. Geom. Methods Mod. Phys., 16 (6), 1950084 (11 pages), 2019.
  • [11] Karadağ M., Sivridağ A., Quaternion valued functions of a single real variable and inclined curves, Erciyes Univ. J. Inst. Sci. Technol 13, 23-36,1997.
  • [12] Keçilioğlu O., İlarslan K. , Quaternionic Bertrand curves in Euclidean 4- space. Bull. Math. Anal. Appl. 5 (3), 27{38, 2013.
  • [13] Matsuda H. and Yorozu S., Notes on Bertrand curves. Yokohama Math. J. 50 (1-2), 41-58, 2003.
  • [14]  Önder M., Quaternionic Salkowski curves and quaternionic similar curves, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci., 90 (3), 447-456, 2020.
  • [15]  Öztürk G., Kişi İ., Büyükkütük S. , Constant ratio quaternionic curves in Euclidean spaces. Adv. Appl. Cli ord Algebr. 27 (2), 1659-1673, 2017.
  • [16] Pears L. R., Bertrand curves in Riemannian space, J. London Math. Soc. 1-10 (2), 180-183, 1935.
  • [17] Şenyurt S., Cevahir C., Altun Y., On spatial quaternionic involute curve a new view. Adv. Appl. Cli ord Algebr. 27 (2), 1815-1824, 2017.
  • [18] Uçum A., İlarslan K., Sasaki M., On (1,3)-Cartan null Bertrand curves in semi-Euclidean 4-space with index 2, J. Geom., 107 (3), 579-591, 2016.
  • [19] Uçum A., Keçilioğlu O., İlarslan K., Generalized Bertrand curves with spacelike (1,3)-normal plane in Minkowski space-time, Turkish J. Math., 40 (3), 487-505, 2016. [20] Uçum A., Keçilioğlu O., İlarslan K., Generalized Bertrand curves with timelike (1,3)-normal plane in Minkowski space-time, Kuwait J. Sci., 42 (3), 10-27, 2015.
  • [21] Yıldız Ö.G., İçer  O., A note on evolution of quaternionic curves in the Euclidean space R4; Konuralp J. Math., 7(2), 462-469, 2019.
  • [22] Yoon D.W. , On the quaternionic general helices in Euclidean 4-space, Honam Mathematical J. 34(3), 381-390, 2012.
  • [23] Yoon D.W., Dae Won, Y. Tuncer, Yilmaz, M.K. Karacan, Generalized Mannheim quaternionic curves in Euclidean 4-space. Appl. Math. Sci. (Ruse) 7, 6583-6592, 2013.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ferdağ Kahraman Aksoyak

Publication Date October 15, 2021
Submission Date January 22, 2021
Acceptance Date October 5, 2021
Published in Issue Year 2021 Volume: 9 Issue: 2

Cite

APA Kahraman Aksoyak, F. (2021). Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$. Konuralp Journal of Mathematics, 9(2), 346-355.
AMA Kahraman Aksoyak F. Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$. Konuralp J. Math. October 2021;9(2):346-355.
Chicago Kahraman Aksoyak, Ferdağ. “Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Konuralp Journal of Mathematics 9, no. 2 (October 2021): 346-55.
EndNote Kahraman Aksoyak F (October 1, 2021) Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$. Konuralp Journal of Mathematics 9 2 346–355.
IEEE F. Kahraman Aksoyak, “Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”, Konuralp J. Math., vol. 9, no. 2, pp. 346–355, 2021.
ISNAD Kahraman Aksoyak, Ferdağ. “Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Konuralp Journal of Mathematics 9/2 (October 2021), 346-355.
JAMA Kahraman Aksoyak F. Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$. Konuralp J. Math. 2021;9:346–355.
MLA Kahraman Aksoyak, Ferdağ. “Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$”. Konuralp Journal of Mathematics, vol. 9, no. 2, 2021, pp. 346-55.
Vancouver Kahraman Aksoyak F. Quaternionic $\left( 1,3\right) -$ Bertrand Curves According to Type 2-Quaternionic Frame in $\mathbb{R}^{4}$. Konuralp J. Math. 2021;9(2):346-55.
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