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

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.

Keywords

• Quaternions
• , Quaternionic frame,
• Bertrand curve

References

1. [1] Bertrand J. M., Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 15, 332-350, 1850.
2. [2] Bharathi K., Nagaraj M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. 18 (6) 507-511.
3. [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. [4] Ersoy S., Tosun M., Timelike Bertrand curves in semi-Euclidean space, Int. J. Math. Stat., 14(2), 78-89, 2013.
5. [5] Gök İ., Okuyucu O.Z., Kahraman F., Hacısalihoğlu H. H., On the quaternionic B2-slant helices in the Euclidean space E4: Adv. Appl. Cliord Algebr., 21, 707-719, 2011.
6. [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. [7] Güngör M. A. and Tosun M., Some characterizations of quaternionic rectifying curves, Dier. Geom. Dyn. Syst. 13, 89-100, 2011.
8. [8] Irmak Y., Bertrand Curves and Geometric Applications in Four Dimensional Euclidean Space, MSc thesis, Ankara University, Institute of Science, 2018.

9. [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. [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. [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. [12] Keçilioğlu O., İlarslan K. , Quaternionic Bertrand curves in Euclidean 4- space. Bull. Math. Anal. Appl. 5 (3), 27{38, 2013.
13. [13] Matsuda H. and Yorozu S., Notes on Bertrand curves. Yokohama Math. J. 50 (1-2), 41-58, 2003.
14. [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. [15]  Öztürk G., Kişi İ., Büyükkütük S. , Constant ratio quaternionic curves in Euclidean spaces. Adv. Appl. Cliord Algebr. 27 (2), 1659-1673, 2017.
16. [16] Pears L. R., Bertrand curves in Riemannian space, J. London Math. Soc. 1-10 (2), 180-183, 1935.
17. [17] Şenyurt S., Cevahir C., Altun Y., On spatial quaternionic involute curve a new view. Adv. Appl. Cliord Algebr. 27 (2), 1815-1824, 2017.
18. [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. [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.
20. [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.
21. [22] Yoon D.W. , On the quaternionic general helices in Euclidean 4-space, Honam Mathematical J. 34(3), 381-390, 2012.
22. [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.