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.
[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. Cliord
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, Dier. 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.
[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. Cliord 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. Cliord 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.
[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. Cliord
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, Dier. 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.
[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. Cliord 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. Cliord 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.
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.