On Generalized Null Bertrand Curves in E^{4}

International Conference on Pure and Applied Mathematics (ICPAM 2015)
ON GENERALIZED NULL BERTRAND CURVES IN E42
2 , Makoto Sakaki
˙
Ali U¸
cum1 , Kazım Ilarslan
3
1
Kırıkkale University, Faculty of Sciences and Arts, Department of
Mathematics, Kırıkkale-Turkey
2 Kırıkkale University, Faculty of Sciences and Arts, Department of
Mathematics, Kırıkkale-Turkey
3 Graduate School of Science and Technology, Hirosaki University, Hirosaki
036-8561, Japan
MSC 2000: 53C50, 53C40
Abstract
In the present paper, generalized Cartan null Bertrand curves in semiEuclidean 4-space with index 2 is considered. Because the (1, 3)-normal
planes of the Cartan null curves is timelike, the (1, 3)-Bertrand mate curves
of the curves can be pseudo null curves, non-null curves or Cartan null
curves, respectively. Thus, we give the necessary and sufficient conditions
for these three cases to be (1, 3)-Bertrand curves and we also give the related
examples.
Keywords: Generalized Bertrand curve, Semi-Euclidean Space, Cartan
null curve.
References
[1] K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of SemiRiemannian Manifolds, World Scientific, London, (2007).
[2] H. Matsuda and S. Yorozu, Notes on Bertrand curves, Yokohama Math.
J. 50 (2003) 41-58.
[3] M. Sakaki, Null Cartan Curves in R42 , Toyama Mathematical Journal 32
(2009) 31-39.
˙
[4] A. U¸cum, O. Ke¸cilio˘
glu and K. Ilarslan,
Generalized Bertrand curves
with timelike (1,3)-normal plane in Minkowski space-time, accepted
(2014).
˙
[5] A. U¸cum, O. Ke¸cilio˘
glu and K. Ilarslan,
Generalized Pseudo Null
Bertrand curves in Semi-Euclidean 4-Space with index 2, to appear
(2014).
1
First Author’s e-mail: aliucum05@gmail.com
Second Author’s e-mail: kilarslan@yahoo.com
3
Third Author’s e-mail: sakaki@hirosaki-u.ac.jp
2
1