On Generalized Null Bertrand Curves in E^{4}
Transcription
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: [email protected] Second Author’s e-mail: [email protected] 3 Third Author’s e-mail: [email protected] 2 1