SpringerIranian Journal of Science and Technology (Sciences)1028-627632120080217INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE 3 S12531223810.22099/ijsts.2008.2238ENM.KAZAZDepartment of Mathematics, Faculty of Art and Sciences, University of Celal Bayar,
Muradiye Campus, 45047, Manisa, TurkeyH. H.UGURLUGazi University, Gazi Faculty of Education, Department of Secondary Education, Science
and Mathematics Teaching, Mathematics Teaching Program, Ankara, TurkeyA.OZDEMIRDepartment of Mathematics, Faculty of Art and Sciences, University of Celal Bayar,
Muradiye Campus, 45047, Manisa, TurkeyJournal Article20060715V. Dannon showed that spherical curves in E4 can be given by Frenet-like equations, and he then<br />gave an integral characterization for spherical curves in E4 . In this paper, Lorentzian spherical timelike and<br />spacelike curves in the space time 4<br />1 R are shown to be given by Frenet-like equations of timelike and<br />spacelike curves in the Euclidean space E3 and the Minkowski 3-space 3<br />1 R . Thus, finding an integral<br />characterization for a Lorentzian spherical 4<br />1 R -timelike and spacelike curve is identical to finding it for E3<br />curves and 3<br />1 R -timelike and spacelike curves. In the case of E3 curves, the integral characterization<br />coincides with Dannon’s.<br />Let {T, N, B}be the moving Frenet frame along the curve α (s) in the Minkowski space 3<br />1 R . Let<br />α (s) be a unit speed C4 -timelike (or spacelike) curve in 3<br />1 R so that α '(s) = T . Then, α (s) is a Frenet<br />curve with curvature κ (s) and torsion τ (s) if and only if there are constant vectors a and b so that<br />(i) { [ ] } 0<br />'( ) ( ) cos ( ) sin ( ) cos ( ) ( ) ( ) ( ) , s T s =κ s a ξ s + b ξ s + ∫ ξ s −ξ δ T δ κ δ dδ T is timelike,<br />(ii) { ( ) } 0<br />'( ) ( ) cosh ( ) ( ) ( ) ( ) s T s =κ s aeξ +be−ξ + ∫ ξ s −ξ δ T δ κ δ dδ , N is timelike,<br />where<br />0<br />( ) ( ) . s ξ s = ∫ τ δ dδhttp://ijsts.shirazu.ac.ir/article_2238_9ccf89393e0c615c0234b48509b20d4e.pdf