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INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE 3 S1 | ||
| Iranian Journal of Science | ||
| مقاله 4، دوره 32، شماره 1، اردیبهشت 2008، صفحه 25-31 اصل مقاله (134.83 K) | ||
| نوع مقاله: Regular Paper | ||
| شناسه دیجیتال (DOI): 10.22099/ijsts.2008.2238 | ||
| نویسندگان | ||
| M. KAZAZ* 1؛ H. H. UGURLU2؛ A. OZDEMIR1 | ||
| 1Department of Mathematics, Faculty of Art and Sciences, University of Celal Bayar, Muradiye Campus, 45047, Manisa, Turkey | ||
| 2Gazi University, Gazi Faculty of Education, Department of Secondary Education, Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey | ||
| چکیده | ||
| V. Dannon showed that spherical curves in E4 can be given by Frenet-like equations, and he then gave an integral characterization for spherical curves in E4 . In this paper, Lorentzian spherical timelike and spacelike curves in the space time 4 1 R are shown to be given by Frenet-like equations of timelike and spacelike curves in the Euclidean space E3 and the Minkowski 3-space 3 1 R . Thus, finding an integral characterization for a Lorentzian spherical 4 1 R -timelike and spacelike curve is identical to finding it for E3 curves and 3 1 R -timelike and spacelike curves. In the case of E3 curves, the integral characterization coincides with Dannon’s. Let {T, N, B}be the moving Frenet frame along the curve α (s) in the Minkowski space 3 1 R . Let α (s) be a unit speed C4 -timelike (or spacelike) curve in 3 1 R so that α '(s) = T . Then, α (s) is a Frenet curve with curvature κ (s) and torsion τ (s) if and only if there are constant vectors a and b so that (i) { [ ] } 0 '( ) ( ) 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, (ii) { ( ) } 0 '( ) ( ) cosh ( ) ( ) ( ) ( ) s T s =κ s aeξ +be−ξ + ∫ ξ s −ξ δ T δ κ δ dδ , N is timelike, where 0 ( ) ( ) . s ξ s = ∫ τ δ dδ | ||
| کلیدواژهها | ||
| Lorentzian 3-sphere؛ Timelike curve؛ spacelike curve؛ curvature | ||
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