%0 Journal Article
%T INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE 3 S1
%J Iranian Journal of Science and Technology (Sciences)
%I Springer
%Z 1028-6276
%A KAZAZ, M.
%A UGURLU, H. H.
%A OZDEMIR, A.
%D 2008
%\ 02/17/2008
%V 32
%N 1
%P 25-31
%! INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE 3 S1
%K Lorentzian 3-sphere
%K Timelike curve
%K spacelike curve
%K curvature
%R 10.22099/ijsts.2008.2238
%X V. Dannon showed that spherical curves in E4 can be given by Frenet-like equations, and he thengave an integral characterization for spherical curves in E4 . In this paper, Lorentzian spherical timelike andspacelike curves in the space time 41 R are shown to be given by Frenet-like equations of timelike andspacelike curves in the Euclidean space E3 and the Minkowski 3-space 31 R . Thus, finding an integralcharacterization for a Lorentzian spherical 41 R -timelike and spacelike curve is identical to finding it for E3curves and 31 R -timelike and spacelike curves. In the case of E3 curves, the integral characterizationcoincides with Dannon’s.Let {T, N, B}be the moving Frenet frame along the curve α (s) in the Minkowski space 31 R . Letα (s) be a unit speed C4 -timelike (or spacelike) curve in 31 R so that α '(s) = T . Then, α (s) is a Frenetcurve 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,where0( ) ( ) . s ξ s = ∫ τ δ dδ
%U http://ijsts.shirazu.ac.ir/article_2238_9ccf89393e0c615c0234b48509b20d4e.pdf