integral characterizations for timelike and spacelike curves on the lorentzian sphere 3 s1

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