INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE S31

Abstract

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