INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE

– 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 ? (s) in the Minkowski space 3 1 R . Let ? (s) be a unit speed C4 -timelike (or spacelike) curve in 3 1 R so that ? ʹ(s) = T . Then, ? (s) is a Frenet curve with curvature ? (s) and torsion ? (s) if and only if there are constant vectors a and b so that (i) { [ ] } 0 ʹ( ) ( ) cos ( ) sin ( ) cos ( ) ( ) ( ) ( ) , s T s =? s a ? s + b ? s + ? ? s ?? ? T ? ? ? d? T is timelike, (ii) { ( ) } 0 ʹ( ) ( ) cosh ( ) ( ) ( ) ( ) s T s =? s ae? +be?? + ? ? s ?? ? T ? ? ? d? , N is timelike, where 0 ( ) ( ) . s ? s = ? ? ? d?