INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE 3 S1

Document Type: Regular Paper

Authors

1 Department of Mathematics, Faculty of Art and Sciences, University of Celal Bayar, Muradiye Campus, 45047, Manisa, Turkey

2 Gazi University, Gazi Faculty of Education, Department of Secondary Education, Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey

Abstract

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

Keywords