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Abstract

SHIRAZ UNIV

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V. Dannon showed that spherical curves in E-4 can be given by Frenet-like equations, and he then gave an integral characterization for spherical curves in E-4. In this paper, Lorentzian spherical timelike and spacelike curves in the space time are shown to be given by Frenet-like equations of timelike and spacelike curves in the Euclidean space E-3 and the Minkowski 3-space R-1(3). Thus, finding an integral characterization for a Lorentzian spherical R-1(4)-timelike and spacelike curve is identical to finding it for E-3 curves and R-1(3)-timelike and spacelike curves. In the case of E-3 curves, the integral characterization coincides with Dannon's. Let {T, N, B} be the moving Frenet frame along the curve alpha(s) in the Minkowski space R-1(3). Let alpha(s) be a unit speed C-4-timelike (or spacelike) curve in R-1(3) so that alpha(s) = T. Then, alpha(s) is a Frenet curve with curvature kappa(s) and torsion tau(s) if and only if there are constant vectors a and b so that (i) T'(s)= kappa(s){acos xi(s)+bsin xi(s) + integral(s)(0) cos[xi(s)-xi(delta)]T(delta)kappa(delta)d delta}, T is timelike, (ii) T'(s) = kappa(s) {ae(xi) + be(-xi) + integral(s)(0) cosh (xi(delta)T(delta)kappa(delta)d delta} N is timelike, where xi(s)= integral(s)(0)tau(delta)d delta.

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http://akademikarsiv.cbu.edu.tr:4000/handle/123456789/6611

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