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A Method of the Determination of a Geodesic Curve on Timelike Ruled Surface with Spacelike Rulings

A curve which is called geodesic on a surface M in Lorentz 3-space is a special curve that its acceleration is everywhere normal to M. In this paper, we analyzed the non-linear differential equation to determine the geodesic curves on ruled surfaces which is obtained by a strictly connected spacelike straight line moving with Frenet's frame along a timelike curve in . When we assume that curvature and torsion of base curve and components with respect to Frenet's frame of spacelike straight-line are constants, for special integration constants, we show that the resulting non-linear differential equation can be integrated exactly. Finally, examples are given to show the geodesic curve on the timelike ruled surfaces with spacelike rulings.