In this study, the E. Study mapping was dermed for the space-like and time-like lines in the Minkowski 3-space R13. Hence, there is one to one correspondence between directed space-like (resp., time-like) lines of R13 and ordered pair of vectors (a,a0.) such that < a,a> = 1 (resp., < a,a> = -1) and <a,a0> = 0.

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In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.Mathematics Subject Classification 2010: 53A25, 53C40, 53C50.

In this paper, we consider the idea of Bertrand partner curves for curves lying on surfaces and by considering the Darboux frames of surface curves, we call these curves as Bertrand partner D -curves and give the characterizations for these curves by means of the geodesic curvatures, the normal curvatures and the geodesic torsions of these associated curves.

In this paper, we give some characterizations of Mannheim partner curves in the Minkowski 3-space E¡. Firstly, we classify these curves in E^. Next, we give some relationships characterizing these curves and we show that the Mannheim theorem is not valid for the Mannheim partner curves in E¡. Moreover, by considering the spherical indicatrix of the Frenet vectors of those curves, we obtain some new relationships between the curvatures and torsions of the Mannheim partner curves in E¡.