Abstract
In this research, spinor descriptions of the curves in surfaces have been given according to the B-Darboux frame in Lorentzian 3-space E_1^3. The relations between B-Darboux and Darboux frames have been specified via their spinor conceptions which expressed in both timelike and spacelike surfaces, separately. Additionally, all these spinor representations have been portrayed in the view of the B-Darboux frame (via the curvatures) in Lorentzian 3-space. The findings are supported by some theorems and corollaries.
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