Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to   Mannheim Offsets of Timelike Ruled Surfaces
  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. 
Mehmet \"Onder, H. H\"useyin U\u{g}urlu
math.DG