The Frenet and Darboux instantaneous rotation vectors for curves on space-like surface

Abstract

In this paper , considering the Darboux instantaneous rotation vector of a solid perpendicular trihedron in the Minkowski 3-space $R_1^3$ , the Frenet instantaneous rotation vector was stated for the Frenet trihedron of a space -like space curve (c) with the binormal b being a time-like vector. The Darboux deri¬vative formulas and the Darboux instantaneous rotation vector were found when the curve (c) is on a space -like surface , A fundamental relation , as a base for the geometry of space-like surfaces, was obtained among the Darboux vectors of the parameter curves $(c_1)$ , $(C_2)$ and an arbitrary curve (c) on a space-like surface.