A Geometric Approach to Timelike Flows in Terms of Anholonomic Coordinates

This paper is devoted to the geometry of vector fields and timelike flows in terms of anholonomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial differential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principal normal vector field. This implies the existence of a surface which contains both s-lines and b-lines. Moreover, we examine a normal congruence of timelike surfaces containing the s-lines and b-lines . Considering the compatibility conditions, we obtain the Gauss-Mainardi-Codazzi equations for this normal congruence of timelike surfaces in the case of the abnormality of normal vector field is zero. Intrinsic geometric properties of this normal congruence of timelike surfaces are obtained. We have deal with important results on these geometric properties.