On differential analysis of spacelike flows on normal congruence of surfaces

The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the $\mathrm{curl}$ of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both $s-lines$ and $b-lines.$ Also, we examine a normal congruence of surfaces containing the $s-lines$ and $b-lines$. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces are given.