M ⊂ M . The (1 , 1) tensor field S on M such that 〈 S ( X ) ,Y 〉 = 〈 II ( X,Y ) ,N 〉 for all X,Y ∈ χ ( M ) is called the shape operator of M ⊂ M derived from N [5]. As usual, S determines a linear operator S : T$_{P}$ ( M ) −→ T$_{P}$ ( M ) at each point P ∈ M . If S is shape operator derived from N , then S ( X ) = − D$_{X}$N and at each point the linear operator S on T$_{P}$ ( M ) is self adjoint. A regular curve r on M is said to be a line of curvature of M if for all p ∈ r the tangent line of r is a principal direction at p . According to this definition, the differential equation of the line of curvature on M is S ( T ) = ωT, ω = 0 , where S is the shape operator of M . A surface in R 3 1 is called a timelike surface if the induced metric on the surface is a Lorentzian metric and is called a spacelike surface if induced metric on the surface is positive definite Riemannian metric, i.e. the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector [9]. A parametric curve r ( s ) is a curve on a surface P = P ( s,t ) in R 3 1 that has a constant s or t parameter value, that is, there exists a parameter s$_{0}$ or t$_{0}$ such that r ( s ) = P ( s,t$_{0}$ ) or r ( t ) = P ( s$_{0}$,t ) . 6