a parameter t$_{0}$ ∈ [ T$_{1}$, T$_{2}$ ] such that P ( s, t$_{0}$ ) = r ( s ) L$_{1}$ ≤ s ≤ L$_{2}$, that is , Let n$_{1}$ ( n$_{1}$ = cosh θN + sinh θB ) be a vector orthogonal to the curve r ( s ), where θ = θ ( s ) is the Lorentzian timelike angle between N and n$_{1}$. The curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if n$_{1}$ is parallel to the normal vector n ( s, t ) of the surface P ( s, t ) and S ( T ) = ωT, ω = 0, where S is the shape operator of the surface. Firstly, we derive the condition for n$_{1}$ to be parallel to the normal vector n ( s, t ) of the surface P ( s, t ) : The normal vector can be expressed as Thus, we get where This follows that n$_{1}$ ( s ) //n ( s, t$_{0}$ ), L$_{1}$ ≤ s ≤ L$_{2}$ , if and only if there exits a φ$_{1}$ ( s, t$_{0}$ ) = ∂v ( s,t 0 ) ∂s ∂w ( s,t 0 ) ∂t − ∂w ( s,t 0 ) ∂s ∂v ( s,t 0 ) ∂t , φ$_{2}$ ( s, t$_{0}$ ) = − ( 1 + ∂u ( s,t 0 ) ∂s ) ∂w ( s,t 0 ) ∂t + ∂w ( s,t 0 ) ∂s ∂u ( s,t 0 ) ∂t , φ$_{3}$ ( s, t$_{0}$ ) = − 1 + ∂u ( s,t 0 ) ∂s ∂v ( s,t 0 ) ∂t + ∂v ( s,t 0 ) ∂s ∂u ( s,t 0 ) ∂t . n ( s, t$_{0}$ ) = φ$_{1}$ ( s, t$_{0}$ ) T ( s ) + φ$_{2}$ ( s, t$_{0}$ ) N ( s ) + φ$_{3}$ ( s, t$_{0}$ ) B ( s ) , n ( s, t ) = ∂P ( s,t ) ∂s × ∂P ( s,t ) ∂t = ( − ( τ ( s ) v ( s, t ) + ∂w ( s,t ) ∂s ) ∂v ( s,t ) ∂t + ( − κ ( s ) u ( s, t ) + τ ( s ) w ( s, t ) + ∂v ( s,t ) ∂s ) ∂w ( s,t ) ∂t ) T ( s ) + (( τ ( s ) v ( s, t ) + ∂w ( s,t ) ∂s ) ∂u ( s,t ) ∂t − (1 − κ ( s ) v ( s, t ) + ∂u ( s,t ) ∂s ) ∂w ( s,t ) ∂t ) N ( s ) + (( − κ ( s ) u ( s, t ) + τ ( s ) w ( s, t ) + ∂v ( s,t ) ∂s ) ∂u ( s,t ) ∂t − (1 − κ ( s ) v ( s, t ) + ∂u ( s,t ) ∂s ) ∂v ( s,t ) ∂t ) B ( s ) . 13 u ( s, t$_{0}$ ) = v ( s, t$_{0}$ ) = w ( s, t$_{0}$ ) ≡ 0 , L$_{1}$ ≤ s ≤ L$_{2}$. (4.2)