Transversal Intersection Curves of Two Surfaces in $\mathbb{R}% _{1}^{3}$

The intersection curve of two spacelike surfaces is a spacelike, the intersection curve of two timelike surfaces may be spacelike or timelike, the intersection curve of two lightlike surfaces may be spacelike or null, the intersection curve of spacelike and timelike surfaces is spacelike, the intersection curve of spacelike and lightlike surfaces is spacelike, and the intersection curve of timelike and lightlike surfaces spacelike or null. We obtain the transversal intersection case, when the normal vectors of the surfaces are linearly independent.

For what follows we suppose $\mathbf{\alpha }(s)$ be a transversal intersection curve of two arbitrary surfaces $\mathbf{S}^{A}$ and $\mathbf{S}^{B}$ and assume that the curve $\mathbf{\beta }(u)=\mathbf{\alpha }(s(u))$ is a regular curve parametrized by a parameter $u$ with the same trace as the curve $\mathbf{% \alpha }(s)$. Let $\mathbf{\zeta }^{A}$ and $\mathbf{\zeta }^{B}$ denote the normal vector fields of the surfaces $S^{A}$and $S^{B},$ respectively. Also, let $\mathbf{N}^{A}$ and $\mathbf{N}^{B}$ denote the unit normal vector fields of the surfaces $S^{A}$and $S^{B},$ respectively. Assume that $% \left\langle \mathbf{\zeta }^{A},\mathbf{\zeta }^{B}\right\rangle _{1}=\lambda$, $\left\langle \mathbf{\zeta }^{A},\mathbf{\zeta }% ^{A}\right\rangle _{1}=\varepsilon ^{A},$ $\left\langle \mathbf{\zeta }^{B},% \mathbf{\zeta }^{B}\right\rangle _{1}=\varepsilon ^{B},$ $\left\langle \mathbf{N}^{A},\mathbf{N}^{B}\right\rangle _{1}=\bar{\lambda}$, $% \left\langle \mathbf{N}^{A},\mathbf{N}^{A}\right\rangle _{1}=\bar{\varepsilon }^{A},$ $\left\langle \mathbf{N}^{B},\mathbf{N}^{B}\right\rangle _{1}=\bar{% \varepsilon}^{B}$.
Since $\mathbf{\zeta }^{A}\times _{1}\mathbf{\zeta }^{B}$ is orthogonal to $% \mathbf{\zeta }^{A}\$and $\ \mathbf{\zeta }^{B},$ so by using proposition 2, we obtain

$$\left\langle \mathbf{\zeta }^{A}\times _{1}\mathbf{\zeta }^{B},\m... ...trix}% =\lambda ^{2}-\varepsilon ^{A}\varepsilon ^{B}. % \tag{5.1} \label{5.1}$$

The surfaces are intersect transversely if $\lambda ^{2}-\varepsilon ^{A}\varepsilon ^{B}\neq 0.$ In particular, if $\lambda =0,$ $\varepsilon ^{A}=0$ and $\varepsilon ^{B}>0,$ in this case the surfaces are intersect transversely at a null curve.