Weierstrass Representation of Lightlike Surfaces in Lorentz-Minkowski 4-Space

We present a Weierstrass-type representation formula which locally represents every regular two-dimensional lightlike surface in Lorentz-Minkowski 4-Space $\mathbb{M}^4$ by three dual functions $(\rho,f,g)$ and generalizes the for surfaces $\mathbb{M}^3$. give necessary sufficient conditions on $\rho$, $f$, $g$ to be minimal, ruled or $l$-minimal. For surfaces, we itself ruled. Furthermore, result totally geodesic half-lightlike holds only $\mathbb{M}^4$.