$L{S}_{\text{lowercase}r}$-valued Gauss maps and spacelike surfaces of revolution in ${\mathbb{R}}_1^4$

To study spacelike surfaces in the Lorentz-Minkowski space ${\mathbb{R}}_1^4,$ we construct a pair of maps whose values are in the lightcone, called $\mathfrak l_r^{\pm} $-Gauss maps. We can use these maps to study umbilical spacelike surfaces and find parametrizations of spacelike surfaces of revolution of hyperbolic and eliptic types in some particular cases.