  To study spacelike surfaces in the Lorentz-Minkowski space $\Bbb R^{4}_1,$ 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.
