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

$LS_{\lowercase {r}}$-valued Gauss maps and spacelike surfaces of revolution in $\mathbb R_1^4$