$H{S}_r$-valued Gauss maps and umbilic spacelike surfaces of codimension two

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1},$ we construct a pair of maps whose values are in $H{S}_r:=H_{+}^n(\mathbf{\text{v}},1)\cap \{x_{n+1}=r\},$ called ${\mathbf{\text{n}}}_r^{\pm }$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umbilic spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$