Let $\mathbb{R}^{2,1}$ denote $\mathbb{R}^{3}$ with the Lorentz metric given by $$\mathrm ds^2=\left(\mathrm dx^1\right)^2+\left(\mathrm dx^2\right)^2-\left(\mathrm dx^{3}\right)^2.$$ The Lorentzian inner product of two vectors $\mathbf v=(v_1,v_2,v_3)$ and $\mathbf w=(w_1,w_2,w_3)$ in $\mathbb{R}^{2,1}$ is given by $$\langle v,w\rangle_1:=v_1 w_1+v_2 w_2 - v_3 w_3.$$ Let $f\colon S\to \mathbb{R}^{2,1}$ be an immersion of the surface $S$. Consider the pullback metric $g:=f^*\bar{g}$ of the Lorentzian metric $\bar{g}$ via $f $on S. For $p\in S$, a vector $v \in T_p(S)$ is called spacelike if $g_p(v,v)\ge 0$ and timelike if $g_p(v,v)\le 0$. The map $f\colon S\to\mathbb{R}^{2,1}$ is said to be a spacelike immersion if $g$ is a positive definite metric on $S$. In this case, $S$ will be called spacelike hypersurface of $\mathbb{R}^{2,1}$.

Then, we will discuss about Gauss map corresponding to a space-like immersion.