  In this work we firstly classify space-like surfaces in Minkowski space
$\mathbb E^4_1$, de-Sitter space $\mathbb S^3_1$ and hyperbolic space $\mathbb
H^3$ with harmonic Gauss map. Then we give a characterization and
classification of space-like surfaces with pointwise 1-type Gauss map of the
first kind. We also give some explicit examples.
