  In this article, we give the integrability conditions for the existence of an
isometric immersion from an orientable simply connected surface having
prescribed Gauss map and positive extrinsic curvature into some unimodular Lie
groups. In particular, we discuss the case when the Lie group is the euclidean
unit sphere $\mathbb{S}^3$ and establish a correspondence between simply
connected surfaces having extrinsic curvature $K$, $K$ different from 0 and -1,
immersed in $\mathbb{S}^3$ with simply connected surfaces having non-vanishing
extrinsic curvature immersed in the euclidean space $\mathbb{R}^3$. Moreover,
we show that a surface isometrically immersed in $\mathbb{S}^3$ has positive
constant extrinsic curvature if, and only if, its Gauss map is a harmonic map
into the Riemann sphere.
