The Gauss map and second fundamental form of surfaces in a Lie group

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.