  The Gauss map of a generic immersion of a smooth, oriented surface into
$\mathbb R^4$ is an immersion. But this map takes values on the Grassmanian of
oriented 2-planes in $\mathbb R^4$. Since this manifold has a structure of a
product of two spheres, the Gauss map has two components that take values on
the sphere. We study the singularities of the components of the Gauss map and
relate them to the geometric properties of the generic immersion. Moreover, we
prove that the singularities are generically stable, and we connect them to the
contact type of the surface and $\mathcal J$-holomorphic curves with respect to
an orthogonal complex structure $\mathcal J$ on $\mathbb R^4$. Finally, we get
some formulas of Gauss-Bonnet type involving the geometry of the singularities
of the components with the geometry and topology of the surface.
