On singularities of the Gauss map components of surfaces in \({\mathbb R}^4\)

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.