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

TL;DR

This paper investigates the singularities of the Gauss map components of surfaces in four-dimensional space, linking their stability and contact properties to the surface's geometry and topology.

Contribution

It characterizes the singularities of the Gauss map components, proves their generic stability, and relates them to contact geometry and $ ext{J}$-holomorphic curves in $ ext{R}^4$.

Findings

Singularities of Gauss map components are generically stable.

Connections established between singularities and contact type of surfaces.

Formulas of Gauss-Bonnet type involving singularities and surface topology.

Abstract

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.