2022/07/20 by W. Domitrz, Domitrz, W., Lucía Ivonne Hernández–Martínez +3
Mathematics · #53A05 #53C42 #58K05 #58K25 #58K30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2207.09630
openalex publication_date 2022/07/20 · openalex created_date 2023/02/14 · openalex updated_date 2026/07/28
The Gauss map of a generic immersion of a smooth, oriented surface into \mathbb R4 is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in \mathbb R4. 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 R4. 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.