2017/08/15 by Raúl Oset Sinha, Sinha, Raúl Oset, Pedro Benedini Riul +1
Mathematics · #53A05 #57R45 #58K05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1708.04651
openalex publication_date 2017/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
At each point in an immersed surface in \mathbb R4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in \mathbb R3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in \mathbb R4 to \mathbb R3 in a tangent direction corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in \mathbb R4 at a certain point to the geometry of the projection of the surface to \mathbb R3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.