A relation between the curvature ellipse and the curvature parabola

TL;DR

This paper explores the relationship between the curvature ellipse of a surface in four-dimensional space and the curvature parabola of its projected singularities in three-dimensional space, revealing geometric connections at singular points.

Contribution

It establishes a link between the curvature ellipse in $\

Findings

The curvature parabola can be derived from the curvature ellipse at singular points.

Projection in asymptotic directions leads to more degenerate singularities.

The geometry of the original surface influences the shape of the curvature parabola.

Abstract

At each point in an immersed surface in $\mathbb R^4$ 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 $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in $\mathbb R^4$ to $\mathbb R^3$ 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 $\mathbb R^4$ at a certain point to the geometry of the projection of the surface to $\mathbb R^3$ at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.