A relation between the curvature ellipse and 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.