Raul Oset (Valencia University)

Friday, August 18 th, 14:20, Room 3-102

**Title: 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 the immersed surface in $mathbb R^4$ to the geometry of the singular projection. In particular, we relate the curvature ellipse of an immersed surface at a certain point to the curvature parabola of the projection at the singular point.