Geometry of surfaces in ${\mathbb{R}}^5$ through projections and normal sections

Abstract

We study the geometry of surfaces in ${\mathbb{R}}^5$ by relating it to the geometry of regular and singular surfaces in ${\mathbb{R}}^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in ${\mathbb{R}}^5$ but are in ${\mathbb{R}}^4$. We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in ${\mathbb{R}}^6$ and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.