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.