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

Abstract

We study the geometry of surfaces in R 5 by relating it to the geometry of regular and singular surfaces in R 4 obtained by orthogonal projections.In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in R 5 but are in 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 R 6 and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.