  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.
