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

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