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.

我们通过将\（\ mathbb {R} ^ 5 \）中的曲面的几何与通过正交投影获得的\（\ mathbb {R} ^ 4 \）中的规则曲面和奇异曲面的几何相关联来研究该曲面的几何。特别是，我们获得渐近方向之间的关系，它们不是\（\ mathbb {R} ^ 5 \）中的曲面的二阶几何，而是在\（\ mathbb {R} ^ 4 \）中的曲面的关系。我们还关联了每种类型的曲面的脐曲率及其与球体的接触。然后，我们将曲面视为\（\ mathbb {R} ^ 6 \）中的3个歧管的法线截面，并通过为3个歧管定义适当的脐曲率，再次关联渐近方向和与球体的接触。