Geometric contacts of surfaces immersed in \(\mathbb R^n,n\geqslant 5\)
The authors study the extrinsic geometry of surfaces immersed in Euclidean space of dimension \(n \geq 5\) by analysing their contacts with different standard geometrical models, such as hyperplanes and hyperspheres. For this purpose they consider the singularities of the distance squared functions and their relation with the behavior of the curvature ellipses at different points on the surface. (The concept of the curvature ellipse of a surface in Euclidean space was first introduced in [\textit{E. B. Wilson} and \textit{C. L. E. Moore}, American Acad. Proc. 52, 269--368 (1916; JFM 46.1133.02)].) The main problem investigated in this paper concerns the generic distribution of the corank two singularities of the distance squared function and of their corresponding focal centers in the focal set of the surface. This leads in a natural way to the concept of umbilical focus and umbilic curvature.
scientific article; zbMATH DE number 5564329
scientific article