Geometric contacts of surfaces immersed in RnRn, n⩾5n⩾5

We study the extrinsic geometry of surfaces immersed in RnRn, n⩾5n⩾5, by analyzing their contacts with different standard geometrical models, such as hyperplanes and hyperspheres. We investigate the relation between different types of contact and the properties of the curvature ellipses at each point. In particular, we focalize our attention on the hyperspheres having contacts of corank two with the surface. This leads in a natural way to the concept of umbilical focus and umbilic curvature.