Isolated roundings and flattenings of submanifolds in euclidean spaces

摘要

We introduce the concepts of rounding and flattening of a smooth map g of an in-dimensional manifold M to the euclidean space R-n with in < n, as those points in M such that the image g(M) has contact of type Sigma(m.....m) with a hypersphere or a hyperplane of R-n, respectively. This includes several known special points such as vertices or flattenings of a curve in R-n, umbilics of a surface in R-3, or inflections of a surface in R-4.