ISOLATED ROUNDINGS AND FLATTENINGS OF SUBMANIFOLDS IN EUCLIDEAN SPACES

Abstract

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