On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space

Abstract

Let $M$ be a smooth surface in ${\mathbb{R}}^3$ (or a complex surface in ${\mathbb{C}}^3$) and $k\ge 2$ be an integer. At any point on $M$ and for any plane in ${\mathbb{R}}^3$, we construct a holomorphic map-germ $({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0)$ of the form $F_k(x,y)=(x,y^k,f(x,y))$, called a $k$-folding map-germ. We study in this paper the local singularities of $k$-folding map-germs and relate them to the extrinsic differential geometry of $M$. More precisely, we (1) stratify the jet space of $k$-folding map-germs so that the strata of codimension $\le 4$ correspond to topologically equivalent $\mathcal{A}$-finitely determined germs; (2) obtain the topological classification of $k$-folding map-germs on generic surfaces in ${\mathbb{R}}^3$ (or ${\mathbb{C}}^3$); (3) generalise the work of Bruce-Wilkinson on folding maps ($k=2$); (4) recover, in a unified way, results obtained by considering the contact of surfaces with lines, planes and spheres; and (5) discover new robust features on smooth surfaces in ${\mathbb{R}}^3$.