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

Abstract

Let M be a smooth surface in R 3 (or a complex surface in C 3 ) and k ≥ 2 be an integer. At any point on M and for any plane in R 3 , we construct a holomorphic map-germ (C 2 , 0) → (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• stratify the jet space of k-folding map-germs so that the strata of codimension ≤ 4 correspond to topologically equivalent A-finitely determined germs; • obtain the topological classification of k-folding map-germs on generic surfaces in R 3 (or C 3 ); • generalise the work of Bruce-Wilkinson on folding maps (k = 2); • recover, in a unified way, results obtained by considering the contact of surfaces with lines, planes and spheres; • discover new robust features on smooth surfaces in R 3 .