2021/02/11 by Sanchis, G. Peñafort, Tari, F.
#53A05 #58K05 #58K65 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2102.06308
Let M be a smooth surface in \mathbb R3 (or a complex surface in \mathbb C3) and k≥ 2 be an integer. At any point on M and for any plane in \mathbb R3, we construct a holomorphic map-germ (\mathbb C2,0)→(\mathbb C3,0) of the form Fk(x,y)= (x,yk,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 ≤ 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 R3 (or \mathbb C3); (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 R3.