  Let $M$ be a smooth surface in $\mathbb R^3$ (or a complex surface in
$\mathbb C^3$) and $k\geq 2$ 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 $\mathbb R^3$.
