Folding maps on a cross-cap

$\mathbb{R}^3$ 中曲面折叠映射的旋转开折的通用性

Versality of Rotation Unfolding of Folding Maps for Surfaces in $\mathbb{R}^3$

本文引入了 $\mathbb{R}^3$ 中曲面折叠映射的旋转开折，证明了其 $\mathcal{A}$-通用性的几何条件，并给出了正则曲面焦集切平面轨迹的微分同胚类型。

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

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 …