Isometric deformations of cuspidal edges

Abstract

Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $\nu$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regular space curve) is not orthogonal to $\nu$. This genericity is equivalent to the condition that its limiting normal curvature $\kappa_\nu$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge can be isometrically deformed preserving $\kappa_\nu$ into a cuspidal edge whose singular set lies in a plane. Such a limiting cuspidal edge is uniquely determined from the initial germ of the cuspidal edge.