Tohoku Mathematical Journal
2016
March
SECOND SERIES VOL. 68, NO. 1

Tohoku Math. J.
68 (2016), 73-90

Title ISOMETRIC DEFORMATIONS OF CUSPIDAL EDGES

Author Kosuke Naokawa, Masaaki Umehara and Kotaro Yamada

(Received August 11, 2014)
Abstract. Along cuspidal edge singularities on a given surface in Euclidean 3-space $\boldsymbol{R}^3$, which can be parametrized by a regular space curve $\hat\gamma(t)$, 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 $\hat\gamma(t)$ 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 $f$ 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.

Mathematics Subject Classification. Primary 57R45; Secondary 53A05.
Key words and phrases. Cuspidal edge, isometric deformation.

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