Tohoku Mathematical Journal

Isometric deformations of cuspidal edges

Kosuke Naokawa, Masaaki Umehara, and Kotaro Yamada

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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.

Article information

Tohoku Math. J. (2), Volume 68, Number 1 (2016), 73-90.

First available in Project Euclid: 17 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings
Secondary: 53A05: Surfaces in Euclidean space

Cuspidal edge isometric deformation


Naokawa, Kosuke; Umehara, Masaaki; Yamada, Kotaro. Isometric deformations of cuspidal edges. Tohoku Math. J. (2) 68 (2016), no. 1, 73--90. doi:10.2748/tmj/1458248863.

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