## Tohoku Mathematical Journal

### Isometric deformations of cuspidal edges

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 17 March 2016

https://projecteuclid.org/euclid.tmj/1458248863

Digital Object Identifier
doi:10.2748/tmj/1458248863

Mathematical Reviews number (MathSciNet)
MR3476137

Zentralblatt MATH identifier
1350.57031

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

#### Citation

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. https://projecteuclid.org/euclid.tmj/1458248863

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