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Geometric deformations of cuspidal $S_1$ singularities

Runa Shimada

TL;DR

This work develops a geometric framework for deformations of cuspidal $S_1$-type singularities by employing source diffeomorphisms and target isometries, yielding refined normal forms whose $2$-jets are $\,\mathcal{A}$-equivalent to $(u,v^2,0)$ or $(u,v^2,uv)$ and incorporating a deformation parameter. It establishes a precise frontality condition, extends the approach to cuspidal $S_k^\\pm$ and introduces minimal frontalization, linking to Mond’s germ classifications. Focusing on $k=1$, the paper analyzes singular point locations, self-intersection curves, and deformation invariants such as bias and secondary cuspidal curvature, revealing how these quantities evolve under deformation and relate to cusp cross caps appearing in the family. It further derives the trajectory of singular points and provides explicit curvature formulas, illuminating how geometric invariants control the deformation geometry of cuspidal $S_1$-type frontals and their transitions.

Abstract

To study a deformation of a singularity taking into consideration their differential geometric properties, a form representing the deformation using only diffeomorphisms on the source space and isometries of the target space plays a crucial role. Such a form for an $S_1$ singularity is obtained by the author's previous work. On this form, we give a necessary and sufficient condition for such a map is being a frontal. The form for an $S_1$ singularity with the frontal condition can be considered such a form for a cuspidal $S_1$ singularity. Using this form, we investigate geometric properties of cuspidal $S_1$ singularities and the cuspidal cross caps appearing in the deformation.

Geometric deformations of cuspidal $S_1$ singularities

TL;DR

This work develops a geometric framework for deformations of cuspidal -type singularities by employing source diffeomorphisms and target isometries, yielding refined normal forms whose -jets are -equivalent to or and incorporating a deformation parameter. It establishes a precise frontality condition, extends the approach to cuspidal and introduces minimal frontalization, linking to Mond’s germ classifications. Focusing on , the paper analyzes singular point locations, self-intersection curves, and deformation invariants such as bias and secondary cuspidal curvature, revealing how these quantities evolve under deformation and relate to cusp cross caps appearing in the family. It further derives the trajectory of singular points and provides explicit curvature formulas, illuminating how geometric invariants control the deformation geometry of cuspidal -type frontals and their transitions.

Abstract

To study a deformation of a singularity taking into consideration their differential geometric properties, a form representing the deformation using only diffeomorphisms on the source space and isometries of the target space plays a crucial role. Such a form for an singularity is obtained by the author's previous work. On this form, we give a necessary and sufficient condition for such a map is being a frontal. The form for an singularity with the frontal condition can be considered such a form for a cuspidal singularity. Using this form, we investigate geometric properties of cuspidal singularities and the cuspidal cross caps appearing in the deformation.

Paper Structure

This paper contains 9 sections, 8 theorems, 48 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Normal forms of cuspidal $S_k^\pm$ singularities including deformations
  3. Minimal frontalization
  4. Geometry on deformations of cuspidal $S_1$ singularities
  5. Description of singular point
  6. Self-intersection curves
  7. Geometric invariants of deformation
  8. Geometry on trajectory of singular points
  9. Geodesic and normal curvatures of even functions

Key Result

Theorem 2.2

(shimada) Let $f : (\boldsymbol{R}^2 \times \boldsymbol{R}, 0) \to (\boldsymbol{R}^3, 0)$ be a deformation of $g:(\boldsymbol{R}^2, 0) \to (\boldsymbol{R}^3, 0)$ such that the $2$-jet of $g$ is $\mathcal{A}$-equivalent to $(u,v^2,0)$ or $(u,v^2,uv)$. Then there exist an orientation preserving diffeo where $f_{32}(0,0,0)=f_{33}(0,0)=0$. If the $2$-jet of $g$ is $\mathcal{A}$-equivalent to $(u,v^2,

Figures (4)

  • Figure 1: Deformation of cuspidal $S_1^+$ singularity $($from left to right $f_{-1}^+, f_{0}^+$ and $f_{1}^+)$
  • Figure 2: Deformation of cuspidal $S_1^-$ singularity $($from left to right $f_{-1}^-, f_{0}^-,$ and $f_{1}^-)$
  • Figure 3: The surfaces in Example \ref{['ex:ex2']} (from left to right $s=-1, 0, 1)$
  • Figure 4: Minimal frontalization of the map in Example \ref{['ex:ex2']} (from left to right $s=-1, 0, 1)$

Theorems & Definitions (18)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Example 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 8 more