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Geometry on deformations of S1 singularities

Runa Shimada

TL;DR

The work addresses the geometry of one-parameter deformations of $S_1^b$ singularities by constructing an $S_1^b$ deformation normal form under $SO(3)$-equivalences and diffeomorphisms, enabling a precise tracking of how Whitney umbrellas arise in deformations. It develops asymptotic expansions for key geometric invariants (e.g., $a_{20}$, $a_{11}$, $a_{02}$) and studies their implications for curvature-related quantities, focal conics, and the curvature parabola. The authors extend the notion of umbilic curvature to Whitney umbrellas within the deformation family and relate these invariants to the trajectory geometry of singular points, offering a cohesive framework to understand appearance/disappearance phenomena of singularities in $S_1^b$ deformations. This contributes a rigorous geometric lens for singularity transitions and may inform broader analyses of frontals and their curvature behavior near codimension-one singularities.

Abstract

To study a one parameter deformation of an $S_1$ singularity taking into consideration its differential geometric properties, we give a form representing the deformation using only diffeomorphisms on the source and isometries of the target. Using this form, we study differential geometric properties of $S_1$ singularities and the Whitney umbrellas appearing in the deformation.

Geometry on deformations of S1 singularities

TL;DR

The work addresses the geometry of one-parameter deformations of singularities by constructing an deformation normal form under -equivalences and diffeomorphisms, enabling a precise tracking of how Whitney umbrellas arise in deformations. It develops asymptotic expansions for key geometric invariants (e.g., , , ) and studies their implications for curvature-related quantities, focal conics, and the curvature parabola. The authors extend the notion of umbilic curvature to Whitney umbrellas within the deformation family and relate these invariants to the trajectory geometry of singular points, offering a cohesive framework to understand appearance/disappearance phenomena of singularities in deformations. This contributes a rigorous geometric lens for singularity transitions and may inform broader analyses of frontals and their curvature behavior near codimension-one singularities.

Abstract

To study a one parameter deformation of an singularity taking into consideration its differential geometric properties, we give a form representing the deformation using only diffeomorphisms on the source and isometries of the target. Using this form, we study differential geometric properties of singularities and the Whitney umbrellas appearing in the deformation.
Paper Structure (7 sections, 11 theorems, 73 equations, 5 figures)

This paper contains 7 sections, 11 theorems, 73 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric deformations of $S_1^\pm$ singularities and their normal forms
  3. Geometric deformations of $S_1^\pm$ singularities
  4. Geometric invariants of deformations
  5. Curvature parabolas
  6. Focal conics
  7. Geometry on trajectories of singular points

Key Result

Theorem 2.3

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 $(u,v^2,0)$. Then there exist an orientation preserving diffeomorphism-germ $\varphi : ( where $f_{32}(0,0,0)=f_{33}(0,0)=(f_{33})_u(0,0)=0$. Furthermore, $f(u,v,0)=g(u,v)$ is an $S_1^+$

Figures (5)

  • Figure 1: Deformation of an $S_1^+$ singularity $($from left to right $f^{-1,+}, f^{0,+}$ and $f^{1,+})$
  • Figure 2: Deformation of an $S_1^-$ singularity $($from left to right $f^{-1,-}, f^{0,-}$ and $f^{1,-}$)
  • Figure 3: The surfaces $f^+$ (left) and $f^-$ (right).
  • Figure 4: The focal conics in Example \ref{['ex:ex4']} (from left to right $s=-1, -1/2, -1/5, 0, 1/5$)
  • Figure 5: The focal conics in Example \ref{['ex:ex5']} (from left to right $s=-1, 0, 1$)

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • Theorem 3.2
  • ...and 16 more