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Criteria for codimension two singularities of surfaces and their applications

Kentaro Saji, Runa Shimada

TL;DR

The paper develops explicit, coordinate-free criteria for recognizing codimension-two singularities of surface map-germs $f:(\mathbb{R}^2,0)\to(\mathbb{R}^3,0)$ under $\mathcal{A}$-equivalence, focusing on the $S_2$, $B_2^\pm$, and $H_2$ types via adapted-vector-field frameworks and jet-coefficient conditions. It provides a practical, jet-based toolkit—using SB-2, SB-3, HP-type adaptations and determinant tests—to determine singularity type and sign, with proofs of invariance under target diffeomorphisms. The methods are then applied to concrete geometric contexts: ruled surfaces, Euclidean center maps, and folded surfaces, yielding explicit geometric criteria in terms of striction data and jet coefficients. The results extend and concretize known classification results (Mond’s list) by giving explicit recognition criteria and by connecting algebraic criteria to geometric singularities. Overall, the work furnishes actionable criteria for identifying and distinguishing codimension-two singularities in central surface-theoretic problems.

Abstract

We give simple criteria for the singularities appearing on surfaces codimension less than or equal to two. As applications, we give conditions for codimension two singularities that appear in ruled surfaces and center maps of surfaces in the Euclidean space.

Criteria for codimension two singularities of surfaces and their applications

TL;DR

The paper develops explicit, coordinate-free criteria for recognizing codimension-two singularities of surface map-germs under -equivalence, focusing on the , , and types via adapted-vector-field frameworks and jet-coefficient conditions. It provides a practical, jet-based toolkit—using SB-2, SB-3, HP-type adaptations and determinant tests—to determine singularity type and sign, with proofs of invariance under target diffeomorphisms. The methods are then applied to concrete geometric contexts: ruled surfaces, Euclidean center maps, and folded surfaces, yielding explicit geometric criteria in terms of striction data and jet coefficients. The results extend and concretize known classification results (Mond’s list) by giving explicit recognition criteria and by connecting algebraic criteria to geometric singularities. Overall, the work furnishes actionable criteria for identifying and distinguishing codimension-two singularities in central surface-theoretic problems.

Abstract

We give simple criteria for the singularities appearing on surfaces codimension less than or equal to two. As applications, we give conditions for codimension two singularities that appear in ruled surfaces and center maps of surfaces in the Euclidean space.
Paper Structure (20 sections, 26 theorems, 91 equations, 4 figures, 1 table)

This paper contains 20 sections, 26 theorems, 91 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Criteria
  3. Preliminaries for brunch to $S_2$ or $B_2$ singularities
  4. Branch to $S_2$ or $B_2$ singularities
  5. The condition for $S_2$ singularity
  6. The condition for $B_2$ singularity
  7. The condition for $H_2$ singularity
  8. Conditions in terms of coefficients of jets
  9. Conditions for $S_2$ and $B_2$ singularities
  10. Condition for $H_2$ singularity
  11. Proof of criteria
  12. Singularities of surfaces
  13. Singularities of ruled surfaces
  14. Singularities of Euclidean center maps
  15. Singularities of folded surfaces
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $f:(\boldsymbol{R}^2,0)\to(\boldsymbol{R}^3,0)$ satisfy $\operatorname{rank} df_0=1$. Then $f$ is

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (61)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 51 more