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Generic singularities of affine distance functions and plane congruences

Igor Chagas Santos

TL;DR

The paper develops a unified Lagrangian-singularity framework to classify generic singularities of 2-parameter plane congruences in $\mathbb{R}^4$ and their affine normal variants. It shows that the affine distance function $\Delta$ is a Morse family and locally $\mathcal{P}$-$\mathcal{R}^{+}$-versal, yielding generic $\Delta_p$ singularities of type $A_1$–$A_5$ and $D_4$, $D_5$, with a Hessian description linking degeneracies to affine focal points. For general plane congruences, transversality and Mather-type stability imply a finite list of generic singularities realized as Lagrangian stable types (Fold, Cusp, Swallowtail, Butterfly, and umbilic variants). In the affine-normal case, the germ of the Blaschke congruence is Lagrangian stable for a residual class of embeddings, with singularities governed by eigenvalues of the affine shape operator within the affine normal plane, providing a concrete corank-1 or corank-2 classification via the operators $S_{\bm{\nu}}$.

Abstract

In this paper, we classify the generic singularities of 2-parameter plane congruences in $\mathbb{R^4}$ and the generic singularities of affine normal plane congruences. We also study the generic singularities of the family of affine distance functions.

Generic singularities of affine distance functions and plane congruences

TL;DR

The paper develops a unified Lagrangian-singularity framework to classify generic singularities of 2-parameter plane congruences in and their affine normal variants. It shows that the affine distance function is a Morse family and locally --versal, yielding generic singularities of type – and , , with a Hessian description linking degeneracies to affine focal points. For general plane congruences, transversality and Mather-type stability imply a finite list of generic singularities realized as Lagrangian stable types (Fold, Cusp, Swallowtail, Butterfly, and umbilic variants). In the affine-normal case, the germ of the Blaschke congruence is Lagrangian stable for a residual class of embeddings, with singularities governed by eigenvalues of the affine shape operator within the affine normal plane, providing a concrete corank-1 or corank-2 classification via the operators .

Abstract

In this paper, we classify the generic singularities of 2-parameter plane congruences in and the generic singularities of affine normal plane congruences. We also study the generic singularities of the family of affine distance functions.

Paper Structure

This paper contains 9 sections, 25 theorems, 81 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results on affine differential geometry
  4. Transversality, Unfoldings, Morse family of functions and Lagrangian singularities
  5. Family of affine distance functions
  6. Generic singularities of the family of affine distance functions
  7. Singularities of plane congruences
  8. General case
  9. Generic singularities of affine normal plane congruences

Key Result

Lemma 2.1

(gg, Lemma 4.6)(Basic Transversality Lemma) Let $X$, $B$ and $Y$ be smooth manifolds with $W$ a submanifold of $Y$. Consider $j: B \rightarrow C^{\infty}\left(X, Y \right)$ a non-necessarily continuous map and define $\Phi: X \times B \rightarrow Y$ by $\Phi(x,b) = j(b)(x)$. Suppose $\Phi$ smooth an is a dense subset of $B$.

Figures (1)

  • Figure 1: Associated diagram

Theorems & Definitions (65)

  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Theorem 2.2
  • ...and 55 more