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Equiaffine structure on frontals

Igor Chagas Santos

TL;DR

The paper broadens affine differential geometry to include frontals by defining an equiaffine transversal field and a Blaschke vector field on fronts, and it establishes precise conditions for the existence and extension of the Blaschke structure across singularities. It develops the notion of a relative affine fundamental form, relative shape operator, and volume element tied to tangent moving bases, and shows how equiaffinity is captured by a vanishing transversal connection form. Through careful analysis of extendable Gaussian curvature and normal curvature, the authors provide canonical forms for certain frontal classes and derive explicit Blaschke-field expressions in examples. The main theoretical advance is a fundamental theorem for frontals, accompanied by compatibility equations that guarantee the integrability of the prescribed data and the affine equivalence of resulting frontal triples, thereby extending classical affine differential geometry to singular hypersurfaces with rigorous consistency conditions.

Abstract

In this paper, we generalize the idea of equiaffine structure to the case of frontals and we define the Blaschke vector field of a frontal. We also investigate some necessary and sufficient conditions that a frontal needs to satisfy to have a Blaschke vector field and provide some examples. Finally, taking the theory developed here into account we present a fundamental theorem, which is a version for frontals of the fundamental theorem of affine differential geometry.

Equiaffine structure on frontals

TL;DR

The paper broadens affine differential geometry to include frontals by defining an equiaffine transversal field and a Blaschke vector field on fronts, and it establishes precise conditions for the existence and extension of the Blaschke structure across singularities. It develops the notion of a relative affine fundamental form, relative shape operator, and volume element tied to tangent moving bases, and shows how equiaffinity is captured by a vanishing transversal connection form. Through careful analysis of extendable Gaussian curvature and normal curvature, the authors provide canonical forms for certain frontal classes and derive explicit Blaschke-field expressions in examples. The main theoretical advance is a fundamental theorem for frontals, accompanied by compatibility equations that guarantee the integrability of the prescribed data and the affine equivalence of resulting frontal triples, thereby extending classical affine differential geometry to singular hypersurfaces with rigorous consistency conditions.

Abstract

In this paper, we generalize the idea of equiaffine structure to the case of frontals and we define the Blaschke vector field of a frontal. We also investigate some necessary and sufficient conditions that a frontal needs to satisfy to have a Blaschke vector field and provide some examples. Finally, taking the theory developed here into account we present a fundamental theorem, which is a version for frontals of the fundamental theorem of affine differential geometry.
Paper Structure (14 sections, 18 theorems, 78 equations, 5 figures)

This paper contains 14 sections, 18 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fixing notations, definitions and some basic results
  3. Equiaffine structure for non-parabolic regular surfaces
  4. Frontals
  5. Frontals with extendable Gaussian curvature
  6. Frontals with extendable normal curvature
  7. Wave fronts with extendable Gaussian curvature
  8. Equiaffine structure on frontals
  9. The Blaschke and the conormal vector fields of a frontal
  10. The Blaschke vector field of a frontal
  11. Examples
  12. Conormal vector field
  13. A fundamental theorem
  14. The compatibility equations

Key Result

Proposition 2.1

(nomizu1994affine, Proposition 1.4) We have Consequently, the following two conditions are equivalent:

Figures (5)

  • Figure 1: Frontal with extendable non-vanishing Gaussian curvature.
  • Figure 2: Singular set $\Sigma(\bm{\xi})$ of the Blaschke vector field.
  • Figure 3: The Blaschke vector field from example \ref{['ex2']}
  • Figure 4: Frontal with extendable non-vanishing Gaussian curvature.
  • Figure 5: Wave front of rank $1$ with extendable non-vanishing Gaussian curvature.

Theorems & Definitions (57)

  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Remark 3.1
  • Definition 3.3
  • Remark 3.2
  • ...and 47 more