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The small growth invariants of Goursat distributions

Susan Jane Colley, Gary Kennedy, Corey Shanbrom

TL;DR

This work advances the local theory of Goursat distributions by introducing and systematizing small growth invariants, building on the monster tower framework and RVT/Goursat code words. It develops explicit charts, focal/vertical order notions, and a detailed structure theorem that describes sections of the small growth sheaves, yielding concrete recursion schemes and calculation pathways. The authors connect the small growth invariants to previously studied structural invariants, via a precise bridge that expresses beta, der, and der^2 in terms of the b-vector, multiplicity vector, and vertical orders, and they establish front-end and back-end recursions, including Jean’s back-end recursions, all the way to the Puiseux-characteristic-based degree of nonholonomy. The results provide both conceptual clarity and practical tools for computing invariants of Goursat distributions across smooth, complex, and algebraic settings, with implications for singularity theory and the geometry of curve projections.

Abstract

This is the second of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. In the prior paper we studied those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In this paper we study invariants arising from the small growth sequence of a Goursat distribution, and relate them to the the structural invariants.

The small growth invariants of Goursat distributions

TL;DR

This work advances the local theory of Goursat distributions by introducing and systematizing small growth invariants, building on the monster tower framework and RVT/Goursat code words. It develops explicit charts, focal/vertical order notions, and a detailed structure theorem that describes sections of the small growth sheaves, yielding concrete recursion schemes and calculation pathways. The authors connect the small growth invariants to previously studied structural invariants, via a precise bridge that expresses beta, der, and der^2 in terms of the b-vector, multiplicity vector, and vertical orders, and they establish front-end and back-end recursions, including Jean’s back-end recursions, all the way to the Puiseux-characteristic-based degree of nonholonomy. The results provide both conceptual clarity and practical tools for computing invariants of Goursat distributions across smooth, complex, and algebraic settings, with implications for singularity theory and the geometry of curve projections.

Abstract

This is the second of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. In the prior paper we studied those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In this paper we study invariants arising from the small growth sequence of a Goursat distribution, and relate them to the the structural invariants.

Paper Structure

This paper contains 12 sections, 16 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The three settings
  3. Small growth
  4. Recollection of the first paper
  5. Charts, coordinates, focal vector fields
  6. Lie derivatives and Lie brackets
  7. Focal order and vertical order
  8. Sections of the small growth sheaves
  9. Calculation pathways
  10. Relating the structural and small growth invariants
  11. Recursions
  12. Degree of nonholonomy via the Puiseux characteristic

Key Result

Lemma 3

For each $i$, there is an alternative basis for $\Delta_i$:

Figures (3)

  • Figure 1: The invariants listed in the top five boxes were studied in our prior paper MR4887124; those listed in the bottom box are invariants of the small growth sequence of a Goursat germ. See Figure \ref{['exdiagram']} for an example.
  • Figure 2: An example of the invariants of Figure \ref{['invdiagram']}. The dashed arrows indicate compatible front-end recursions.
  • Figure 3: Obtaining the proximity diagram for the Goursat word $RRVTVVR$ from the proximity diagram for its lifted Goursat word $RRRVVR$; the new multiplicity is $m_1=3+3+2$.

Theorems & Definitions (45)

  • Example 1
  • Example 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Example 6
  • Lemma 7
  • ...and 35 more