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On Affinely Homogeneous Submanifolds: The Power Series Method of Equivalence

Julien Heyd, Joel Merker

Abstract

We determine all affinely homogeneous models for surfaces $S^2 \subset \mathbb{R}^4$, including the simply transitive models. We employ an improved power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. We find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models, sometimes parametrized by a certain invariant algebraic variety. Three main features may be emphasized: 1) Iterated single-pointed jet bundles; 2) Cartan-enhanced power series method of equivalence; 3) Constant ping-pong between normal forms (nf) and vector fields (vf).

On Affinely Homogeneous Submanifolds: The Power Series Method of Equivalence

Abstract

We determine all affinely homogeneous models for surfaces , including the simply transitive models. We employ an improved power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. We find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models, sometimes parametrized by a certain invariant algebraic variety. Three main features may be emphasized: 1) Iterated single-pointed jet bundles; 2) Cartan-enhanced power series method of equivalence; 3) Constant ping-pong between normal forms (nf) and vector fields (vf).
Paper Structure (31 sections, 19 theorems, 554 equations)

This paper contains 31 sections, 19 theorems, 554 equations.

Table of Contents

  1. Introduction
  2. Differential Invariants and Homogeneous Models
  3. Fibers Over Group Transversals Versus Full Jet Bundle
  4. Example: Parabolic Surfaces $S^2 \subset \mathbb{C}^3$
  5. General Setting: Induction on Jet Order
  6. Non-Reduced Linear Representation
  7. Setting Up Dependent Jets
  8. Reduced Linear Representation and Branch Creation
  9. Determination of Homogeneous Models
  10. Creations of Geometries
  11. General Method in Affine Context
  12. A Classification of Affinely Homogeneous $S^2 \subset \mathbb{C}^3$
  13. Termination: Moduli Spaces of Homogeneous Models
  14. Termination: Jacobi Identities and Zero Remainders
  15. Lie Algebras of Vector Fields Versus Closed Forms
  16. ...and 16 more sections

Key Result

Theorem 12.1

Up to the action of a finite subgroup of ${\sf Aff}(\mathbb{C}^3)$, there are 12 (families of) affinely homogeneous model surfaces $S^2 \subset \mathbb{C}^3$, as represented by the diagram on p. diag-arbre-S2-R3. The list of (truncated) normal forms together with their associated transitive Lie alge

Theorems & Definitions (21)

  • Definition 9.2
  • Theorem 12.1
  • Lemma 18.2
  • Proposition 19.1
  • Proposition 19.4
  • Proposition 20.1
  • Lemma 21.1
  • Proposition 21.3
  • Proposition 21.4
  • Theorem 23.1
  • ...and 11 more