Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reductions of path structures and classification of homogeneous structures in dimension three

Elisha Falbel, Martin Mion-Mouton, Jose M. Veloso

TL;DR

This work studies path structures on 3-manifolds by constructing a Cartan connection on a canonical principal bundle $Y$ with structure group $B_{\mathbb{R}}$, and by introducing curvature invariants $Q^1$ and $Q^2$ that control local equivalence to the flat model $SL(3,{\mathbb R})/B_{\mathbb R}$. It proves canonical reductions: non-flat path structures admit a $\mathbb{Z}/2\mathbb{Z}$-reduction, and under further curvature conditions often a parallelism, which yields Tresse's classical bound that the automorphism group has dimension at most $3$. The paper then classifies invariant path structures on all three-dimensional Lie groups by analyzing the adjoint action and by computing explicit curvatures for homogeneous models, with detailed results organized in tables. It also specializes to SL$(2,{\mathbb R})$, connecting the path-structure data to the Lorentzian geometry of the Killing form and a cross-ratio invariant that parametrizes left-invariant structures, thus providing a complete moduli description in this fundamental case. Overall, the work advances the classification of non-flat path geometries in dimension three, connecting Cartan-invariant reductions to concrete homogeneous models and yielding concrete invariants for canonical comparisons and SEO-friendly descriptions.

Abstract

In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a Z/2Z-structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896). We also classify the invariant path structures on three-dimensional Lie groups.

Reductions of path structures and classification of homogeneous structures in dimension three

TL;DR

This work studies path structures on 3-manifolds by constructing a Cartan connection on a canonical principal bundle with structure group , and by introducing curvature invariants and that control local equivalence to the flat model . It proves canonical reductions: non-flat path structures admit a -reduction, and under further curvature conditions often a parallelism, which yields Tresse's classical bound that the automorphism group has dimension at most . The paper then classifies invariant path structures on all three-dimensional Lie groups by analyzing the adjoint action and by computing explicit curvatures for homogeneous models, with detailed results organized in tables. It also specializes to SL, connecting the path-structure data to the Lorentzian geometry of the Killing form and a cross-ratio invariant that parametrizes left-invariant structures, thus providing a complete moduli description in this fundamental case. Overall, the work advances the classification of non-flat path geometries in dimension three, connecting Cartan-invariant reductions to concrete homogeneous models and yielding concrete invariants for canonical comparisons and SEO-friendly descriptions.

Abstract

In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a Z/2Z-structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896). We also classify the invariant path structures on three-dimensional Lie groups.
Paper Structure (25 sections, 17 theorems, 185 equations)

This paper contains 25 sections, 17 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The Cartan connection of a path structure
  3. The coframe bundle $Y$ over the bundle $E$ of contact forms
  4. The connection form on the bundle $Y$
  5. Bianchi identities
  6. Higher order Bianchi identities
  7. Strict path structures
  8. Bianchi identities
  9. Reductions
  10. Reduction of the structure: $Q^1\neq 0$ and $Q^2\neq 0$.
  11. Reduction of the structure: $Q^1=0$ and $Q^2\neq 0$.
  12. Further reductions of the structure: $Q^1=0$ and $Q^2\neq 0$.
  13. Homogeneous structures
  14. Three dimensional Lie groups with path structures
  15. $\hbox{ad}_Y\mathfrak{p}\subset \mathfrak{p}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

If the path structure is not flat, there exists a canonical reduction of the fiber bundle $Y$ to a ${{\mathbb Z}}/2{{\mathbb Z}}$-structure.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 3.1
  • Proposition 3.2
  • Theorem 4.1: M. A. Tresse tresse
  • Theorem 4.2
  • Proposition 4.3
  • ...and 12 more