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Existence and classification of the Cartan $(2,3,5)$-distribution

Jiro Adachi

TL;DR

This work shows that the existence of Cartan $(2,3,5)$-distributions on a $5$-manifold is governed by purely topological data, reframed through an almost Cartan structure as a formal model. Using Gromov’s convex integration, the authors prove an $h$-principle that lifts formal almost Cartan structures to genuine Cartan structures, and they derive a one-parameter homotopy classification: two Cartan distributions are homotopic through genuine structures whenever they are homotopic through almost Cartan structures. For closed orientable $5$-manifolds, the obstructions reduce to spin-structure and the vanishing Kervaire semicharacteristic, together with the condition $ frac{1}{2}p_1(M)=e( ext{D})^2$ for the Euler class $e( ext{D})$ of a rank-$2$ distribution. The results yield concrete examples and connect differential topology with Cartan geometry, enabling a topological criterion for existence and a principled classification framework via the $h$-principle.

Abstract

The Cartan $(2,3,5)$-distribution is a tangent distribution of rank~$2$ on a $5$-dimensional manifold satisfying certain generic conditions. The necessary and sufficient condition for a manifold to admit such a structure is established in this paper. The condition obtained is purely topological. In addition, the classification of such structures, up to homotopy as formal Cartan distributions, is obtained.

Existence and classification of the Cartan $(2,3,5)$-distribution

TL;DR

This work shows that the existence of Cartan -distributions on a -manifold is governed by purely topological data, reframed through an almost Cartan structure as a formal model. Using Gromov’s convex integration, the authors prove an -principle that lifts formal almost Cartan structures to genuine Cartan structures, and they derive a one-parameter homotopy classification: two Cartan distributions are homotopic through genuine structures whenever they are homotopic through almost Cartan structures. For closed orientable -manifolds, the obstructions reduce to spin-structure and the vanishing Kervaire semicharacteristic, together with the condition for the Euler class of a rank- distribution. The results yield concrete examples and connect differential topology with Cartan geometry, enabling a topological criterion for existence and a principled classification framework via the -principle.

Abstract

The Cartan -distribution is a tangent distribution of rank~ on a -dimensional manifold satisfying certain generic conditions. The necessary and sufficient condition for a manifold to admit such a structure is established in this paper. The condition obtained is purely topological. In addition, the classification of such structures, up to homotopy as formal Cartan distributions, is obtained.

Paper Structure

This paper contains 23 sections, 17 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cartan distribution and almost Cartan structure
  3. Tangent distributions and their derivations
  4. The Cartan $(2,3,5)$-distribution and $(3,5)$-distribution
  5. Almost Cartan structure
  6. (1) The bundle $\mathcal{D}$.
  7. (2) The bundle $\mathcal{E}/\mathcal{D}$.
  8. (3) The bundle $TM/\mathcal{E}$.
  9. Algebraic topology on almost Cartan structures
  10. Homotopy principle and Gromov's convex integration method
  11. Differential relations and homotopy principle
  12. Convex integration method
  13. The h-principles for distributions of type $(3,5)$
  14. Claims for $(3,5)$-distributions
  15. Differential relation and its ampleness
  16. ...and 8 more sections

Key Result

Theorem 1

Let $M$ be a possibly closed $5$-dimensional manifold. Then $M$ admits a Cartan $(2,3,5)$-distribution if and only if it admits an almost Cartan structure.

Figures (4)

  • Figure 4.1: cone type
  • Figure 4.2: hyperboloid type ($+$)
  • Figure 4.3: hyperboloid type ($-$)
  • Figure 4.4: non-central case

Theorems & Definitions (30)

  • Theorem 1
  • Theorem : Dave-Haller
  • Theorem 2
  • Theorem 3
  • Definition
  • Proposition 2.1
  • Theorem : Cartan
  • Proposition 2.2
  • Definition 2.3: almost Cartan structure
  • Proposition 2.4
  • ...and 20 more