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Bi-contact structures with symmetry: local normal forms

Connor Jackman

TL;DR

The paper develops a complete local classification of bi-contact structures $(M, \xi, \tilde{\xi})$ on 3-manifolds that admit infinitesimal symmetries, via a Cartan-type invariant framework built from the axis $A = \xi \cap \tilde{\xi}$, the Schwartzian invariant $\mathcal{S}$, and the generating invariant $I$. It separates the analysis into the vanishing and non-vanishing Schwartzian cases, deriving explicit local normal forms for the pair and detailing the associated symmetry algebras. Two concrete realizations are discussed: orientable Anosov flows realized by item I$_1$ and integrable contact fields giving item I$_2$, illustrating the geometric and dynamical significance of the classification. The results connect bi-contact structures to path geometries, Riccati equations, and Cartan's equivalence method, offering a structured pathway to understand projective Anosov dynamics through local normal forms and invariants.

Abstract

A pair of transverse contact distributions on a 3-manifold will in general admit no 1-parameter families of symmetries: a flow preserving both contact distributions. Here, we will determine local normal forms for such pairs admitting symmetries. In particular, we observe that orientable Anosov flows may be globally given by the intersection of a pair of oppositely oriented contact distributions admitting, around any point, maximal local symmetries.

Bi-contact structures with symmetry: local normal forms

TL;DR

The paper develops a complete local classification of bi-contact structures on 3-manifolds that admit infinitesimal symmetries, via a Cartan-type invariant framework built from the axis , the Schwartzian invariant , and the generating invariant . It separates the analysis into the vanishing and non-vanishing Schwartzian cases, deriving explicit local normal forms for the pair and detailing the associated symmetry algebras. Two concrete realizations are discussed: orientable Anosov flows realized by item I and integrable contact fields giving item I, illustrating the geometric and dynamical significance of the classification. The results connect bi-contact structures to path geometries, Riccati equations, and Cartan's equivalence method, offering a structured pathway to understand projective Anosov dynamics through local normal forms and invariants.

Abstract

A pair of transverse contact distributions on a 3-manifold will in general admit no 1-parameter families of symmetries: a flow preserving both contact distributions. Here, we will determine local normal forms for such pairs admitting symmetries. In particular, we observe that orientable Anosov flows may be globally given by the intersection of a pair of oppositely oriented contact distributions admitting, around any point, maximal local symmetries.

Paper Structure

This paper contains 10 sections, 23 theorems, 101 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Invariants
  3. Two examples
  4. Anosov flows
  5. Integrable contact vector fields
  6. Computations
  7. Vanishing Schwartzian
  8. Associated path geometries
  9. Non-zero Schwartzian
  10. proof of theorem \ref{['thm:lnf']}

Key Result

Theorem 1.1

Suppose a pair of transverse contact distributions $\xi, \tilde{\xi}$ on a 3-manifold $M$ admits an infinitesimal symmetry around $m\in M$ transverse to $\xi\cap\tilde{\xi}$. Then around $m$, the pair is given in some coordinates by one of the following local normal forms:

Figures (1)

  • Figure 1: Locally, integral curves $\mathcal{A}$ of a Legendrian foliation on $(M,\xi)$ are canonically identified with $\mathbb{RP}^1$ (up to projective transformations). For $\Sigma \cong_{loc} M/A$, a local slice, send $m\in \mathcal{A}$ to $\pi_*(\xi_m) \in \mathbb{P}(T_{\pi(m)}\Sigma)$ (by the contact condition, a local diffeomorphism).

Theorems & Definitions (68)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Remark 1
  • Remark 2: On the transversal condition to the axis.
  • Remark 3
  • Definition 3
  • Remark 4
  • Definition 4
  • Definition 5
  • ...and 58 more