On singular foliations tangent to a given hypersurface

Abstract

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order $k$ foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter $k$ encodes the "order of tangency" of the leafwise vector fields to $L$. We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of $(k-1)$-jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.

Figures (5)

• Figure 1: Leaves of some foliations of $S^1 \times \mathbb{R}$.
• Figure 2: Equivalence relations of some foliations of $S^1 \times \mathbb{R}$, restricted to $T=\{0\} \times \mathbb{R}$.
• Figure 3: Holonomy groupoids of several foliations of $S^1 \times \mathbb{R}$ whose leaves are $S^1 \times \mathbb{R}_+$, $S^1 \times \{0\}$ and $S^1 \times \mathbb{R}_-$, restricted to $T=\{0\} \times \mathbb{R}$.
• Figure 4: The geometry behind Lemma \ref{['graphnbhd']}.
• Figure 5: Separating one component of $J^k_\mathbb{R}$ from the rest inside $G_\textup{full}(\mathcal{F}^k_\mathbb{R})$.

Theorems & Definitions (183)

• Definition 1.1
• Example 1.2
• Example 1.3
• Theorem : Definition \ref{['invdef']}
• Theorem : Theorem \ref{['holcompl']}
• Theorem : Theorem \ref{['holrange']}
• Theorem : Sections \ref{['principalbundles']}, \ref{['sec:gaugefull']}, \ref{['sec:gaugemin']}
• Theorem : Theorem \ref{['thm:topprop']}
• Theorem : Theorem \ref{['isotrope']}
• Theorem : Theorem \ref{['thm:extract']}, Corollary \ref{['C*L']}