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The closure of linear foliations

Mateus de Melo, Ivan Struchiner

TL;DR

The paper provides a direct geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem by developing a framework of $\pi$-projectable foliations, compatible Ehresmann connections, and linearization around leaf closures, and by lifting linear foliations to frame bundles to obtain invariant, Riemannian structures. It proves a practical criterion for when a projectable foliation is Riemannian, shows how linear foliations can be extended via a compatible foliated affine connection, and deduces smoothness of the closure of linearized foliations, yielding an alternative MAR proof. This geometric approach bypasses analytic orbit-like methods, offering a clearer structural understanding of how leaf closures acquire smooth singular foliations. The results supply a versatile toolkit for analyzing leaf-closure smoothness in singular Riemannian foliations and have potential implications for the broader structure theory and holonomy of foliations.

Abstract

This paper presents a simplified geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem, which states that the closure of a singular Riemannian foliation on a complete Riemannian manifold is itself a smooth singular Riemannian foliation. Our approach circumvents several technical and analytical tools employed in the previous proof of the Theorem, resulting in a more direct geometric demonstration. We first establish conditions for a projectable foliation to be Riemannian, focusing on compatible connections. We then apply these results to linear foliations on vector bundles and their lifts to frame bundles. Finally, we use these findings to the linearization of singular Riemannian foliations around leaf closures. This method allows us to prove the smoothness of the closure directly for the linear semi-local model, bypassing the need for intermediate results on orbit-like foliations.

The closure of linear foliations

TL;DR

The paper provides a direct geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem by developing a framework of -projectable foliations, compatible Ehresmann connections, and linearization around leaf closures, and by lifting linear foliations to frame bundles to obtain invariant, Riemannian structures. It proves a practical criterion for when a projectable foliation is Riemannian, shows how linear foliations can be extended via a compatible foliated affine connection, and deduces smoothness of the closure of linearized foliations, yielding an alternative MAR proof. This geometric approach bypasses analytic orbit-like methods, offering a clearer structural understanding of how leaf closures acquire smooth singular foliations. The results supply a versatile toolkit for analyzing leaf-closure smoothness in singular Riemannian foliations and have potential implications for the broader structure theory and holonomy of foliations.

Abstract

This paper presents a simplified geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem, which states that the closure of a singular Riemannian foliation on a complete Riemannian manifold is itself a smooth singular Riemannian foliation. Our approach circumvents several technical and analytical tools employed in the previous proof of the Theorem, resulting in a more direct geometric demonstration. We first establish conditions for a projectable foliation to be Riemannian, focusing on compatible connections. We then apply these results to linear foliations on vector bundles and their lifts to frame bundles. Finally, we use these findings to the linearization of singular Riemannian foliations around leaf closures. This method allows us to prove the smoothness of the closure directly for the linear semi-local model, bypassing the need for intermediate results on orbit-like foliations.

Paper Structure

This paper contains 4 sections, 5 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Riemannian Foliations on Bundles and Compatible Connections
  3. Linear Foliations
  4. Closure of Linear Semi-local Models for SRFs

Key Result

Theorem 2.6

Let $\pi:P \rightarrow B$ be a submersion with $\mathcal{F}$ a regular $\pi$-projectable foliation whose projection is a regular foliation $\mathcal{F}_B$ in $B$. Suppose $\mathcal{H}$ is a connection compatible with $\mathcal{F}$, $\eta_B$ is a metric on $B$, and $\eta^{\mathbf{v}}$ is a metric on Moreover, if $\mathcal{H}$, $\eta_B$ and the restrictions of $\eta^{\mathbf{v}}$ to the fibers are

Theorems & Definitions (17)

  • Definition 2.1: Projectable Foliation
  • Definition 2.2: Compatible Ehresmann Connection
  • Remark 2.3
  • Remark 2.4: Induced Metric
  • Definition 2.5: $\mathcal{F}$-foliated Connection
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • proof
  • Definition 3.1: Metric-Preserving Linear Foliation
  • ...and 7 more