The closure of linear foliations

TL;DR

This paper offers a simplified geometric proof of the Molino-Alexandrino-Radeschi Theorem, demonstrating that the closure of a singular Riemannian foliation remains a smooth singular Riemannian foliation, using a direct geometric approach.

Contribution

It provides a more direct geometric proof of the theorem, avoiding complex analytical tools and focusing on linear foliations and their linearization.

Findings

Proved the smoothness of the closure of singular Riemannian foliations directly for linear models.

Established conditions for projectable foliations to be Riemannian based on compatible connections.

Applied results to linearize singular Riemannian foliations around leaf closures.

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.