Regularisation of Lie algebroids and Applications

TL;DR

The paper introduces a regularisation method for Lie algebroids that translates their geometric structures into foliated structures on higher-dimensional manifolds, enabling new proofs of the Weinstein conjecture for various Lie algebroids.

Contribution

It develops a regularisation procedure applicable to diverse Lie algebroids, simplifying the proof of the Weinstein conjecture without technical hypotheses.

Findings

Proved Weinstein conjecture for overtwisted b^k-contact structures.

Extended the conjecture's proof to other Lie algebroids.

Introduced tangent distributions as new objects of study.

Abstract

We describe a procedure, called regularisation, that allows us to study geometric structures on Lie algebroids via foliated geometric structures on a manifold of higher dimension. This procedure applies to various classes of Lie algebroids; namely, those whose singularities are of b^k, complex-log, or elliptic type, possibly with self-crossings. One of our main applications is a proof of the Weinstein conjecture for overtwisted b^k-contact structures. This was proven by Miranda-Oms using a certain technical hypothesis. Our approach avoids this assumption by reducing the proof to the foliated setting. As a by-product, we also prove the Weinstein conjecture for other Lie algebroids. Along the way we also introduce tangent distributions, i.e. subbundles of Lie algebroids, as interesting objects of study and present a number of constructions for them.