Atiyah-Molino class of the foliated Lie algebroid (Q1896725)

This note presents an introduction to the study of the relations between the projectable structures on a foliation and the theory of Lie groupoids with their linearizations. The author ``introduces basic notions which are necessary in the theory of Lie algebroids with foliated bases, and constructs obstructions to the existence of projectable connections in a foliated Lie algebroid'' and states without demonstration the two following theorems: Theorem 1. --- Let \((A, \tau)\) be the foliated Lie algebroid over a foliated base \((B,F)\), and assume \(S = \text{Im} \tau\) be a sub-bundle of \(A\). Then the induced exact sequence of vector bundles: \(0 \to {\mathcal G} @>j'>> A/S @>q'>> TB/TF \to 0\) consists of foliated vector bundles and foliated morphisms. For foliated sections of the bundle \(A/S\) there is defined a bracket having properties of the Lie algebroid bracket with respect to the morphism \(q'\). Theorem 2. --- Let \(A\) be a foliated Lie algebroid. Then for an adapted connection \(\omega : A \to {\mathcal G}\) with the curvature form \(\Omega : A \otimes A \to {\mathcal G}\) we have \(d_F \Omega_{1,1} = 0\). The class \([\Omega_{1, 1}] \in H_F^{1,1} (A,{\mathcal G})\) does not depend on a choice of adapted connection. This class equals zero if and only if a projectable connection exists on \(A\).