Atiyah-Molino class of the foliated Lie algebroid
type: Item
instance of: scholarly article
title: Atiyah-Molino class of the foliated Lie algebroid
zbMATH Open document ID: 0838.57018
author: I. V. Bel'ko
published in: Russian Mathematics
publication date: 1995-11-06T00:00:00Z
review text: 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 \(A\).
reviewed by: J.L.Viviente
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 20L05
zbMATH DE Number: 795240
zbMATH Keywords: curvature
zbMATH Keywords: Lie algebroid
zbMATH Keywords: Lie groupoid
zbMATH Keywords: obstruction class
zbMATH Keywords: projectable foliation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 795240