Lie foliations transversely modeled on nilpotent Lie algebras
type: Item
instance of: scholarly article
title: Lie foliations transversely modeled on nilpotent Lie algebras
zbMATH Open document ID: 1332.57027
author: Naoki Kato
published in: Geometriae Dedicata
publication date: 2015-12-16T00:00:00Z
review text: Let \(M\) be a \(n\)-dimensional closed orientable smooth manifold and let \(\mathcal F\) be a codimension \(q\) transversely orientable smooth foliation on M. Let \(g\) be some \(q\)-dimensional Lie algebra over \(\mathbb R\). The foliation \(\mathcal F\) is a Lie \(g\)-foliation if there exists a nonsingular Maurer-Cartan form \(\omega \in \Omega^1(M,g)\) such that \(T \mathcal F = Ker(\omega)\). For each Lie foliation \(\mathcal F\) there are associated two Lie algebras, the model Lie algebra \(g\) and the structure Lie algebra \(h \subset g\). There is a natural question to determine the pairs of Lie algebras \((g, h)\) which can be realized as a Lie \(g\)-foliations \(\mathcal F\) on some closed manifolds \(M\) with structure Lie algebras \(h\). Such pairs \((g,h)\) are called realizable. In this paper, two problems of realization for Lie foliations are studied: (1) Which pair of Lie algebras \((g,h)\) can be realized? and (2) Which pair \(g,m\) \((m \in \mathbb N)\) can be realized as a Lie \(g\)-foliation on a closed manifold with the structure Lie algebra \(\mathbb R^m\)? An answer to (1) in the case, when \(g\) is a nilpotent Lie algebra, is given: let \(h\) be a subalgebra of such Lie algebra \(g\), then the pair \((g, h)\) is realizable if and only if \(h\) is an ideal of \(g\) and the quotient Lie algebra \(g/h\) has a rational structure. An answer is given to (2) in the case when \(g\) is a nilpotent Lie algebra which has a rational structure: the pair \(g,m\) is realizable if and only if \(m \leq \dim c(g)\), where \(c(g)\) is the center of \(g\). Here the assumption that \(g\) has a rational structure can't be avoided.
reviewed by: V. V. Gorbatsevich
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 6521446
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lie foliation
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: nilpotent Lie group
MaRDI profile type: Publication
full work available at URL: s10711-015-0065-9
OpenAlex ID: W2099014167
cites work: Q3900695
cites work: Uncountably many nonisomorphic nilpotent Lie algebras
cites work: Q4692884
cites work: Q5622951
cites work: Lie Flows of Codimension 3
cites work: Q3803880
cites work: Transverse structure of Lie foliations
cites work: Solvable Lie foliations
cites work: Unimodular Lie foliations
cites work: Q3892996
cites work: Q3664067
cites work: Q3977313
cites work: Deux remarques sur les flots Riemanniens. (Two remarks on Riemannian flows)
cites work: Q5669004
DOI: 10.1007/S10711-015-0065-9
description: scientific article; zbMATH DE number 6521446