×

About Lie flags. (Sur les drapeaux de Lie.) (French) Zbl 1044.57010

Summary: A flag of foliations on manifold \(M\) is a sequence of foliations \(\{{\mathcal F}_1,\dots,{\mathcal F}_p\}\) where for \(i\in \{1,\dots, p\}\dim{\mathcal F}_i= i\) and, for \(i\neq p\), \(T{\mathcal F}_i\subset T{\mathcal F}_{i+1}\).
In this paper we establish a “universal lifting theorem” on Lie flags.
Let \(\{{\mathcal F}_1,\dots,{\mathcal F}_p\}\) be a Lie flag on a compact manifold \(M\) , where each \({\mathcal F}_i\) is a \(G_i\)-Lie foliation admitting \(\Gamma_i\) as holonomy group. Then on the universal covering \(\widetilde M\) of \(M\) the flag can be lifted to a flag of simple Lie foliations such that for all \(i\in \{1,\dots, p-1\}\), there exists a submersion \(\theta_i\) from \(G_i\) onto \(G_{i+1}\), and a developing map \(D_i\) of \({\mathcal F}_i\) on \(\widetilde M\) satisfying the following conditions
1. \(\theta_i(\gamma.g)= \theta_i(\gamma.\theta_i(g))\) for each \(\gamma\in\Gamma_i\) and each \(g\in G_i\),
2. \(D_{i+1}= \theta_i\circ D_i\) and \(\Gamma_{i+1}= \theta_i(\Gamma_i)\).
From this theorem we deduce a characterization of Lie flags with a minimal flow, and the classification of Lie flags on three-dimensional compact, connected and orientable manifolds.

MSC:

57R30 Foliations in differential topology; geometric theory
PDF BibTeX XML Cite