## 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