Feuilletages riemanniens à croissance polynômiale. (Riemannian foliations with polynomial growth) (Q1113097)

The author gives necessary and sufficient conditions for a Riemannian foliation \({\mathcal F}\) of a compact manifold M to be closed at infinity and to have a polynomial growth in terms of algebraic properties of the structural Lie algebra \({\mathfrak g}\) of \({\mathcal F}\). It is also shown that if \({\mathfrak g}\) is nilpotent, then \(\delta\) (\({\mathfrak g})\leq d({\mathcal F})\) where \(\delta\) (\({\mathfrak g})\) is the degree of nilpotence of \({\mathfrak g}\) and d(\({\mathcal F})\) is the degree of polynomial growth of \({\mathcal F}\). The following facts are obtained as corollaries: the structural Lie algebra of a Riemannian flow on a compact manifold is abelian; a Riemannian foliation \({\mathcal F}\) with polynomial growth on a compact manifold M is minimizable if and only if the basic cohomology of \({\mathcal F}\) of maximal degree does not vanish.