Abstract
We give here some examples of solvable Lie foliation with some interesting properties on compact manifolds. Every manifold in this paper is compact and our Lie group is simply connected.
Highlights
Ghys (1988) is the Lie foliation, which is called homogeneous Lie foliation, obtained in the following way: G and H two connected, connected, Lie groups, Γ a cocompact discrete subgroup of H, and φ a surjective morphism of Lie groups of H in G; the classes on the left of H modulo ker φ reprojected on H/Γ are the leaves of a G-foliation of H/Γ whose holonomy morphism is the restriction of φ to Γ and the developing application is φ
The naturel question is in which condition a Lie foliation can be reduced, using naturals operations, to a homogeneous Lie foliation.In the case where the Lie group is nilpotent Haefliger (1984) has shown that every nilpotent Lie foliation on compact manifolds is an inverse image of a homogeneous Lie foliation
Since Γ is not polycyclic this foliation is not inverse image of homogeneous foliation. Using this method we show Theorem 4 There is a compact manifold with boundary, of dimension 2n + 1 that carries a Lie GA-foliation which is not inverse image of homogeneous Lie foliation, where GA is the affine group
Summary
Else on the other hand, if M and V are two compact manifolds, F a Lie G- foliation on M and f : V −→ M a transverse differentiable application to F ; the inverse images of leaves of F define a new Lie G-foliation Fon V. We say that Fis an inverse image of F , in this case the holonomy group of Fis a subgroup of that of F. The naturel question is in which condition a Lie foliation can be reduced, using naturals operations, to a homogeneous Lie foliation.In the case where the Lie group is nilpotent Haefliger (1984) has shown that every nilpotent Lie foliation on compact manifolds is an inverse image of a homogeneous Lie foliation. The main purpose of this article is to build foliations that are not inverse image of a homogeneous Lie foliation
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