A classification of certain codimension one locally free actions of nilpotent Lie groups up to a differentiable orbital conjugacy

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Let G be a simply connected nilpotent Lie group, let M be a compact connected manifold with dim(M) = dim(G)+1 and let Φ be a C∞ locally free action of G on M. If G admits no lattice and if the centralizer in G of its derived group G′ is the center of G′, then Φ is differentiably orbitally conjugated to a homogeneous action of the same kind.