Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ℱ, there exists one more foliation (M, $$\mathcal{O}$$ ), which is generally singular and is called an aureole foliation; moreover, the foliations ℱ and $$\mathcal{O}$$ have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ℊ/ℎ with a compactly embedded Lie subalgebra ℊ in ℎ, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ℱ). We prove that for such foliations, there exists a unique minimal set ℱ, and ℱ is contained in the closure of any leaf. If the foliation (M, ℱ) is proper, then ℳ is a unique closed leaf of this foliation.