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 $\mathscr F$, there exists one more foliation $(M,\mathscr O)$, which is generally singular and is called an aureole foliation; moreover, the foliations $\mathscr F$ and $\mathscr O$ have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type $\mathfrak g/\mathfrak h$ with a compactly embedded Lie subalgebra $\mathfrak h$ in $\mathfrak g$, 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 $(M,\mathscr F)$. We prove that for such foliations, there exists a unique minimal set $\mathscr M$, and $\mathscr M$ is contained in the closure of any leaf. If the foliation $(M,\mathscr F)$ is proper, then $\mathscr M$ is a unique closed leaf of this foliation.