Global attractors of complete conformal foliations (Q2901887)

The present paper deals with the existence of global attractors for complete non-Riemannian conformal foliations and describes the structure of such foliations. It is proved that every complete conformal foliation \((M,\mathcal{F})\) of codimension \(q\geq 3\) is either Riemannian or a \((\text{Conf}\,(S^q),S^q)\)-foliation. Without the assumption of compactness of \(M\), it is proved that if \((M,\mathcal{F})\) is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In particular, every proper conformal non-Riemannian foliation \((M,\mathcal{F})\) has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all nonclosed leaves is a connected \(q\)-dimensional orbifold. Also, it is shown that every countable group of conformal transformations of the sphere \(S^q\) can be realized as the global holonomy group of a complete conformal foliation. Examples that illustrate the present work are also constructed.