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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2012 Volume 203, Number 3, Pages 79–106 (Mi sm7821)

This article is cited in 18 papers

Global attractors of complete conformal foliations

N. I. Zhukova

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: We prove that every complete conformal foliation $(M,\mathscr F)$ of codimension $q\geqslant 3$ is either Riemannian or a $(\operatorname{Conf}(S^q), S^q)$-foliation. We further prove that if $(M,\mathscr F)$ 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 this theorem we do not assume that the manifold $M$ is compact. In particular, every proper conformal non-Riemannian foliation $(M,\mathscr F)$ 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. We show 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 of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.
Bibliography: 20 titles.

Keywords: conformal foliation, global holonomy group, minimal set, global attractor.

UDC: 514.77

MSC: Primary 37C85, 57R30; Secondary 22F05, 53C12

Received: 18.11.2010 and 12.05.2011

DOI: 10.4213/sm7821


 English version:
Sbornik: Mathematics, 2012, 203:3, 380–405

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