Ends of generic leaves of complete Cartan foliations
type: Item
instance of: scholarly article
title: Ends of generic leaves of complete Cartan foliations
zbMATH Open document ID: 1201.53033
DOI: 10.1134/S0001434610010384
author: N. I. Zhukova
published in: Mathematical Notes
publication date: 2010-09-24T00:00:00Z
review text: Suppose that \(\left( M,\mathcal F\right) \) is a complete Cartan foliation. The author gives the following theorems: If the structure Lie algebra \(\mathfrak{g}_0=\mathfrak{g}_0\left( M,\mathcal F\right)\) is nonzero, then the set of ends of the generic leaves of \(\left( M,\mathcal F\right)\) is either finite and consists of one or two points or is Cantor. If, in addition, \(\dim\mathcal F =2\), then either all generic leaves are noncompact and have genus 0 or all generic leaves are orientable and have only nonplanar ends, or all ends of generic leaves are nonorientable. For \(\mathfrak{g}_0=0\), any finite, countable, or Cantor set can be realized as the set of ends of the generic leaves of \(\left( M, \mathcal F\right)\). If \(\left( M,\mathcal F\right)\) has a compact leaf, then either the set of ends of a generic leaf \(L_0\) is Cantor, or \(e\left( L_0\right)\) is finite and equal to 0,1 or 2; moreover, \(e\left( L_0\right) =0\) if and only if \(\left( M,\mathcal F\right)\) is a compact Riemannian foliation and the space of leaves \(M/\mathcal F\) is a smooth orbifold. Suppose that \(\left( M,\mathcal F\right)\) is of type \(\mathfrak{g}/\mathfrak{h}\) with nonzero structure Lie algebra and at least one of the following two conditions holds: {\parindent7mm \begin{itemize}\item[(1)] the structure algebra \(\mathfrak{g}_0\left( M,\mathcal F\right)\) is solvable; \item[(2)] the algebra \(\mathfrak{g}\) is solvable and the foliation \(\left( M,\mathcal F\right)\) has zero transverse curvature. \end{itemize}} Then the standard generic leaf of this foliation has precisely one or precisely two ends. If \(M\) is a simply connected compact manifold, then \(e\left( L_0\right)\) is equal to 0, 1 or 2. Moreover \(e\left( L_0\right) =0\) if and only if \(\left( M,\mathcal F\right)\) is a compact Riemannian foliation and \(M/\mathcal F\) is a smooth simply connected compact orbifold.
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 5791050
zbMATH Keywords: complete Cartan foliation
zbMATH Keywords: end of a manifold
zbMATH Keywords: orbifold
zbMATH Keywords: generic leaf of a foliation
zbMATH Keywords: structure Lie algebra
zbMATH Keywords: Lie derivative
zbMATH Keywords: adjoint representation
zbMATH Keywords: Cantor set
reviewed by: Andrzej Piatkowski
MaRDI profile type: Publication
cites work: Cartan connections in foliated bundles
cites work: Parabolic geometries and canonical Cartan connections
cites work: Ends of leaves of Lie foliations
cites work: The end point compactification of manifolds
cites work: Generic leaves
cites work: Topology of generic leaves
cites work: Enden offener Räume und unendliche diskontinuierliche Gruppen
full work available at URL: s0001434610010384
OpenAlex ID: W2048228619
description: scientific article; zbMATH DE number 5791050