Splitting of nonnegatively curved leaves in minimal sets of foliations
type: Item
instance of: scholarly article
title: Splitting of nonnegatively curved leaves in minimal sets of foliations
zbMATH Open document ID: 0793.53032
DOI: 10.1215/S0012-7094-93-07104-9
author: Scot Adams
author: Garrett Stuck
published in: Duke Mathematical Journal
publication date: 1994-08-16T00:00:00Z
review text: Let \(L\) be a leaf in a minimal set \({\mathcal L}\) of a smooth foliation of a compact Riemannian manifold, \(G\) an isometry group of universal covering \(\widetilde{L}\) of \(L\). The authors prove, that 1) if \(G\) is cocompact (\(\widetilde{L}= G*K\) for some compact subset \(K\subseteq L\)) then there exist: (A) a compact Hausdorff space \(X\), (B) a locally free minimal action of \(G\) on \(X\), (C) a continuous map \(\eta: X\to {\mathcal L}\) which carries \(G\)-orbits to leaves of \({\mathcal L}\), such that the image of \(\eta\) intersects every leaf of \({\mathcal L}\), and for every \(x\in X\), the leaf through \(\eta(x)\) is isometric to \(\widetilde{L}/ G_ x\), where \(G_ x\) denotes the stabilizer in \(G\) of \(x\); 2) if \(L\) has nonnegtive Ricci curvature, then \(\widetilde{L}\) is isometric to the Riemannian product of a compact manifold by a flat Euclidean space, moreover, any two leaves of \({\mathcal L}\) a) have isometric universal cover, b) which split are isometric, and some other statements. The proofs are based on the theory of measure-preserving and topological actions, Cheeger and Gromoll splitting theorem and the results of the first author and \textit{A. Freire} [J. Differ. Geom. 34, No. 3, 681-700 (1991; Zbl 0751.53010)].
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 495437
zbMATH Keywords: minimal set
zbMATH Keywords: isometry group
zbMATH Keywords: Ricci curvature
zbMATH Keywords: Riemannian product
zbMATH Keywords: splitting
reviewed by: Vladimir Yu. Rovenskij
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07104-9
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description: scientific article; zbMATH DE number 495437