Foliations with leaves of nonpositive curvature
type: Item
instance of: scholarly article
title: Foliations with leaves of nonpositive curvature
zbMATH Open document ID: 0872.53024
DOI: 10.1007/BF02774030
published in: Israel Journal of Mathematics
publication date: 1997-09-21T00:00:00Z
review text: The author answers a question of \textit{M. Gromov} [Geom. Funct. Anal. 1, 253-320 (1991; Zbl 0768.53012)] in the codimension one case: if \(F\) is a codimension one foliation of a compact manifold \(M\) with leaves of negative curvature, then \(\pi_1(M)\) has exponential growth. He proves a result analogous to \textit{R. J. Zimmer} [Am. J. Math. 105, 1011-1022 (1983; Zbl 0527.57015)]: if \(F\) is a codimension one foliation on a compact manifold \(M\) with leaves of nonpositive curvature, and if \(\pi_1(M)\) has subexponential growth, then almost every leaf is flat. Finally, he gives a foliated version of the Hopf theorem on surfaces without conjugate points: if \(F\) is an orientable foliation on a compact manifold \(M\) with 2-dimensional leaves having no conjugate points, if the Euler class of the tangent bundle of the foliation is zero in real homology, and if all leaves have subexponential growth, then all leaves are flat.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1011392
zbMATH Keywords: foliations
zbMATH Keywords: negative curvature
zbMATH Keywords: fundamental group
zbMATH Keywords: exponential growth
author: Chengbo Yue
reviewed by: James Hebda
MaRDI profile type: Publication
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Wikidata QID: Q125946952
description: scientific article; zbMATH DE number 1011392