Topology of codimension-one foliations of nonnegative curvature
type: Item
instance of: scholarly article
title: Topology of codimension-one foliations of nonnegative curvature
zbMATH Open document ID: 1283.53030
published in: Sbornik: Mathematics
publication date: 2013-09-26T00:00:00Z
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 6212100
zbMATH Keywords: foliation
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: curvature
zbMATH Keywords: holonomy
zbMATH Keywords: resilient leaves
author: D. V. Bolotov
MaRDI profile type: Publication
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DOI: 10.1070/SM2013V204N05ABEH004314
review text: The author shows topological results on the structure of codimension one \(C^2\)-foliations on closed manifolds whose leaves have non-negative (sectional/Ricci) curvature. In these conditions, this is a foliation with almost no holonomy (any noncompact leaf has trivial holonomy), this fact allows a suitable decomposition into blocks with a good description. It is proved that the five-dimensional sphere does not admit a codimension one \(C^2\)-foliation with non-negative sectional curvature, that is a partial answer to an open question by \textit{G. Stuck} [C. R. Acad. Sci., Paris, Sér. I 313, No. 8, 519--522 (1991; Zbl 0735.53019)]. NEWLINENEWLINENEWLINEThe paper is organized in three sections (excluding the introduction). NEWLINENEWLINENEWLINEThe first section introduces known facts in the theory of codimension one foliations like stability theorems of Reeb and Thurston, relation between growth of leaves and transverse dynamics [\textit{J. F. Plante} and \textit{W. P. Thurston}, Comment. Math. Helv. 51, 567--584 (1976; Zbl 0348.57009)] or the structure of a block (a saturated connect manifold with boundary) with almost no holonomy [\textit{H. Imanishi}, J. Math. Kyoto Univ. 16, 93--99 (1976; Zbl 0335.57019)]. Therefore in \(C^2\)-foliations with non-negative Ricci curvature, leaves are dense or they contain a compact leaf in their closure and there are no resilient leaves. The author also introduces known results about the topological structure of complete manifolds with non-negative curvature [\textit{J. Cheeger} and \textit{D. Gromoll}, Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] and some results in the topology of four-dimensional manifolds which will be useful in the special case of codimension one foliation in the five-dimensional sphere [\textit{J. A. Hillman}, Four-manifolds, geometries and knots. Coventry: Geometry \& Topology Publications (2002; Zbl 1087.57015)]. NEWLINENEWLINENEWLINEThe second section shows that transversely orientable codimension…
reviewed by: Carlos Meniño
description: scientific article; zbMATH DE number 6212100