Topology of codimension-one foliations of nonnegative curvature
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 one \(C^2\)-foliations with non-negative Ricci curvature are foliations with almost no holonomy. In addition, these foliations satisfy one of the following claims:NEWLINE{\parindent=0.5cm\begin{itemize}\item[(1)] All leaves are dense, and the ambient manifold is a fiber bundle over \(S^1\). \item[(2)]The foliation contains at least one compact leaf and it is divided by a finite number of blocks of the following three types: NEWLINENEWLINE\end{itemize}}NEWLINE{\parindent=1cm\begin{itemize}\item[--] an exceptional block, homeomorphic to a direct product of a compact manifold and a closed interval, where one of the boundary leaves of this product is a limit leaf for the set of compact leaves.NEWLINE\item[--] a dense block. NEWLINE\item[--] a proper block, that is topologically similar to a Reeb component. NEWLINENEWLINE\end{itemize}}NEWLINEThe fundamental group of dense and proper leaves is also described. In the case of non-negative sectional curvature, it is possible to bound the number of connected components of the boundary of these blocks by \(2\). In this latest hypothesis, the embedding of each boundary component in the block is a homotopy equivalence. NEWLINENEWLINENEWLINEThe third section deals with the existence of such foliations in five-dimensional manifolds. The author uses the introduced tools in four-dimensional topology to obtain extra information on the leaves in this restricted case. Results of the previous section allow to deal with the different structures that could appear and the obstructions imposed by an ambient manifold homeomorphic to a five-dimensional sphere. It is concluded that five-dimensional Riemannian manifolds homeomorphic to the five-sphere do not admit a \(C^2\)-foliation of codimension \(1\) with non-negative sectional curvature.NEWLINENEWLINENEWLINEWe remark that the author makes a good use of algebraic topology, providing clean and readable proofs.
scientific article; zbMATH DE number 6212100
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