Twisted submersions in nonnegative sectional curvature
type: Item
instance of: scholarly article
title: Twisted submersions in nonnegative sectional curvature
zbMATH Open document ID: 1277.53033
author: Pablo Angulo Ardoy
author: Luis Guijarro
author: Gerard Walschap
published in: Archiv der Mathematik
publication date: 2013-09-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.2569
review text: Let \(F\) be a metric foliation with complete leaves on a Riemannian manifold \((M,g)\) of nonnegative sectional curvature. A curve is said to be horizontal if it is orthogonal to the leaves of \(F\) at every point. Given any point \(P\in M\), consider the subset of \(M\) that can be reached by the horizontal curves from \(P\); the work of \textit{B. Wilking} [Geom. Funct. Anal. 17, No. 4, 1297--1320 (2007; Zbl 1139.53014)] shows that they form a singular metric foliation \(F^{\#}\) which is called the dual foliation; if the metric is positively curved, then the dual foliation contains only a single leaf. The authors consider the case in which the foliation appears as the collection of fibers of a Riemannian submersion; a Riemannian submersion \(\pi:M\rightarrow B\) with fiber \(F\) is said to be twisted if the dual foliation only contains one leaf. The authors show: \smallbreak\noindent{Theorem 1.} Let \(\pi:M\rightarrow B\) be a Riemannian submersion. Assume that \(M\) is complete and has nonnegative sectional curvature. Let \(\delta\) be the connecting homomorphism from \(\pi_q(B)\) to \(\pi_{q-1}(F)\) in the long exact sequence in homotopy. If \(\pi\) is not twisted then any element in the range of \(\delta\) can be represented by a map \(f\) from \(S^{q-1}\) to the fiber \(F\) which is not surjective. \smallbreak\noindent Various applications of this result are given, for example: \smallbreak\noindent{Theorem 3.} Let \(M\) be a compact simply connected manifold. Assume that \(N=M\times S^2\) admits a nonnegatively curved metric such that there exists a Riemannian submersion \(\pi:N\rightarrow M\) with totally geodesic fibers. Then either \(\pi\) is twisted or \(N\) is isometric to the Riemannian product of \(M\times S^2\). \noindent The homotopy conditions in Theorem 1 can be reformulated in cohomological terms using transgressive elements.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 6210419
zbMATH Keywords: dual foliation
zbMATH Keywords: Riemannian submersion
zbMATH Keywords: twisted submersion
zbMATH Keywords: horizontal curve
zbMATH Keywords: long exact sequence in homotopy
zbMATH Keywords: Hopf fibration
zbMATH Keywords: transgressive elements
zbMATH Keywords: principal torus bundle
MaRDI profile type: Publication
OpenAlex ID: W2008059147
arXiv ID: 1212.2569
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DOI: 10.1007/S00013-013-0550-Z
description: scientific article; zbMATH DE number 6210419