Totally umbilic Riemannian foliations
type: Item
instance of: scholarly article
title: Totally umbilic Riemannian foliations
zbMATH Open document ID: 0695.53024
author: Grant Cairns
published in: Michigan Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The author proves the following theorem: Let F be an umbilicalisable Riemannian foliation of a compact connected manifold M. Then F is dense, and the following conditions are equivalent: (i) the basic cohomology of F verifies Poincaré duality; (ii) F is taut; (iii) F is geodesible. Moreover, if F does not satisfy these conditions then there is a Riemannian metric on M for which F is totally umbilic and isoparametric, and for which the leaves of F have zero sectional curvature. Furthermore, there is an open dense subset of M that is saturated by simply connected leaves. The above theorem implies some sufficient conditions for the geodesibility of F. A classification of umbilical and non-geodesible Riemannian foliations of codimension 2 of a compact connected manifold M, is given.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4138690
zbMATH Keywords: umbilicalisable Riemannian foliation
zbMATH Keywords: Poincaré duality
zbMATH Keywords: taut
zbMATH Keywords: geodesible
zbMATH Keywords: totally umbilic
zbMATH Keywords: isoparametric
Wikidata QID: Q115238989
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004072
OpenAlex ID: W2070489305
DOI: 10.1307/MMJ/1029004072
description: scientific article; zbMATH DE number 4138690