Comparing Riemannian foliations with transversally symmetric foliations
type: Item
instance of: scholarly article
title: Comparing Riemannian foliations with transversally symmetric foliations
zbMATH Open document ID: 0613.53013
author: Franz W. Kamber
author: Ernst A. Ruh
author: Philippe Tondeur
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: In this paper Riemannian foliations are compared with transversally homogeneous foliations, where the model transverse structure is of the type of an irreducible compact symmetric space G/K. The datum needed for this comparison is a connection in the normal bundle, having similar properties as the canonical connection in the case of a transversally symmetric foliation. This similarity is most conveniently formulated in terms of the corresponding Cartan connections. For the symmetric model case the curvature of the Cartan connection vanishes. An almost transversally symmetric foliation is one where this curvature is small in an appropriate Sobolev norm. In the spirit of Rauch's comparison theorem, and more specifically, the comparison theorem of Min-Oo and Ruh, one wishes to conclude that this assumption already implies the existence of a transversally symmetric structure of type G/K. The main result of this paper is that it is true for tense Riemannian foliations with small mean curvature. Here the tenseness means that the mean curvature form of the Riemannian foliation is a basic 1-form. A weaker form of this result was announced earlier, where the mean curvature form was assumed to vanish. The idea of the proof is to construct on the bundle P of normal frames of \({\mathcal F}^ a \)Cartan connection \({\bar \omega}\) with vanishing curvature. This yields a developing map \(\Phi\) : \~P\(\to G\) on the universal covering \~P, equivariant with respect to a homomorphism \(\pi _ 1(P)\to \Gamma \subset G\) (holonomy of \({\bar \omega}\)). It induces in turn a map \(\phi\) : \~M\(\to G/K\), possibly after an averaging process. This map defines the transversally symmetric structure of type G/K for the foliation \({\mathcal F}\) as desired, via its lift to the universal covering \~M. The construction of a Cartan connection with vanishing curvature makes use of the de Rham-Hodge theory for the basic cohomology recently developed [see \textit{F. Kamber} and \textit{Ph. Tond…
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3992349
zbMATH Keywords: Riemannian foliations
zbMATH Keywords: transversally homogeneous foliations
zbMATH Keywords: symmetric space
zbMATH Keywords: connection in the normal bundle
zbMATH Keywords: Cartan connections
zbMATH Keywords: small mean curvature
zbMATH Keywords: de Rham-Hodge theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442004
OpenAlex ID: W1527361024
Wikidata QID: Q115183820
DOI: 10.4310/JDG/1214442004
description: scientific article; zbMATH DE number 3992349