Classification des feuilletages totalement géodésiques de codimension un
type: Item
instance of: scholarly article
title: Classification des feuilletages totalement géodésiques de codimension un
zbMATH Open document ID: 0534.57015
DOI: 10.1007/BF02564652
published in: Commentarii Mathematici Helvetici
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139954
review text: The aim of this paper is to construct the totally geodesic differentiable foliations of codimension 1. The author constructs some model foliations \((M_ D,F_ D)\), which are torus bundles, and he proves that if m is a compact orientable manifold, and F is a \(C^{\infty}\) foliation of codimension 1 and transversally orientable on M, then M has a Riemannian metric such that the leaves of F are totally geodesic hypersurfaces, iff either F is transversal to a locally free action of \(S^ 1\) on M or M is diffeomorphic to \(M_ D\) and F is conjugate to the corresponding model \(F_ D\). Similarly, if M is noncompact but either F is \(C^{\infty}\) and M has a fundamental group of finite type or F is \(C^{\omega}\), one has a complete metric making F totally geodesic in the same cases as for compact M, and in the case where F is transversal to a trivial fibration of M with fibre \({\mathbb{R}}\), and whose restriction to every leaf is a covering map.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 3848243
zbMATH Keywords: circle
zbMATH Keywords: totally geodesic differentiable foliations of codimension 1
zbMATH Keywords: totally geodesic hypersurfaces
zbMATH Keywords: locally free action
author: Étienne Ghys
reviewed by: Izu Vaisman
MaRDI profile type: Publication
OpenAlex ID: W378113248
description: scientific article; zbMATH DE number 3848243