Applied geodesic foliations
type: Item
instance of: scholarly article
title: Applied geodesic foliations
zbMATH Open document ID: 0794.53020
DOI: 10.1007/BF01459759
published in: Mathematische Annalen
publication date: 1994-09-01T00:00:00Z
full work available at URL: https://eudml.org/doc/165194
review text: This paper contains applications to many problems of geometry, of results on (non existence of) geodesic foliations. Some results on geodesic foliations belong to (\textit{A. Zeghib}: ``Feuilletages géodésiques des variétés localement symétriques'', Lyon: ENS, Preprint 1992), but most of the applications can be derived from a result of this paper itself. That is a classification of geodesic ``partially hyperbolic'' foliations, with complete leaves, on finite volume (not necessarily complete) Riemannian manifolds. Here partially hyperbolic means that the curvature tensor applied to three vectors, for which one (at least) is tangent to the foliation, has the same form as in the hyperbolic space of curvature \(-1\). A typical application is the following. Let \(V^ n\) and \(W^{n+q}\) be two hyperbolic manifolds (i.e. of curvature \(-1\)), and \(f:V\to W\), a \(C^ 3\) isometric immersion. Suppose \(q<n\), and \(V\) of finite volume and complete; then \(f(V)\) is a geodesic submanifold of \(W\).
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 37C85
zbMATH DE Number: 562485
zbMATH Keywords: partially hyperbolic foliations
zbMATH Keywords: geodesic foliations
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: isometric immersion
zbMATH Keywords: geodesic submanifold
author: Abdelghani Zeghib
MaRDI profile type: Publication
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OpenAlex ID: W2913327558
description: scientific article; zbMATH DE number 562485