The geometry of totally geodesic foliations admitting Killing field
type: Item
instance of: scholarly article
title: The geometry of totally geodesic foliations admitting Killing field
zbMATH Open document ID: 0657.53019
author: Carlos Currás-Bosch
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: This paper is concerned with results of preservation of codimension-one totally geodesic foliations admitting at least one compact leaf. By using De Rham's Decomposition Theorem for foliate manifolds and studying the expression of the Killing vector fields on the universal covering of the manifold, we give, first of all, a result of preservation for geodesic flows on surfaces. With similar techniques another proof of a recent preservation result due to \textit{G.-I. Oshikiri} [ibid. 38, 351-356 (1986; Zbl 0585.53025)] for closed manifolds is given, and finally we prove that on a 3-dimensional complete Riemannian manifold, with a codimension-one totally geodesic foliation, admitting at least one compact leaf, any Killing vector field preserves this foliation.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 4073879
zbMATH Keywords: totally geodesic foliations
zbMATH Keywords: De Rham's Decomposition Theorem
zbMATH Keywords: Killing vector fields
zbMATH Keywords: compact leaf
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227921
OpenAlex ID: W1982463963
cites work: De Rham decomposition theorems for foliated manifolds
cites work: Killing Vector Fields and Holonomy Algebras
cites work: Introduction to the Geometry of Foliations, Part A
cites work: Q3999045
cites work: Totally geodesic foliations
cites work: Totally geodesic foliations and Killing fields
cites work: Jacobi fields and the stability of leaves of codimension-one minimal foliations
cites work: Totally geodesic foliations and Killing fields. II
DOI: 10.2748/TMJ/1178227921
description: scientific article; zbMATH DE number 4073879