Feuilletages de Killing. (Killing foliations)
type: Item
instance of: scholarly article
title: Feuilletages de Killing. (Killing foliations)
zbMATH Open document ID: 0614.53030
published in: Collectanea Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/38462
review text: The main result of the paper is the following: Let \((M,F,g_ T)\) be a Killing foliation of codimension q in a closed connected manifold of dimension \(n=p+q\). If the minimal dimension of the closure of the leaves is \(p+r\), then: (i) There are r global transverse fields \(X_ 1,X_ 2,...,X_ r\) which commute and are linearly independent at each point. (ii) The orbits of the Lie algebra of the transverse fields spanned by \(X_ 1,...,X_ r\) give a new Riemannian foliation F' of codimension q-r where the closure of the leaves is the same as in F. Moreover, F' has at least one compact leaf. Here \(g_ T\) is a transverse metric defining a Riemannian foliation F. A Killing foliation is defined to be a Riemannian foliation with a trivial central transversal sheaf. An interesting consequence of this theorem is: If the span of M is k (maximal number of linearly independent vector fields) then every Killing foliation of codimension q in M admits closures of the leaves of codimension at least q-r. If \(k=0\) (or \(\chi\) (M)\(\neq 0)\) the foliation has a compact leaf, and this is a result proved by \textit{E. Ghys} [Ann. Inst. Fourier 34, No.4, 203-223 (1984; Zbl 0525.57024)].
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3994458
zbMATH Keywords: Killing foliation
zbMATH Keywords: Riemannian foliation
author: Witold Mozgawa
reviewed by: Fabiano G. B. Brito
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3994458