Transversely homogeneous foliations

Blumenthal, Robert A.

doi:10.5802/aif.771

Cited by 46 publications

(46 citation statements)

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“…Le probleme est donc de demontrer que la condition (1) implique la condition (2). Nous allons demontrer le theoreme 1 dans chaque cas du theoreme de P. Molino ([14]) qui classifie les feuilletages riemanniens de codimension 2 sur une variete fermee.…”

Section: Introductionunclassified

Sur la classe caractéristique exotique de Lazarov-Pasternack en codimension 2, II

Yamato¹

1981

Jpn. j. math

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Section: Introductionunclassified

Sur la classe caractéristique exotique de Lazarov-Pasternack en codimension 2, II

Yamato¹

1981

Jpn. j. math

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“…Then H = E / ® Q / and so T(F(Q)) = E' ®Q' ®V where V is the bundle of vertical vectors. Hence we may regard Q' ® V as the normal bundle of 3F 1 . Let El be the q X q matrix with a 1 in the h th column and £ th row and 0 elsewhere and let σ(E%) be the corresponding fundamental vector field on…”

Section: = T(m)/e Of ^ As a Subbundle Of T(m) Satisfying T(m) = E® Qmentioning

confidence: 99%

“…It is easy to see that such a codimension one foliation admits a flat Riemannian structure and so we obtain Reeb , q where {e l9 , βj is the standard basis of i? 9 and let Q'dT(F(Q)) be the g-plane bundle spanned by JS 1 , , E q . Then H = E / ® Q / and so T(F(Q)) = E' ®Q' ®V where V is the bundle of vertical vectors.…”

Section: = T(m)/e Of ^ As a Subbundle Of T(m) Satisfying T(m) = E® Qmentioning

confidence: 99%

Riemannian foliations with parallel curvature

Blumenthal

1983

Nagoya Mathematical Journal

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There is a unique Riemannian metric g on V such that f*(g) = g\ U. Let F be the Riemannian connection on V. Then P\U = f~\F). It is natural to study the relationship between the curvature of V and the structure of the foliated manifold (M, &).In the present work we study the case of parallel curvature, that is FR = 0 where R(X, Y)Z denotes the curvature tensor of F.Let 2F be a Riemannian foliation with parallel curvature of a compact manifold M. THEOREM 1. Let M be the universal cover of M and let # be the lift

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“…The case (c) will be the primary focus of this paper. In [1], Blumenthal has given a differential form characterization of transversely homogeneous structures. We give it first for cases (a)-(c) above, and then refer to [1] for a general formulation.…”

Section: The Godbillon-vey Invariant Of a Transversely Homogeneous Fomentioning

confidence: 99%

“…A general characterization of transversely homogeneous foliations in terms of differential forms is carried out in [1], to which the reader is referred for details.…”

Section: The Godbillon-vey Invariant Of a Transversely Homogeneous Fomentioning

confidence: 99%

The Godbillon-Vey Invariant of a Transversely Homogeneous Foliation

Brooks¹,

Goldman²

1984

Transactions of the American Mathematical Society

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ABSTRACT. A real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on RP1 (so that the coordinate changes are real linear fractional transformations).Given a closed manifold M, there is but a finite set of values in f/3(M;R) which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For M an oriented circle bundle over a surface, this finite set is explicitly computed.In [5], C. Godbillon and J. Vey introduced a characteristic class associated to a codimension-one foliation which lies in the third real cohomology group of the ambient manifold. The question was then raised whether one could assign some geometric significance to this invariant. The problem of understanding the GodbillonVey invariant in such a manner was underscored by a result of Thurston [12], who showed that for any closed 3-manifold there are families of codimension-one foliations whose Godbillon-Vey invariant varies continuously in a parameter, and which take on arbitrarily large values.It is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous (see §1). To explain the geometric picture which emerges, it is worthwhile reviewing briefly Gromov's notion of simplicial volume [7].Given a compact manifold M, Gromov assigns to M a volume ||A7||, defined topologically, which measures the complexity of representing the fundamental class of M by singular simplices. For many manifolds, for instance spheres, it is easily seen that ||M|| = 0. However, inspired by the theorem of Milnor [9] and Wood [15], Gromov and Thurston (see [7]) were able to show that when M is a hyperbolic manifold, ||M|| is equal to the volume of M in its hyperbolic metric, up to a constant depending only on dim(M). This is done in part by comparing ||M|| to a certain volume invariant associated to representations from 7Ti(M) to the isometry group of hyperbolic space. At the heart of the argument is the elementary fact that a geodesic simplex in hyperbolic space has bounded volume.In the present case, we consider foliations of a manifold M which are transversely projective, that is, which are locally modelled on the action of PSL(2, R) on the circle 5 '. The holonomy of such a foliation is described by a homomorphism 7Ti(M) -» (SL(2,R))~, the universal covering of PSL(2,R), as is discussed in § §1and 2.

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Transversely homogeneous foliations

Cited by 46 publications

References 9 publications

Sur la classe caractéristique exotique de Lazarov-Pasternack en codimension 2, II

Sur la classe caractéristique exotique de Lazarov-Pasternack en codimension 2, II

Riemannian foliations with parallel curvature

The Godbillon-Vey Invariant of a Transversely Homogeneous Foliation

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