Variations of the Godbillon-Vey invariant of foliated 3-manifolds
type: Item
instance of: scholarly article
title: Variations of the Godbillon-Vey invariant of foliated 3-manifolds
zbMATH Open document ID: 1435.53025
author: Vladimir Yu. Rovenskij
author: Paweł Walczak
published in: Complex Analysis and Operator Theory
publication date: 2019-11-22T00:00:00Z
review text: This paper is a continuation of [Differ. Geom. Appl. 66, 212--230 (2019; Zbl 1447.53026)] of the same authors which studies the so-called invariant of Godbillon-Vey type for a 3-dimensional manifold with a given plane field. The Godbillon-Vey cohomology class \(gv (\mathcal{F}\)) of a transversely oriented codimension-one foliation \(\mathcal{F}\) of a compact manifold \(M\) has been defined by \textit{C. Godbillon} and \textit{J. Vey} [C. R. Acad. Sci., Paris, Sér. A 273, 92--95 (1971; Zbl 0215.24604)] as the de Rham cohomology class of a certain 3-form on the foliated manifold \(M\). In the paper, the authors consider a 3-dimensional smooth manifold \(M\) which is equipped with a 1-form \(\omega\) and a vector field \(T\) such that \(\omega (T) = 1\). Denote the plane field \(\ker \omega\) on \(M\) by \(\mathcal{D}\). Of course, \(T\) is transverse to \(\mathcal{D}\) but the distribution \(\mathcal{D}\) is, in general, non-integrable. A Riemannian metric \(g\) on \(M\) is called compatible with a pair \((\mathcal{D}, T)\) if and only if \(T\) is the unit normal to \(\mathcal{D}\). In [the authors, loc. cit], the authors constructed a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and studied its dependence on \(\mathcal{D}, T\) and the compatible Riemannian metric \(g\). Namely, applying the inner product \(\iota\) and the Lie derivative \(\mathcal{L}\), the authors defined a 1-form as follows: \[ \eta = {\iota}_{T} d \omega \equiv \eta = {\mathcal{L}}_{T}\omega.\] The 1-form \(\eta\) does not depend on any compatible Riemannian metric \(g\) on the considered manifold \(M\) but strongly depends on the pair \((\mathcal{D}, T)\). Moreover, the 3-form \(\eta \wedge d \eta\) defines the de Rham cohomology class and represents the Godbillon-Vey type invariant of a pair \((\mathcal{D}, T)\) by the following formula \[(\omega, T) \mapsto gv (\omega, T) : = \int_{M} \eta \wedge d \eta . \] Of course, when \(\mathcal{D} = \ker \omega\) is an inte…
reviewed by: Le Anh Vu
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 7135380
zbMATH Keywords: foliation
zbMATH Keywords: plane field
zbMATH Keywords: Godbillon-Vey invariant
zbMATH Keywords: singularity
zbMATH Keywords: variation
zbMATH Keywords: Reeb foliation
zbMATH Keywords: twisted product
zbMATH Keywords: transversely holomorphic flow
MaRDI profile type: Publication
full work available at URL: s11785-018-0871-9
OpenAlex ID: W2903542781
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DOI: 10.1007/S11785-018-0871-9
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description: scientific article; zbMATH DE number 7135380