Variations of the Godbillon-Vey invariant of foliated 3-manifolds
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 integrable distribution, i.e., there exists a transversely oriented codimension-one foliation \(\mathcal{F}\) such that \(\mathcal{D} = T\mathcal{F}\), then \(gv (\omega, T)\) is exactly the Godbillon-Vey invariant \(gv (\mathcal{F})\) of \(\mathcal{F}\). Following the ideas of [\textit{F. G. B. Brito} and the second author, Ann. Pol. Math. 73, No. 3, 269--274 (2000; Zbl 0997.53025); \textit{M. Lużyńczyk} and the second author, Ann. Global Anal. Geom. 48, No. 2, 195--209 (2015; Zbl 1323.53027)], the authors continue to study in this paper the variations of \((\omega, T)\) when the 3-form \(\omega\) and, therefore, the distribution \(\mathcal{D} = \ker \omega\) are defined outside a ``singularity set'', i.e., a finite union \(\sum\) of pairwise disjoint closed submanifolds of codimension greater than 2 under an additional assumption of convergence of certain improper integrals. The main results of the paper are the following ones: -- Find derivatives of the functional \(gv\) and characterize critical foliations for different types of variations \(({\omega}_t, T_t)\) when \(\mathcal{D}\) is integrable on \(M \setminus \sum\) (see Theorems 2.3, 2.6) and when \(\mathcal{D}\) is non-integrable on \(M \setminus \sum\) (see Theorem 4.2) -- Introduce sufficient conditions for the criticality of pairs \((\omega, T)\) when \(\mathcal{D}\) is integrable distributions on \(M\) (see Corollaries 2.7 and 2.10). -- Find the index form of the variation problem, and introduce some examples with Reeb foliations and twisted products (see Propositions 3.2 and 3.3).
scientific article; zbMATH DE number 7135380
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