Godbillon-Vey Class Research Articles

Godbillon-Vey invariants of Non-Lorentzian spacetimes and Aristotelian hydrodynamics

We study the geometry of foliated non-Lorentzian spacetimes in terms of the Godbillon-Vey class of the foliation. We relate the intrinsic torsion of a foliated Aristotelian manifold to its Godbillon-Vey class, and interpret it as a measure of the local spin of the spatial leaves in the time direction. With this characterisation, the Godbillon-Vey class is an obstruction to integrability of the -structure defining the Aristotelian spacetime. We use these notions to formulate a new geometric approach to hydrodynamics of fluid flows by endowing them with Aristotelian structures. We establish conditions under which the Godbillon-Vey class represents an obstruction to steady flow of the fluid and prove new conservation laws.

Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class

The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions satisfying certain conditions. All these foliations are pairwise homeomorphic and have trivial Godbillon-Vey class. We show that the modified Godbillon-Vey is non-trivial for some Reeb foliations and it is trivial for some other Reeb foliations. In particular, the modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and it provides more information than the classical Godbillon-Vey class. We also show that this class is non-trivial for some foliations on the two-dimensional surfaces.

LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS

AbstractFollowing Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

We consider a 3-dimensional smooth manifold M equipped with an arbitrary, a priori non-integrable, distribution (plane field) D and a vector field T transverse to D. Using a 1-form ω such that D=ker⁡ω and ω(T)=1 we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on ω and T. For a compatible Riemannian metric on M, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of D. We deduce Euler-Lagrange equations of associated functionals: for variable (D,T) on M, and for variable Riemannian metric on (M,D). We show that for a geodesic field T (e.g., for a contact structure) such (D,T) is critical, characterize critical pairs when D is integrable (with examples among twisted products) and prove that these critical pairs are not extrema.

Variations of the Godbillon–Vey Invariant of Foliated 3-Manifolds

Let M be a smooth three-dimensional manifold equipped with a vector field T transverse to a plane field $${\mathcal {D}}$$ —the kernel of a one form $$\omega $$ such that $$\omega (T)=1$$ . Recently, in Rovenski and Walczak (A Godbillon–Vey type invariant for a 3-dimensional manifold with a plane field, 2017. arXiv:1707.04847 ), we constructed a three-form analogous to that defining the Godbillon–Vey class of a foliation, showed how does this form depend on $$\omega $$ and T, and deduced Euler–Lagrange equations of the associated functional. In this paper, we continue our study when distributions/foliations and forms are defined outside a “singularity set” (a finite union of pairwise disjoint closed submanifolds of codimension $$\ge 2$$ ) under additional assumption of convergence of certain integrals. We characterize critical pairs $$(\omega ,T)$$ and foliations for different types of variations, find sufficient conditions for critical pairs when variations are among foliations, consider applications to transversely holomorphic flows, calculate the index form and present examples of critical foliations among Reeb foliations and twisted products.

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Let $\mathcal{F}$ be a codimension-one, $C^2$-foliation on a manifold $M$ without boundary. In this work we show that if the Godbillon–Vey class $GV(\mathcal{F}) \in H^3(M)$ is non-zero, then $\mathcal{F}$ has a hyperbolic resilient leaf. Our approach is based on methods of $C^1$-dynamical systems, and does not use the classification theory of $C^2$-foliations. We first prove that for a codimension-one $C^1$-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points $E(\mathcal{F})$ has positive Lebesgue measure. We then prove that if $E(\mathcal{F})$ has positive measure for a $C^1$-foliation, then $\mathcal{F}$ must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The first statement then follows, as a $C^2$-foliation with non-zero Godbillon–Vey class has non-trivial Godbillon measure. These results apply for both the case when $M$ is compact, and when $M$ is an open manifold.

Notes on 'Infinitesimal derivative of the Bott class and the Schwarzian derivatives'

The derivatives of the Bott class and those of the Godbillon-Vey class with respect to infinitesimal deformations of foliations, called infinitesimal derivatives, are known to be represented by a formula in the projective Schwarzian derivatives of holonomies [3], [1]. It is recently shown that these infinitesimal derivatives are represented by means of coefficients of transverse Thomas-Whitehead projective connections [2]. We will show that the formula can be also deduced from the latter representation.

A heat flow approach to the Godbillon-Vey class

A heat flow approach to the Godbillon-Vey class

Characteristic classes of foliations via SAYD-twisted cocycles

We find the first non trivial “SAYD-twisted” cyclic cocycle over the groupoid action algebra under the symmetry of the affine linear transformations of the Euclidian space. We apply the cocycle to construct a characteristic map by which we transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. In codimension 1, our result matches with the (only explicit) computation done by Connes–Moscovici. We carry out the explicit computation in codimension 2 to present the transverse fundamental class, the Godbillon–Vey class, and the other four residual classes as cyclic cocycles on the groupoid action algebra.

Transverse projective structures of foliations and infinitesimal derivatives of the Godbillon–Vey class

We study transverse projective structures of foliations and construct an invariant, which is a homomorphism from a foliated cohomology to the ordinary one. It is shown that the infinitesimal derivatives of the Godbillon–Vey class and the Bott class are determined by the invariant. As a corollary, a rigidity theorem for the Godbillon–Vey class and the Bott class is shown.

On some Godbillon–Vey classes of a family of regular foliations

The aim of the paper is to construct some Godbillon–Vey classes of a family of regular foliations, defined in the paper. These classes are cohomology classes on the manifold or on suitable open subsets. Some examples are also considered.

Intrinsic characteristic classes of a local Lie group

For a local Lie group M we define cohomology classes [w_{2k+1}] \in H^{2k+1}_{dR} (M,ℝ) . We show that [w_1] is an obstruction to globalizability and give an example where [w_1] ≠ 0 . We also show that [w_3] coincides with Godbillon–Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes.

Infinitesimal derivative of the Bott class and the Schwarzian derivatives

An infinitesimal derivative of the Bott class is defined by generalizing Heitsch'es construction. We prove a formula relating the infinitesimal derivative to the Schwarzian derivatives, which gives a generalization of the Maszczyk formula for the Godbillon-Vey class of real codimension-one foliations. As an application, a residue of infinitesimal derivatives with respect to the Julia set in the sense of Ghys, Gomez-Mont and Saludes is introduced.

The Godbillon–Vey cyclic cocycle for PL-foliations

We define a cyclic cocycle which corresponds to the piecewise linear Godbillon–Vey class of Ghys and Sergiescu [E. Ghys, V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987) 185–239]. Using Connes's pairing [A. Connes, Non-commutative differential geometry. Part II: De Rham homology and noncommutative algebra, Publ. Math. Inst. Hautes Études Sci. 62 (1985) 257–360; A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in: H. Araki, G. Effros (Eds.), Geometric Methods in Operator Algebras, Pitman Res. Notes Math. Ser., vol. 123, Longman, Harlow, 1986, pp. 52–144] between cyclic cohomology and K-theory, we then evaluate this cocycle on a suitable K-theory class and obtain a nontrivial result, for foliations of the 3-torus by slope components.

Canonical Foliations of Certain Classes of Almost Contact Metric Structures

The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation ℱ defined by the contact distribution is Riemannian and tangentially almost Kahler of codimension 1 and that ℱ is tangentially Kahler if the manifold M is normal. Furthermore, we show that a semi–invariant submanifold N of such a manifold M admits a canonical foliation ℱN which is defined by the antiinvariant distribution and a canonical cohomology class c(N) generated by a transversal volume form for ℱN. In addition, we investigate the conditions when the even–dimensional cohomology classes of N are non–trivial. Finally, we compute the Godbillon–Vey class for ℱN.

Complexification of foliations and complex secondary classes

Some properties of complex secondary classes are discussed. It is shown that the Godbillon-Vey class and the Bott class are related via complexification.

Localization and residue of the Bott class

The Bott class of transversely holomorphic foliations is studied. We first introduce a formula which relates the Bott class and the Godbillon–Vey class. Then a ‘localizable part’ of the Bott class is defined. It is indeed localizable and written in terms of the Godbillon measure studied by Heitsch and Hurder. The above-mentioned formula is reviewed in terms of localizable parts. Finally, complex codimension-one foliations are considered. A version of residue is introduced and it is shown that the Bott class is ‘localized’ near the Julia set in the sense of Ghys–Gomez-Mont–Saludes. Some examples of calculation of the residue are presented.

Godbillon–Vey classes of symplectic foliations

Godbillon–Vey classes of symplectic foliations

Invariance of the Godbillon-Vey class by C1-diffeomorphisms for higher codimensional foliations

G. Raby proved in [4] that the Godbillon-Vey invariant for codimension-one foliations is invariant by C1-diffeomorphisms. In this paper we generalize the result for foliations of arbitrary codimension.

Foliations with rigid Godbillon-Vey class

The general formula for the variation of the Godbillon-Vey class is given in terms of the obstruction to the existence of a projective transversal structure (when a foliation arises by gluing of level sets of local functions with fractional linear transition maps). Using the above formula one obtains (under the technical condition of separability of some topological space of cohomology) that the Godbillon-Vey number of a foliation \({\cal F}\) of codimension one on a compact orientable 3-fold is topologically rigid (i.e. constant under infinitesimal singular deformations) iff \({\cal F}\) admits a projective transversal structure.