Camille Laurent-Gengoux
We construct the equivalent of the Godbillon-Vey class and its generalizations for regular foliations on super-manifolds. We interpret these classes as classes of foliated flat connections.
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Godbillon-Vey type functional for almost contact manifolds
本文将 Godbillon-Vey 泛函引入三维近切触流形，推导出保持 Reeb 向量场的 Euler-Lagrange 方程，并构造了具有双重扭曲积结构且属于 Chinea-Gonzalez 分类 $C_{5}\oplus C_{12}$ 类的临界解。
Let 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(F) \in H^3(M) is non-zero, then 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(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resil…
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 dist…
Godbillon-Vey classes of Lie subalgebroids
本文通过固定Lie代数胚并相对定义其Lie子代数胚的Godbillon-Vey类，研究其性质，推广了正则叶状结构的Godbillon-Vey类到奇异情形。
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.
We construct a groupoid equivariant Kasparov class for transversely oriented foliations in all codimensions. In codimension 1 we show that the Chern character of an associated semifinite spectral triple recovers the Connes-Moscovici cyclic cocycle for the Godbillon-Vey secondary characteristic class.
We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $ω$ such that ${\cal D} = \ker\,ω$ 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 ${\cal D}$. We deduce Euler-La…
Godbillon-Vey classes of regular Jacobi manifolds
针对具有正则叶状结构的 Jacobi 流形，本文研究其 Godbillon-Vey 示性类，并利用 Jacobi 结构给出该类的显式表达。
We consider a $(2q+1)$-dimensional smooth manifold $M$ equipped with a $(q+1)$-dimensional, a priori non-integrable, distribution ${\cal D}$ and a $q$-vector field ${\bf T}=T_1\wedge\ldots\wedge T_q$, where $\{T_i\}$ are linearly independent vector fields transverse to~${\cal D}$. Using a $q$-form $ω$ such that ${\cal D} = \ker\,ω$ and $ω({\bf T})=1$, we construct a $(2q+1)$-form analogous to that defining the Godbillon--Vey class of a $(q+1)$-dimensional foliation, and show how does this form depend on $ω$ and ${\bf T}$. For a compatible Riemannian metric $g$ on $M$, we express this $(2q+1)$-…
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the…
Let F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective outside a compact hypersurface, improving our previous work (see version 1). Such a vector field insures the existence of a global meromorphic Godbillon-Vey sequence for the foliation F. We derive sufficient conditions on this sequence insuring such alternative. For instance, if the…
Godbillon-Vey Invariants of Non-Lorentzian Spacetimes and Aristotelian Hydrodynamics
本文利用Godbillon-Vey类刻画非洛伦兹时空的几何，将其与亚里士多德流形的内蕴挠率关联，并应用于流体动力学，建立稳态流动的障碍条件及新守恒律。
This paper studies the interaction of $π_1(M)$ for a $C^\infty$ manifold $M$ with Bott's original obstruction to integrability, and with differential geometric invariants such as Godbillon-Vey and Cheeger-Simons invariants of a foliation. We prove that the ring of higher Pontrjagin and higher Chern classes of an integrable subbundle $E$ of the tangent bundle of a manifold vanishes above dimension $2k$ where $k=dim(TM/E)$, and where the higher Pontrjagin and Chern rings are rings generated by $i^*y \cup p_j(TM/E)$ and by $i^*y \cup c_j(TM/E)$ respectively, with $p_j$ the $j$-th Pontrjagin class…
We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the classic Atiyah-Patodi-Singer index formula for $(X,\F)$. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relativ…
For a local Lie group M we define odd order cohomology classes. The first class is an obstruction to globalizability of the local Lie group. The third class coincides with Godbillon-Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes.
Following 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 non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, 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. I…
On Losik classes of diffeomorphism pseudogroups
本文为微分同胚伪群定义了 Godbillon-Vey-Losik 类与第一 Chern-Losik 类，给出显式表达式，并构造相应丛以用体积形式和辛形式表示，附二维示例。
We prove a generalisation of Bott's vanishing theorem for the full transverse frame holonomy groupoid of any transversely orientable foliated manifold. As a consequence we obtain a characteristic map encoding both primary and secondary characteristic classes. Previous descriptions of this characteristic map are formulated for the Morita equivalent étale groupoid obtained via a choice of complete transversal. By working with the full holonomy groupoid we obtain novel geometric representatives of characteristic classes. In particular we give a geometric, non-étale analogue of the codimension 1 G…
We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in codimension 1 matches with the only existing explicit computation done by Connes-Moscovici. In codimension 2 case, we carry out a constructive and explicit computation, by which we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual cla…
Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse structures are modeled on the G-action on S, except the case where G=SO(n+1,1) for even n. For this exceptional case, we construct examples of foliations on a manifold which break the finiteness and show a weaker form of the finiteness. These are generalizations of a finiteness …
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