Melanie Bertelson
A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.
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We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unim…
Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold $M$ is equivalent to to give a Kähler-Riemann foliation on $M$ such that the leafwise symplectic form is invariant with respect to all local foliate perpendicular vector fields. We show also that the sum of the vector space of leafwise cohomology and the vector space of basic forms is a sub…
We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.
On an orientable manifold M, we consider a regular even dimensional foliation F which is globally defined by a set of k-independent 1-forms. We give necessary and sufficient conditions for the existence of a regular Poisson structure on M whose Characteristic foliation is precisely F. Moreover, introducing a special class of the foliated 1-cohomology we describe obstructions for the existence of unimodular Poisson structures with a given characteristic foliation. In the same lines, we also give conditions for the existence of transversally constant Poisson structures.
We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spirit of Haefliger's criterion for homotoping a distribution to a foliation. We give an example to show that this criterion is not too unsharp.
We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its deformation problem. Indeed, viewing symplectic foliations as regular Poisson structures, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated $L_\infty$-algebra. Using this, we show that infinitesimal deformations of symplectic foliations can be obstructed. …
Numerically flat foliations and holomorphic Poisson geometry
本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。
$C^0$-Poisson geometry, coisotropic submanifolds, and clean intersection points
本文研究了Poisson同胚的刚性与非刚性，证明了共切子流形的$C^0$刚性，并引入了$C^0$-特征划分作为拓扑不变量。
In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we consider in a later paper.
Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras
本文研究了对称泊松结构与雅可比-乔丹代数的关系，揭示了其在几何和动力学中的应用。
The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be p-related (p is the projection of TM onto M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fibe…
Invariant Poisson Structures on Homogeneous Manifolds: Algebraic Characterization, Symplectic Foliation, and Contravariant Connections
本文通过经典Yang-Baxter方程的解给出齐性流形上不变泊松结构的代数刻画，建立其与特定李子代数类的双射对应，并研究了相关辛叶状结构与不变反变联络。
Geometric quantization of a Poisson manifold need not imply quantization of its symplectic leaves. We provide the leafwise geometric quantization of a Poisson manifold, seen as a foliated one, whose quantum algebra restricted to each leaf is quantized.
We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provided.
We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singular…
We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian fu…
In this paper we prove h-principal for regular Symplectic Foliations on Closed manifolds.
We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space.
Singular Riemannian foliations and $\mathcal{I}$-Poisson manifolds
本文在黎曼度量框架下建立奇异黎曼叶状结构的模版本及其Morita等价理论，并引入$\mathcal{I}$-泊松流形范畴，证明Morita等价的奇异叶状结构具有同构的约化泊松代数。
We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras…
A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling Poisson structures which describe the interaction between the tangential and transversal data of the characteristic distributions. As a consequence, we derive some criteria of vanishing of the first Poisson cohomology groups and apply the general splitting formulas to some particular classes of Poisson structures associated with singular symplectic foliations.
Collapsing regular Riemannian foliations with flat leaves
本文研究了如何保持截面曲率有界的情况下，对具有平坦叶的正则黎曼流形进行坍缩，并证明在特定条件下流形由环面作用给出。
We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.
We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foli…
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.
Basic Albanese maps of regular Riemannian foliations
本文提出正则黎曼流的基本阿贝尔映射概念，通过基本1形式定义，并将其与环境流形的经典阿贝尔映射联系起来，同时展示了多个重要例子。
We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions.
It is known that there exist singular Poisson structures in $4$-manifolds, whose symplectic foliation is given by singular fibrations over surfaces. In this work, we describe the effect of the monodromy of the fibration on the Poisson cohomology groups.
In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed.
The modular class of a singular foliation
本文利用李代数胚与正则叶状结构的关系，通过泛李∞-代数胚处理奇点，将模类概念推广到奇异叶状结构，阐明了该方法的合理性，为定义更多奇异叶状结构的示性类奠定基础。
Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pair groupoids.
Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of the s.r.f are compact, then, for each small epsilon, the…
We prove $h$-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on $h$ principle of contact foliations in terms of the regular Jacobi structures.
We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the \emph{additivity property}, analogous to the additivity of Clausius--Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the char…
Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a flow ([9]), of a regular foliation ([11]), of a regular distribution ([5]) and of a geometrical structure ([22]). The essential results of this paper is the nullity of the Finsler entropy for a controllable distribution and for a singular Riemannian foliation.
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。
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. A…
A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures. The thesis consists of two parts. In the first part we classify foliations on open manifolds whose leaves are either locally conformal symplectic or contact manifolds. These foliations can be described by some higher geometric structures - namely the Poisson and the Jacobi structures. In the second part of the thesis, we consider foliations on open contact manifolds whose lea…
We prove an analogue of Thurston's h-principle for $2$-dimensional foliations on manifolds of dimension bigger or equal to $4$, in the presence of a fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of rank $2$ regular Poisson structures on open manifolds with dimension bigger or equal to $4$ and it also helps us understand the flexibility of Poisson structures (not regular) on closed $4$-manifolds.
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.
If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's super-vector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid $A$ as a bracket-close…
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ …
We extend the problem of finding Hamiltonian-invariant volume forms on a Poisson manifold to the problem of construction of Hamiltonian-invariant generalized functions. For this we introduce the notion of generalized center of a Poisson algebra, which is the space of generalized Casimir functions. We study as the case when the set of test-objects for generalized functions is the space of compactly supported smooth functions, so the case when the test-objects are n-forms, where n is the dimension of the Poisson manifold. We describe the relations of this problem with the homological properties …
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on $S^4$ associated with a holomorphic Poisson structure on $\mathbb{CP}^3$. The space of the Poisson structures on $S^4$ is a real algebraic variety in the space of holomorphic Poisson structures on $\mathbb{CP}^3$. We generalize it to $\mathbb{HP}^n$ by using the twistor method. Furthermore, we provide examples of Poisson structures on $S^4$ associated with codimension one holomorphic foliations of degree 2 on $\mathbb{CP}^3$.
Given a regular curve in Minkowski spacetime, we describe necessary and sufficient conditions for this curve to admit a family of pairwise-disjoint crooked planes. Using this criterion, we describe crooked foliations along orbit curves of one-parameter groups of Lorentzian isometries.
A singular riemannian foliation on a complete riemannian manifold is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such singular foliation is the partition by orbits of a polar action,e.g. the orbits of the adjoint action of a compact Lie group on itself. We pro…
A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previou…
First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.
A singular foliation is a partition of a manifold into leaves of perhaps varying dimension. Stefan and Sussmann carried out fundamental work on singular foliations in the 1970s. We survey their contributions, show how diffeological objects and ideas arise naturally in this setting, and highlight some consequences within diffeology. We then introduce a definition of transverse equivalence of singular foliations, following Molino's definition for regular foliations. We show that, whereas transverse equivalent singular foliations always have diffeologically diffeomorphic leaf spaces, the converse…
In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidian space. This result gives us local information about the singular foliation and in particular about the singular stratification of the foliation. It also allows us to describe the plaques of the foliation as level sets of a t…
We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The linear holonomy is closely related to the modular class studied by A. Weinstein. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector fi…
Lagrangian Foliations on compact Kähler manifolds
该研究证明带有正则黎曼叶状结构且叶为拉格朗日极小叶的紧Kähler流形是平坦的，为此建立了Ros积分公式的叶状版本，丰富了Kähler流形的相关理论。
The derivation $d_T$ on the exterior algebra of forms on a manifold $M$ with values in the exterior algebra of forms on the tangent bundle $TM$ is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.
We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4- and 5-manifolds with positive or nonnegative sectional curvature.
Transversely projectively flat foliations
研究紧致凯勒流形上的横截射影平坦正则叶状结构，探讨其几何性质与分类。
We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity in relation with the perimeter and the area.
We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a vector bundle, and give the full description of the family of Poisson structures for this algebra.
In this paper we investigate the spectral sequence associated to a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples.
An analogue of the Hofer metric $\varrho_H$ on the Hamiltonian group $Ham(M,Λ)$ of a Poisson manifold $(M,Λ)$ can be defined but there is the problem of its non-degeneracy. First we observe that $\varrho_H$ is a genuine metric on $Ham(M,Λ)$ when the union of all closed leaves (as subsets of $M$) of the corresponding symplectic foliation is dense. Next we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable if it can be realized as the space of units of a symplectic groupoid. Our main result states that $\varrho_H$ is a Hofer type me…
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