Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems.
It is shown that Nambu-Poisson and Nambu-Jacobi brackets can be defined inductively: a n-bracket, n>2, is Nambu-Poisson (resp. Nambu-Jacobi) if and only if fixing an argument we get a (n-1)-Nambu-Poisson (resp. Nambu-Jacobi) bracket. As a by-product we get relatively simple proofs of Darboux-type theorems for these structures.
We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.
We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.
In this paper, we present a relation between Jacobi-Reeb dynamics and the dynamics associated with a mechanical Hamiltonian system with respect to a linear Poisson structure on a vector bundle. For this purpose, we will use the so-called Jacobi bundle metrics induced by the mechanical Hamiltonian system. These constructions extend classical results on the relation between standard mechanical Hamiltonian systems on cotangent bundles and Reeb dynamics.
From 2-covariant tensor fields to Poisson and Jacobi structures
Instituto de Ciencias Matemáticas (ICMAT–CSIC)（数学科学研究所）；Real Academia de Ciencias Exactas, Físicas y Naturales de España（西班牙皇家精确、物理和自然科学院）；Great Bay University（大湾区大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Universitat Rovira i Virgili（罗维拉-伊维尔基里大学）
The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras are understood to be the algebraic counterpart of Poisson manifolds. We propose a formalism, based on the dimensioned algebra technology recently developed by the author, to capture the algebraic counterparts of Jacobi manifolds as dimensioned Poisson algebras. Particularly, we give a generalisation of the functor $\text{C}^\infty:\textsf{Smooth}\to \textsf…
In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.
We define gauge transformations of Jacobi structures on a manifold. This is related to gauge transformations of Poisson structures via the Poissonization. Finally, we discuss how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base.
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…
Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space $V$ a non-abelian cohomological type object ${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ is constructed: it classifies all Jacobi algebras containing $A$ as a subalgebra of codimension equal to ${\rm dim} (V)$. Representations of $A$ are used in order to give the decomposi…
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
We generalize Poisson-Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson-Nijenhuis structures and their relations with Jacobi structures.
In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton-Jacobi theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton-Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poiss…
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi-Nijenhuis structures.
The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $Δ$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $Δ$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization…
We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, 1/2-Kenmotsu structures and locally conformally Kahler structures.
We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\frac{1}{ 2}$-Kenmotsu structures and locally conformally Kähler structures.
In this note we show that given an exact QS-manifold (a natural generalisation of an exact Poisson manifold) one can associate a family of odd Jacobi structures on the same underlying supermanifold.
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under reciprocal transformations. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermani…
In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, (1/2)-Kenmotsu structures and locally conformally Kähler structures. In this paper we are generalizing this work to the framework of Lie algebroids.
We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.
We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissonization.
Jacobi groupoids are introduced as a generalization of Poisson and contact groupoids and it is proved that generalized Lie bialgebroids are the infinitesimal invariants of Jacobi groupoids. Several examples are discussed.
Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.
We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu \cite{prequan}. The methods used are those of Crainic-Fernandes on $A$-paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable case.
We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure.
The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alternative proof of the splitting theorem of homogeneous Poisson structures.
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential operators. We present problems which are natural. We give some new partial solutions for these natural problems.
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.
We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also discuss a new example of a Lie bialgebroid on Poisson manifolds.
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.
Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main too…
We define certain extensions of Jacobi groups of $A_n$, prove an analogue of Chevalley Theorem for their invariants, and construct a Dubrovin Frobenius structure on it orbit space.
We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which case our method sheds light on the role of the Poisson condition as an obstruction to realization; and the Maurer-Cartan structure equation associated with a Lie algebroid, in which case we obtain an explicit formula for a solution to the equation which generalizes the well known…
As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.
The notion of a Dirac submanifold of a Poisson manifold was studied by Xu (arXiv:math.SG/0110326). We give an interpretation of Xu's definition in terms of a general notion of tensor fields soldered to a normalized submanifold. Then, this interpretation is used to define Dirac submanifolds of a Jacobi manifold. Several properties and examples are discussed.
Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in ${\mathbb R}^3$ is proposed. Compatible Poisson structures and the corresponding bi-Hamiltonian N=3 systems are also discussed.
We use the supergeometric formalism, more precisely, the so-called "big bracket" (for which brackets and anchors are encoded by functions on some graded symplectic manifold) to address the theory of Jacobi algebroids and bialgebroids (following mainly Iglesias-Marrero and Grabowski-Marmo as a guideline). This formalism is in particular efficient to define the Jacobi-Gerstenhaber algebra structure associated to a Jacobi algebroid, to define its Poissonization, and to express the compatibility condition defining Jacobi bialgebroids. Also, we claim that this supergeometric language gives a simple…
It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible. In particular result can be applied directly to known classes of almost (para-) contact metric manifolds when considered Jacobi operator of characteristic vector field has maximal rank. There is effective algorithmic procedure which resolves the problem of existence of such vect…
We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite dimensional representations of reductive Lie groups. Moreover, we will explicitly generate a family of degree-preserving Poisson transforms whose restriction to real valued differential forms has coclosed images. In addition, as a transform on sections of density bundles it can be related to the classical Poisson transform, proving that we produced a natura…
We show that one can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, a n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a n-ary version of the Jacobi identity must be skew-symmetric. A similar result holds for a non-antisymmetric version of Lie algebroids.
New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.
