Jacobi Structures Research Articles

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

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 tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.

Classical r-matrices of real low-dimensional Jacobi–Lie bialgebras and their Jacobi–Lie groups

In this paper, we obtain the classical [Formula: see text]-matrices of real two- and three-dimensional Jacobi–Lie bialgebras. In this way, we classify all non-isomorphic real two- and three-dimensional coboundary Jacobi–Lie bialgebras and their types (triangular and quasi-triangular). Also, we obtain the generalized Sklyanin bracket formula by use of which, we calculate the Jacobi structures on the related Jacobi–Lie groups. Finally, we present a new method for constructing classical integrable systems using coboundary Jacobi–Lie bialgebras.

Multivariate topology simplification

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on “lip”-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph.

REAL HYPERSURFACES OF NON-FLAT COMPLEX SPACE FORMS WITH GENERALIZED ξ-PARALLEL JACOBI STRUCTURE OPERATOR

AbstractThe aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space TpM of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξl)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξl = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.

Real hypersurfaces of non--flat complex space forms in terms of the Jacobi structure operator

The aim of the present paper is the study of some classes of real hypersurfaces equipped with the condition \phi l = l \phi, (l = R(., \xi, \xi))

Jacobi structures and Spencer operators

This paper reveals the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs [20,13]; in particular, we provide a direct and much simpler/geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multiplicative forms on Lie groupoids with non-trivial coefficients [5,19].

Odd Jacobi Manifolds and Loday-Poisson Brackets

We construct a nonskew symmetric 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.

On existence of regular jacobi structures

We prove \(h\)-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.

Graded contact manifolds and contact Courant algebroids

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal R×-bundles. Gradings compatible with the R×-action lead to the concept of a graded contact manifold, in particular a linear (more generally, n-linear) contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid, for which the corresponding cohomology operator is represented not by a vector field (de Rham derivative) but by a first-order differential operator. It is shown that one can view Kirillov or Jacobi brackets as homological Hamiltonians on linear contact manifolds. Contact manifolds of degree 2, as well as contact analogs of Courant algebroids are studied. We define lifting procedures that provide us with constructions of canonical examples of the structures in question.

Differential Graded Contact Geometry and Jacobi Structures

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.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS SOME OF WHOSE JACOBI OPERATORS ARE ξ-INVARIANT

We classify Hopf real hypersurfaces in complex two-plane Grassmannians for which either the Jacobi structure operator or the Jacobi operators corresponding to the directions in the distribution [Formula: see text] are Lie ξ-parallel if they satisfy a further condition.

On the Geometry of Almost -Manifolds

An -structure on a manifold is an endomorphism field satisfying . We call an f-structure regular if the distribution is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost -structure, it determines a torus fibration of M over a symplectic manifold. When rank , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with -structure or -structure, we do not assume that the f-structure is normal. We also show that given an almost -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

Regularity of weakly harmonic maps from a Finsler surface into ann-sphere

We study harmonic maps on Finsler surfaces. Using Berwald frames on Finsler surfaces, we prove conformal invariance for the energy of Finsler harmonic maps. As an application, we show that weakly harmonic maps from a Finsler surface to a sphere S n are in fact smooth by establishing a new Jacobi structure, generalizing the regularity result previously known for the case of a harmonic map from a Riemannian surface.

Jacobi structures in supergeometric formalism

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 the previous works of Iglesias–Marrero [9] and Grabowski–Marmo [10]. This formalism is 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. It also yields a simple description of the Jacobi bialgebroid associated to a Jacobi structure, and conversely, of the Jacobi structure associated to a Jacobi bialgebroid.

Jacobi structures of evolutionary partial differential equations

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 the action of reciprocal transformations that only change the spatial variable. 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 supermanifolds. These constructions are used to compute the Lichnerowicz–Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.

Generalized geometrical structures of odd dimensional manifolds

We define an almost-cosymplectic-contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost-coPoisson–Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated algebras of functions. As examples of the above structures, we present geometrical dynamical structures of the phase space of a general relativistic particle, regarded as the 1st jet space of motions in a spacetime. We describe geometric conditions by which a metric and a connection of the phase space yield cosymplectic and dual coPoisson structures, in case of a spacetime with absolute time (a Galilei spacetime), or almost-cosymplectic-contact and dual almost-coPoisson–Jacobi structures, in case of a spacetime without absolute time (an Einstein spacetime).

GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

Dirac structures and generalized complex structures on

Abstract We consider Courant and Courant–Jacobi brackets on the stable tangent bundle TM × ℝ h of a differentiable manifold and corresponding Dirac, Dirac–Jacobi and generalized complex structures. We prove that Dirac and Dirac–Jacobi structures on TM × ℝ h can be prolonged to TM × ℝ k , k > h, by means of commuting infinitesimal automorphisms. Some of the stable generalized complex structures are a natural generalization of the normal almost contact structures; they are expressible by a system of tensors (P, θ, F, Za , ξ a ) (a = 1, …, h), where P is a Poisson bivector field, θ is a 2-form, F is a (1, 1)-tensor field, Za are vector fields and ξ a are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal almost contact structure (F, Z, ξ). We prove that such a generalized structure projects to a generalized complex structure of a space of leaves and we characterize the structure by means of the projected structure and of a normal bundle of the foliation. Like in the Boothby–Wang theorem about contact manifolds, principal torus bundles with a connection over a generalized complex manifold provide examples of this kind of generalized normal almost contact structures.

Integrability of Jacobi and Poisson structures

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. 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.

Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids

We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and we prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted Dirac-Jacobi structures are presented.