Jacobi Manifolds Research Articles

On the Jacobi Equation and Manifolds with Multiple Conjugate Points

We investigate the phenomenon of multiple conjugate points along a geodesic. In the first instance, we investigate conjugate points in the context of the Jacobi equation, a second order ordinary differential equation, which captures precisely the geometry of conjugate points on surfaces. We then construct geometric examples which exhibit similar properties in higher dimensions.

De Rham cohomology and homotopy Frobenius manifolds

We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.

Symplectic Lie–Rinehart–Jacobi Algebras and Contact Manifolds

AbstractWe give a characterization of contact manifolds in terms of symplectic Lie–Rinehart–Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

Jacobi–Jacobi Commuting Models and Manifolds

Let \({\mathcal{J}}(\pi)\) be the higher order Jacobi operator. We study algebraic curvature tensors where \({\mathcal{J}} (\pi){\mathcal{J}} (\pi^{\bot}) = {\mathcal{J}} (\pi^{\bot}){\mathcal{J}} (\pi)\). In the Riemannian setting, we give a complete characterization of such tensors; in the pseudo-Riemannian setting, partial results are available. We present non-trivial geometric examples of Riemannian manifolds with this property.

On Quasi-Jacobi and Jacobi-Quasi Bialgebroids

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.

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.

Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one–one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold ( M , Λ , E ) for which 1 is an admissible function and Jacobi quotient manifolds of M. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.

Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map J : M → Γ 0 J:M \rightarrow \Gamma _0 from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of Γ 0 \Gamma _0 on M M , and we show that the quotients of fibers J − 1 ( x ) J^{-1}(x) by suitable Lie subgroups Γ x \Gamma _x are either contact or locally conformal symplectic manifolds with structures induced by the one on M M . We show that Willett’s reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett’s reduction can be performed only at distinguished points. As an application we obtain Kostant’s prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

Poisson–Jacobi reduction of homogeneous tensors

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 $\Delta$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\Delta$ 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. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case -- a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\Delta$-homogeneous symplectic structures on $M$ and contact structures on $N$.

Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with irreducible quasi-permutation monodromy representation outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of P1 . The solution is given in terms of a generalization of Szego kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices. Results of this work generalize the results of papers [13] and [14] where the 2× 2 case was solved.

On the computation of the Lichnerowicz–Jacobi cohomology

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples.

On a new relation between Jacobi and homogeneous Poisson manifolds

We establish a new, very close, relationship which links Jacobi structures and homogeneous Poisson structures defined on the same manifold and study the characteristic foliations of the related structures. Several examples of this construction are also given.

Star calculus on Jacobi manifolds

We study the Gerstenhaber bracket on differential forms induced by the two main examples of Jacobi manifolds: contact manifolds and l.c.s. manifolds. Moreover, we obtain explicit expressions of the generating operators and the derivations on the algebra of multivector fields. We define star operators for contact manifolds and l.c.s. manifolds and we study some of its properties.

Local structure of Jacobi–Nijenhuis manifolds

After a brief review on the basic notions and the principal results concerning the Jacobi manifolds, the relationship between homogeneous Poisson manifolds and conformal Jacobi manifolds, and also the compatible Jacobi manifolds, we give a generalization of some of these results needed for the contents of this paper. We introduce the notion of Jacobi–Nijenhuis structure and we study the relation between Jacobi–Nijenhuis manifolds and homogeneous Poisson–Nijenhuis manifolds. We present a local classification of homogeneous Poisson–Nijenhuis manifolds and we establish some local models of Jacobi–Nijenhuis manifolds.

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

The BV-algebra of a Jacobi manifold

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 give new examples of Lie bialgebroids over Poisson ma

Nambu-Jacobi and generalized Jacobi manifolds

The geometry of Nambu-Jacobi and generalized Jacobi manifolds is studied. A large collection of examples is given. The characteristic distribution generated by the Hamiltonian vector fields on a Nambu-Jacobi manifold is proved to be completely integrable, and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated.