A Generalization of Poisson-Nijenhuis Structures

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.Comment: 21 pages, Late

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.Comment: 32 page

Jacobi structures revisited

Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.Comment: 20 page

Morse-Novikov cohomology of locally conformally K\"ahler manifolds

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse-Novikov class and the Bott-Chern class of a Vaisman manifold vanishes. Moreover, for any LCK-structure on a Vaisman manifold, we prove that its Morse-Novikov class vanishes. We show that a compact LCK-manifold $M$ with vanishing Bott-Chern class admits a holomorphic embedding to a Hopf manifold, if \dim_\C M \geq 3, a result which parallels the Kodaira embedding theorem.Comment: 22 pages. Version 4.0, minor corrections, clarifications and typos. To appear in Journal of Geometry and Physic

The graded Jacobi algebras and (co)homology

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.Comment: 20 pages, a few typos corrected; final version to be published in J. Phys. A: Math. Ge

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 GL(1,R)-bundles. Gradings compatible with the GL(1,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 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 are studied, as well as contact analogs of Courant algebroids. We define lifting procedures that provide us with constructions of canonical examples of the structures in question.Comment: 39 pages, some changes of the terminology, references extended and update

Lie algebroid foliations and E1(M){\cal E}^1(M)-Dirac structures

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid $A$ over $M$ and the leaves of the Lie algebroid foliation on $M$ associated with $A$. Using these results, we show that a ${\cal E}^1(M)$-Dirac structure $L$ induces on every leaf $F$ of its characteristic foliation a ${\cal E}^1(F)$-Dirac structure $L_F$, which comes from a precontact structure or from a locally conformal presymplectic structure on $F$. In addition, we prove that a Dirac structure $\tilde{L}$ on $M\times \R$ can be obtained from $L$ and we discuss the relation between the leaves of the characteristic foliations of $L$ and $\tilde{L}$.Comment: 25 page

Étude vibrationnelle du 3,4'-bitriazole et de quelques-uns de ses dérivés C-monosubstitués

L'étude vibrationnelle des 3,4'-bitriazole, 5-méthyl-3,4'-bitriazole et 5-bromo-3,4'- bitriazole a été effectuée et une attribution de leurs vibrations fondamentales a été proposée sur la base de l'existence d'une seule forme à l'état solide. La substitution de l'hydrogène en position 3 du cycle triazolique {1H} par un groupement 4-triazolyle entraîne une diminution de la force de la liaison hydrogène NH...N comparativement au 1,2,4-triazole et à ses dérivés C-monosubstitués. L'introduction d'un substituant en position 5 du bihétérocycle augmente la force de l'autoassociation surtout dans le cas où le substituant est un groupement attracteur d'électrons. A partir des fréquences νNH, l'estimation des distances N...N dans ces dérivés a été effectuée. La distinction entre les vibrations du cycle triazolique et celles du groupement 4-triazolyle semble impossible, probablement à cause de la conjugaison des deux cycles. La fréquence plus élevée obtenue pour la vibration de valence de la liaison intercyclique dans le cas du BrbTA est explicable par une conjugaison plus forte et une liaison C-N plus courte dans ce dernier