19 research outputs found
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
Effect of genotype, gelling agent, and auxin on the induction of somatic embryogenesis in sweet potato (<i>Ipomoea batatas</i> Lam.)
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 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 -Dirac structures
We prove some general results about the relation between the 1-cocycles of an
arbitrary Lie algebroid over and the leaves of the Lie algebroid
foliation on associated with . Using these results, we show that a
-Dirac structure induces on every leaf of its
characteristic foliation a -Dirac structure , which comes
from a precontact structure or from a locally conformal presymplectic structure
on . In addition, we prove that a Dirac structure on can be obtained from and we discuss the relation between the leaves of
the characteristic foliations of and .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