Raúl Ibáñez

Leibniz algebroid associated with a Nambu-Poisson structure

The notion of a Leibniz algebroid is introduced, and it is shown that each Nambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact permits one to define the modular class of a Nambu-Poisson manifold as an appropriate cohomology class, extending the well known modular class of Poisson manifolds.

Dynamics of generalized Poisson and Nambu–Poisson brackets

A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.

On certain geometric and homotopy properties of closed symplectic manifolds

Abstract It is well known that closed Kahler manifolds have certain homotopy properties which do not hold for symplectic manifolds. Here we survey interconnections between those properties.

Duality and modular class of a Nambu-Poisson structure

In this paper we introduce cohomology and homology theories for Nambu-Poisson manifolds. Also we study the relation between the existence of a duality for these theories and the vanishing of a particular Nambu-Poisson cohomology class, the modular class. The case of a regular Nambu-Poisson structure and some singular examples are discussed.

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.

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.

The canonical spectral sequences for poisson manifolds

For a compact symplectic manifoldM of dimension 2n, Brylinski proved that the canonical homology groupHkcan(M) is isomorphic to the de Rham cohomology groupH2n-k(M), and the first spectral sequence {Er(M)} degenerates atE1(M). In this paper, we show that these isomorphisms do not exist for an arbitrary Poisson manifold. More precisely, we exhibit an example of a five-dimensional compact Poisson manifoldM5 for whichH1can(M5) is not isomorphic toH4(M5), andE1(M5) is not isomorphic toE2(M5).

Reduction of generalized poisson and Nambu-Poisson manifolds

Abstract The Marsden-Ratiu geometrical reduction for Poisson manifolds is extended to Nambu-Poisson and generalized Poisson manifolds. If in addition a set of Hamiltonians on a Nambu-Poisson manifold is given, the corresponding Hamiltonian system is also reduced. Several applications are given.

A Nomizu's theorem for the coeffective cohomology

This work has been partially supported through grants DGICYT (Spain), Projects PB91-0142 and PB89-0571, and through grants UPV, Project 127.310-EA 191/94

Co-isotropic and Legendre - Lagrangian submanifolds and conformal Jacobi morphisms

The notion of a co-isotropic and Legendre - Lagrangian submanifold of a Jacobi manifold is given. A characterization of conformal Jacobi morphisms and conformal Jacobi infinitesimal transformations is obtained as co-isotropic and Legendre - Lagrangian submanifolds of Jacobi manifolds.