Kaehler-Nijenhuis Manifolds

A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is compatible with the Poisson stucture defined by F in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M is foliated by im A into non degenerate Kaehler-Nijenhuis submanifolds. If A is a non degenerate (1,1)-tensor field on M, (M,g,J,A) is a Kaehler-Nijenhuis manifold iff one of the following two properties holds: 1) A is associated with a symplectic structure of M that defines a Poisson structure compatible with the Poisson structure defined by F; 2) A and its inverse are associated with closed 2-forms. On a Kaehler-Nijenhuis manifold, if A is non degenerate and AJ=-JA, A must be a parallel tensor field.Comment: LaTex, 10 page

Hamiltonian vector fields on almost symplectic manifolds

Let $(M,\omega)$ be an almost symplectic manifold ($\omega$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_X\omega=0,d(i(X)\omega)=0$, and it is Hamiltonian if, furthermore, the 1-form $i(X)\omega$ is exact. Such vector fields were considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on $M$ and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.Comment: LaTex, 18 page

Foliation-coupling Dirac structures

We extend the notion of "coupling with a foliation" from Poisson to Dirac structures and get the corresponding generalization of the Vorobiev characterization of coupling Poisson structures. We show that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf, give new proofs of the results of Dufour and Wade on the transversal Poisson structure, and compute the Vorobiev structure of the total space of a normal bundle of the leaf. Finally, we use the coupling condition along a submanifold, instead of a foliation, in order to discuss submanifolds of a Dirac manifold which have a differentiable, induced Dirac structure. In particular, we get an invariant that reminds the second fundamental form of a submanifold of a Riemannian manifold.Comment: Correction of the expression of the second fundamental for

Reduction and submanifolds of generalized complex manifolds

We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.Comment: LaTex, 33 pages, Contains added result

A construction of Courant algebroids on foliated manifolds

For any transversal-Courant algebroid $E$ on a foliated manifold $(M,\mathcal{F})$, and for any choice of a decomposition $TM=T\mathcal{F}\oplus Q$, we construct a Courant algebroid structure on $T\mathcal{F}\oplus T^*\mathcal{F}\oplus E$.Comment: LaTeX, 6 page

Generalized para-K\"ahler manifolds

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-K\"ahler structure. This class of structures contains both the classical para-K\"ahler structure and the classical K\"ahler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple $(\gamma,\psi,F)$, where $\gamma$ is a (pseudo) Riemannian metric, $\psi$ is a $2$-form and $F$ is a complex $(1,1)$-tensor field such that $F^2=Id,\gamma(FX,Y)+\gamma(X,FY)=0$. We deduce integrability conditions similar to those of the generalized K\"ahler structures and give several examples of generalized para-K\"ahler manifolds. We discuss submanifolds that bear induced para-K\"ahler structures and, on the other hand, we define a reduction process of para-K\"ahler structures.Comment: LaTeX, 22 pages. The second version includes an additional materia

Locally Lagrangian Symplectic and Poisson Manifolds

We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Exemples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold. Finally, we indicate the generalization of this type of structures to Poisson manifolds.Comment: LaTex, 21 pages. Lecture at ``Poisson 2000'', CIRM, Luminy, France, June 26-30, 200

Dirac Structures and Generalized Complex Structures on TM\times\mathds{R}^h

We consider Courant and Courant-Jacobi brackets on the stable tangent bundle TM\times\mathds{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\times\mathds{R}^h can be prolonged to TM\times\mathds{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 $(P,\theta,F,Z_a,\xi^a)$ $(a=1,...,h)$, where $P$ is a bivector field, $\theta$ is a 2-form, $F$ is a $(1,1)$-tensor field, $Z_a$ are vector fields and $\xi^a$ are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal, almost contact structure $(F,Z,\xi)$. We prove that such a generalized structure projects to a generalized, complex structure of a space of leaves and 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.Comment: LaTex, 27 page

Soldered tensor fields of normalized submanifolds

In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost complex structure of an almost K\"ahler manifold is soldered to a submanifold iff the latter is an invariant, totally geodesic submanifold.Comment: LaTeX, 8 page

Towards a double field theory on para-Hermitian manifolds

In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-K\"ahler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangent bundle of the manifold has a natural, metric-compatible bracket that extends the C-bracket of double field theory. In the para-K\"ahler case this bracket is equal to the sum of the Courant brackets of the two Lagrangian foliations of the manifold. Then, we define a canonical connection and an action of the field that correspond to similar objects of double field theory. Another section is devoted to the Marsden-Weinstein reduction in double field theory on para-Hermitian manifolds. Finally, we give examples of fields on some well-known para-Hermitian manifolds.Comment: LaTex, 36 pages. Language improvements, references added and the definition of action changed. To appear in J. of Math. Phy