Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost complex structures of F'. The existence of a Bott connection which yields a structure I that is projectable to the space of leaves is equivalent to the fact that F is a transversally projective foliation. A Bott connection which yields a projectable structure J exists iff F is a transversally projective foliation which satisfies a supplementary cohomological condition, and, in this case, I is projectable as well. J is never integrable. The essential integrability condition of I is the flatness of the transversal projective structure of F.Comment: LaTex, 33 p

Dirac submanifolds of Jacobi manifolds

The notion of a Dirac submanifold of a Poisson manifold was studied by Xu (arXiv:math.SG/0110326). We give an interpretation of Xu's definition in terms of a general notion of tensor fields soldered to a normalized submanifold. Then, this interpretation is used to define Dirac submanifolds of a Jacobi manifold. Several properties and examples are discussed.Comment: Latex, 22 page

A note on submanifolds and mappings in generalized complex geometry

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.Comment: LaTex, 18 pages, style changes and misprint corrections, to appear in Monatshefte f\"ur Mathemati

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

On hypersurfaces of generalized K\"ahler manifolds

We establish the conditions for the induced generalized metric F structure of an oriented hypersurface of a generalized K\"ahler manifold to be a generalized CRFK structure. Then, we discuss a notion of generalized almost contact structure on a manifold $M$ that is suggested by the induced structure of a hypersurface. Such a structure has an associated generalized almost complex structure on M\times\mathds{R}. If the latter is integrable, the former is normal and we give the corresponding characterization. If the structure on M\times\mathds{R} is generalized K\"ahler, the structure on $M$ is said to be binormal. We characterize binormality and give an example of binormal structure.Comment: LaTeX, 30 pages. References and corresponding comments adde

Nambu-Lie Groups

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several examples of Nambu-Lie groups and algebras.Comment: 17 pages, LaTe

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