arXiv subjects
Publications and source records attributed to Izu Vaisman.
The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization.
A generalized F-structure is a complex, isotropic subbundle $E$ of $T_cM\oplus T^*_cM$ ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ and the metric is defined by pairing) such that $E\cap\bar E^{\perp}=0$. If $E$ is also closed by the Courant bracket, $E$ is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism $Φ$ of $TM\oplus T^*M$ that satisfies the condition $Φ^3+Φ=0$ and we express the CRF-condition by means of the Courant-Nijenhuis torsion of $Φ$. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair $(\mathcal{V},σ)$ where $\mathcal{V}$ is an integrable subbundle of $TM$ and $σ$ is a 2-form on $M$, a generalized, normal, almost contact structure of codimension $h$. We show that a generalized complex structure on a manifold $\tilde M$ induces generalized CRF-structures into some submanifolds $M\subseteq\tilde M$. Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kähler structures and are equivalent with quadruples $(γ,F_+,F_-,ψ)$, where $(γ,F_\pm)$ are classical, metric CRF-structures, $ψ$ is a 2-form and some conditions expressible in terms of the exterior differential $dψ$ and the $γ$-Levi-Civita covariant derivative $\nabla F_\pm$ hold. If $dψ=0$, the conditions reduce to the existence of two partially Kähler reductions of the metric $γ$. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.
A big-isotropic structure $E$ is an isotropic subbundle of $TM\oplus T^*M$, endowed with the metric defined by pairing. The structure $E$ is said to be integrable if the Courant bracket $[\mathcal{X},\mathcal{Y}]\inΓE$, $\forall\mathcal{X},\mathcal{Y}\inΓE$. Then, necessarily, one also has $[\mathcal{X},\mathcal{Z}]\inΓE^\perp$, $\forall\mathcal{Z}\inΓE^\perp$ \cite{V-iso}. A weak-Hamiltonian dynamical system is a vector field $X_H$ such that $(X_H,dH)\in E^\perp$ $(H\in C^\infty(M))$. We obtain the explicit expression of $X_H$ and of the integrability conditions of $E$ under the regularity condition $dim(pr_{T^*M}E)=const.$ We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) \cite{{BR},{DS}} may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems.
We define integrable, big-isotropic structures on a manifold $M$ as subbundles $E\subseteq TM\oplus T^*M$ that are isotropic with respect to the natural, neutral metric (pairing) $g$ of $TM\oplus T^*M$ and are closed by Courant brackets (this also implies that $[E,E^{\perp_g}]\subseteq E^{\perp_g}$). We give the interpretation of such a structure by objects of $M$, we discuss the local geometry of the structure and we give a reduction theorem.
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 $(P,θ,F,Z_a,ξ^a)$ $(a=1,...,h)$, where $P$ is a bivector field, $θ$ is a 2-form, $F$ is a $(1,1)$-tensor field, $Z_a$ are vector fields and $ξ^a$ are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal, almost contact structure $(F,Z,ξ)$. 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.
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.
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.
We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure iff N is a Dirac submanifold. In the presymplectic case, it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, we make new remarks about it, and we establish characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.
We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum $E\oplus C$ where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, i.e., a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection. In particular, this class contains all the transitive Courant algebroids of minimal rank; for these, the flat term mentioned above is zero. The results extend to regular Courant algebroids, i.e., Courant algebroids with a constant rank anchor. The paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.
Let M be a differentiable manifold endowed with a foliation F. A Poisson structure P on M is F-coupling if the image of the annihilator of TF by the sharp-morphism defined by P is a normal bundle of the foliation F. This notion extends Sternberg's coupling symplectic form of a particle in a Yang-Mills field. In the present paper we extend Vorobiev's theory of coupling Poisson structures from fiber bundles to foliations and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. Then we discuss the extension of the coupling condition to Jacobi structures on foliated 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.
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples, and find the cohomological obstractions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index free presentation of these connections.
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.
We discuss hamiltonian structures of the Gelfand-Dorfman complex of projectable vector fields and differential forms on a foliated manifold. Such a structure defines a Poisson structure on the algebra of foliated functions, and embeds the given foliation into a larger, generalized foliation with presymplectic leaves. In a so-called tame case, the structure is induced by a Poisson structure of the manifold. Cohomology spaces and classes relevant to geometric quantization are also considered.
The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be p-related (p is the projection of TM onto M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fiberwise polynomial algebra, and are transversal Poisson structures of the foliation by fibers. Finally, such bivector fields are produced by a process of horizontal lifting of Poisson structures from M to TM via connections.
We expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poisson manifolds with non regular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.
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.
In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.