The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.     

Reduction of Poisson manifolds

Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket.Research supported by DOE contract DE-AT03-85ER 12097.Supported by an A. P. Sloan Foundation fellowship.     

Zero-lapse loci in asymptotically flat maximally foliated spacetime manifolds

We settle the question of the occurrence of zero-lapse loci for maximal foliations {S_λ},S_λ ? R³, in an asymptotically flat spacetime.     

Separation of variables for integrable systems on Poisson manifolds

For an integrable system on Poisson manifolds, a construction of separated variables is discussed. We suppose that, for a given integrable system, we know a realization of the corresponding Lagrangian submanifold as the product of plane curves. In this case, we can use properties of the foliation of the initial Poisson manifold on symplectic leaves and values of the Casimir functions in order to construct separated variables.     

Commutative partially integrable systems on Poisson manifolds

We show that, as distinct from completely integrable Hamiltonian systems, a commutative partially integrable system admits different compatible Poisson structures on a phase manifold that are related by a recursion operator. The existence of action–angle coordinates around an invariant submanifold of such a partially integrable system is proved.     

Morita equivalence of Poisson manifolds

Poisson manifolds are the classical analogue of associative algebras. For Poisson manifolds, symplectic realizations play a similar role as representations do for associative algebras. In this paper, the notion of Morita equivalence of Poisson manifolds, a classical analogue of Morita equivalence of algebras, is introduced and studied. It is proved that Morita equivalent Poisson manifolds have equivalent categories of complete symplectic realizations. For certain types of Poisson manifolds, the geometric invariants of Morita equivalence are also investigated.     

WZW–Poisson manifolds

We observe that a term of the WZW-type can be added to the Lagrangian of the Poisson σ-model in such a way that the algebra of the first class constraints remains closed. This leads to a natural generalization of the concept of Poisson geometry. The resulting “WZW–Poisson” manifold M is characterized by a bivector Π and by a closed three-form H such that 1/2[Π,Π]_Schouten=H,ΠΠΠ.     

A Lagrangian for Hamiltonian vector fields on singular Poisson manifolds

On a manifold equipped with a bivector field, we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points project onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least defines an integrable distribution—a class of bivector fields generalizing twisted Poisson structures that we study in detail.     

On some involution theorems on twofold Poisson manifolds

We point out some involution theorems which are consequences of the existence of two compatible Poisson structures on a manifold. Using a theorem of Lichnerowicz on local triviality of the Schouten-Nijenhuis cohomology, we show that local exactness of the second Poisson structure with respect to the ground one is equivalent to involutivity of the algebra of invariant functions of the ground structure. Then an involution theorem of Mishchenko and Fomenko is given, founded on global exactness of the second structure. Finally a generalization of a recurrence operator is given to obtain a set of traces which are in involution.     

Poisson manifolds and their associated stacks

We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case.     

On the geometric prequantization of Poisson manifolds

The geometric prequantization of Poisson manifolds is described using the Weinstein theory of local symplectic groupoids.     

Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

Poisson structures on double Lie groups

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauss-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.     

Poisson structures on the Lorentz group

Poisson-Lie structures on the Lorentz group are completely classified. A method applicable to an arbitrary semisimple complex Lie group (treated as real) is developed.     

The background field method and the linearization problem for Poisson manifolds

The background field method (BFM) for the Poisson sigma model (PSM) is studied as an example of the application of the BFM technique to open gauge algebras. The relationship with Seiberg–Witten maps arising in non-commutative gauge theories is clarified. It is shown that the implementation of the BFM for the PSM in the Batalin–Vilkovisky formalism is equivalent to the solution of a generalized linearization problem (in the formal sense) for Poisson structures in the presence of gauge fields. Sufficient conditions for the existence of a solution and a constructive method to derive it are presented.     

On quadratic Poisson structures

We provide a general study on quadratic Poisson structures on a vector space. In particular, we obtain a decomposition for any quadratic Poisson structures. As an application, we classify all the three-dimensional quadratic Poisson structures up to a Poisson diffeomorphism.Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of the University of Pennsylvania.