Reduction of symplectic manifolds with symmetry

We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.     

Ruled CR-submanifolds of locally conformal Kähler manifolds

The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal Kähler manifold.     

Induction for weak symplectic Banach manifolds

The symplectic induction procedure is extended to the case of weak symplectic Banach manifolds. Using this procedure, one constructs hierarchies of integrable Hamiltonian systems related to the Banach Lie–Poisson spaces of k$k$-diagonal trace class operators.     

Serre duality for polarised symplectic manifolds

Let F be a polarisation, in the sense of Kostant, of a 2n-dimensional symplectic manifold M, and let L be a complex line bundle with flat connection along F. We prove, under suitable conditions, that the r-dim. cohomology of C^∞L-valued half forms normal to F with forms on F as coefficients, has as dual space the (n?r)-dim. cohomology of compactly supported distributional L¹-valued half forms normal to F with forms on F as coefficients. We deduce that the spectrum of an F-preserving function ? is the same when quantised on the highest C^∞ cohomology as when quantised on the compactly supported distributional sections.     

Distinguished Hamiltonian theorem for homogeneous symplectic manifolds

A diffeomorphism of a finite-dimensional flat symplectic manifold which is canonoid with respect to all linear and quadratic Hamiltonians, preserves the symplectic structure up to a factor: so runs the quadratic Hamiltonian theorem. Here we show that the same conclusion holds for much smaller sufficiency subsets of quadratic Hamiltonians, and the theorem may thus be extended to homogeneous infinite-dimensional symplectic manifolds. In this way, we identify the distinguished Hamiltonians for the Kähler manifold of equivalent quantizations of a Hilbertizable symplectic space.     

Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.     

Star products on compact pre-quantizable symplectic manifolds

We define in this Letter, a notion of representation for a star product (equipped with a star-compatible trace) and show that every compact pre-quantizable symplectic manifold admits a representable star product.Supported by NSF grant DMS 890771.     

Note on duality on polarized symplectic manifolds

In this note we use some of the results of [3] to derive a general duality theorem for the cohomologies of foliated structures on a manifold. The result is applied to the special case of a symplectic manifold M on which the foliation is given by a complex polarization F in the sense of geometric quantization. We obtain, for example, a rigorous proof of the fact that for a smooth function ƒ on M whose Hamiltonian vector field leaves F invariant, the spectrum of the corresponding prequantization operator v(ƒ) coincides with the spectrum of its transpose, under the above duality. This latter result was obtained by Simms in [12] under certain hypotheses. Proofs of the validity of those hypotheses are now available in the literature; cf. [3] and [7].     

Some rigidity theorems for locally symmetrical Finsler manifolds

In this paper we study some rigidity properties for locally symmetrical Finsler manifolds and obtain some results. We obtain the local equivalent characterization for a Finsler manifold to be locally symmetrical and prove that any locally symmetrical Finsler manifold with nonzero flag curvature must be Riemannian. We also generalize a rigidity result due to Akbar-Zadeh.     

Coeffective and de Rham cohomologies of symplectic manifolds

We discuss the relation of the coeffective cohomology of a symplectic manifold with the topology of the manifold. A bound for the coeffective numbers is obtained. The lower bound is got for compact Kähler manifolds, and the upper one for non-compact exact symplectic manifolds. A Nomizu's type theorem for the coeffective cohomology is proved. Finally, the behaviour of the coeffective cohomology under deformations is studied.     

Capacities as conformal quasi-invariants on pseudo-riemannian manifolds

A class of capacities is introduced on pseudo-riemannian manifolds. They arise as a natural counterpart of the well-known plane quasiconformal capacities and their higher dimensional analogues which have been studied extensively in the recent years by F.W. Gehring, R. Kühnau and others. The capacities in question are shown to be either conformal invariants or conformal quasi-invariants, and, in the latter case, exact bounds are established. We thus arrive at the notion of quasiconformal mappings of pseudo-riemannian manifolds, which correspond to the inhomogeneous media. These mappings are studied briefly and the physical interpretation of some of the capacities in question is also given.     

Hamiltonian circle actions on symplectic manifolds and the signature

Let M be a symplectic manifold with a Hamiltonian circle action with isolated fixed points. We prove that σ (M) = b₀(M) − b₂(M) + b₄(M) − b₆(M) + … where σ (M) is the signature of M and b_i(M) is the ith Betti number of M.     

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.     

Numerical studies of hyperbolic manifolds supporting diffusion in symplectic mappings

Diffusion in generic quasi integrable systems at small values of the perturbing parameters has been a very studied subject since the pioneering work of Arnold [3]. For moderate values of the perturbing parameter a different kind of diffusion occurs, the so called Chirikov diffusion, since the Chirikov’s papers [11, 13]. The two underlying mechanisms are different, the first has an analytic demonstration only on specific models, the second is based on an heuristic argument. Even if the relation between chaos and diffusion is far to be completely understood, a key role is played by the topology of hyperbolic manifolds related to the resonances. Different methods can be found in the literature for the detection of hyperbolic manifolds, at least for two dimensional systems. For higher dimensional ones some sophisticated methods have been recently developed (for a review see [55]). In this paper we review some of these methods and an easy tool of detection of invariant manifolds that we have developed based on the Fast Lyapunov Indicator. The relation between the topology of hyperbolic manifolds and diffusion is discussed in the framework of Arnold diffusion.