612 research outputs found
Coupling Poisson and Jacobi structures on foliated manifolds
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.Comment: LateX, 38 page
Weak-Hamiltonian dynamical systems
A big-isotropic structure is an isotropic subbundle of ,
endowed with the metric defined by pairing. The structure is said to be
integrable if the Courant bracket ,
. Then, necessarily, one also has
, \cite{V-iso}. A weak-Hamiltonian dynamical system is a vector field
such that . We obtain the
explicit expression of and of the integrability conditions of under
the regularity condition 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.Comment: 19 pages, minor improvement
On the geometry of double field theory
Double field theory was developed by theoretical physicists as a way to
encompass -duality. In this paper, we express the basic notions of the
theory in differential-geometric invariant terms, in the framework of
para-Kaehler manifolds. We define metric algebroids, which are vector bundles
with a bracket of cross sections that has the same metric compatibility
property as a Courant bracket. We show that a double field gives rise to two
canonical connections, whose scalar curvatures can be integrated to obtain
actions. Finally, in analogy with Dirac structures, we define and study
para-Dirac structures on double manifolds.Comment: The paper will appear in J. Math. Phys., 201
Basics of lagrangian foliations
The paper is an exposition of basic known local and global results on Lagrangian foliations such as the theorems of Darboux-Lie, Weinstein, Arnold-Liouville, a global characterization of cotangent bundles, higher order Maslov classes, etc
Isotropic subbundles of
We define integrable, big-isotropic structures on a manifold as
subbundles that are isotropic with respect to the
natural, neutral metric (pairing) of and are closed by
Courant brackets (this also implies that ). We give the interpretation of such a structure by objects of
, we discuss the local geometry of the structure and we give a reduction
theorem.Comment: LaTex, 37 pages, minimization of the defining condition
On invariants of almost symplectic connections
We study the irreducible decomposition under Sp(2n, R) of the space of
torsion tensors of almost symplectic connections. Then a description of all
symplectic quadratic invariants of torsion-like tensors is given. When applied
to a manifold M with an almost symplectic structure, these instruments give
preliminary insight for finding a preferred linear almost symplectic connection
on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections.
Properties of torsion of the vectorial kind are deduced
Jacobi Structures in
The most general Jacobi brackets in are constructed after
solving the equations imposed by the Jacobi identity. Two classes of Jacobi
brackets were identified, according to the rank of the Jacobi structures. The
associated Hamiltonian vector fields are also constructed
Geometric quantization of mechanical systems with time-dependent parameters
Quantum systems with adiabatic classical parameters are widely studied, e.g.,
in the modern holonomic quantum computation. We here provide complete geometric
quantization of a Hamiltonian system with time-dependent parameters, without
the adiabatic assumption. A Hamiltonian of such a system is affine in the
temporal derivative of parameter functions. This leads to the geometric Berry
factor phenomena.Comment: 20 page
Reduction and construction of Poisson quasi-Nijenhuis manifolds with background
We extend the Falceto-Zambon version of Marsden-Ratiu Poisson reduction to
Poisson quasi-Nijenhuis structures with background on manifolds. We define
gauge transformations of Poisson quasi-Nijenhuis structures with background,
study some of their properties and show that they are compatible with reduction
procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis
structures with background.Comment: to appear in IJGMM
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