1,011 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
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
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
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
Linearizing Generalized Kahler Geometry
The geometry of the target space of an N=(2,2) supersymmetry sigma-model
carries a generalized Kahler structure. There always exists a real function,
the generalized Kahler potential K, that encodes all the relevant local
differential geometry data: the metric, the B-field, etc. Generically this data
is given by nonlinear functions of the second derivatives of K. We show that,
at least locally, the nonlinearity on any generalized Kahler manifold can be
explained as arising from a quotient of a space without this nonlinearity.Comment: 31 pages, some geometrical aspects clarified, typos correcte
Tri-hamiltonian vector fields, spectral curves and separation coordinates
We show that for a class of dynamical systems, Hamiltonian with respect to
three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are
provided by the common roots of a set of bivariate polynomials. These
polynomials, which generalise those considered by E. Sklyanin in his
algebro-geometric approach, are obtained from the knowledge of: (i) a common
Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu
P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to
its symplectic leaves. The frameworks is applied to Lax equations with spectral
parameter, for which not only it unifies the separation techniques of Sklyanin
and of Magri, but also provides a more efficient ``inverse'' procedure not
involving the extraction of roots.Comment: 49 pages Section on reduction revisite
Gauge field theories with covariant star-product
A noncommutative gauge theory is developed using a covariant star-product
between differential forms defined on a symplectic manifold, considered as the
space-time. It is proven that the field strength two-form is gauge covariant
and satisfies a deformed Bianchi identity. The noncommutative Yang-Mills action
is defined using a gauge covariant metric on the space-time and its gauge
invariance is proven up to the second order in the noncommutativity parameter.Comment: Dedicated to Ioan Gottlieb on the occasion of his 80th birthday
anniversary. 12 page
Regular Poisson structures on massive non-rotating BTZ black holes
We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian
symmetric space context. Using classical symmetric space techniques we find
that every such space intrinsically carries a regular Poisson structure whose
symplectic leaves are para-hermitian symmetric surfaces. We also obtain a
global expression of the metric yielding a dynamical description of the black
hole from its initial to its final singularity.Comment: LaTex, 18 pages, 3 figures, version published in Nucl. Phys.
- …