791 research outputs found
On the concept of a filtered bundle
We present the notion of a filtered bundle as a generalisation of a graded
bundle. In particular, we weaken the necessity of the transformation laws for
local coordinates to exactly respect the weight of the coordinates by allowing
more general polynomial transformation laws. The key examples of such bundles
include affine bundles and various jet bundles, both of which play fundamental
roles in geometric mechanics and classical field theory. We also present the
notion of double filtered bundles which provide natural generalisations of
double vector bundles and double affine bundles. Furthermore, we show that the
linearisation of a filtered bundle - which can be seen as a partial
polarisation of the admissible changes of local coordinates - is well defined.Comment: 23 page
Remarks on Contact and Jacobi Geometry
We present an approach to Jacobi and contact geometry that makes many facts,
presented in the literature in an overcomplicated way, much more natural and
clear. The key concepts are Kirillov manifolds and linear Kirillov structures,
i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear
Poisson manifolds. The difference with the existing literature is that the
homogeneity of the Poisson structure is related to a principal -bundle structure on the manifold and not just to a vector
field. This allows for working with Jacobi bundle structures on nontrivial line
bundles and drastically simplifies the picture of Jacobi and contact geometry.
Our results easily reduce to various basic theorems of Jacobi and contact
geometry when the principal bundle structure is trivial, while giving new
insights into the theory
Polarisation of Graded Bundles
We construct the full linearisation functor which takes a graded bundle of
degree (a particular kind of graded manifold) and produces a -fold
vector bundle. We fully characterise the image of the full linearisation
functor and show that we obtain a subcategory of -fold vector bundles
consisting of symmetric -fold vector bundles equipped with a family of
morphisms indexed by the symmetric group . Interestingly, for
the degree 2 case this additional structure gives rise to the notion of a
symplectical double vector bundle, which is the skew-symmetric analogue of a
metric double vector bundle. We also discuss the related case of fully
linearising -manifolds, and how one can use the full linearisation functor
to "superise" a graded bundle
Odd Connections on Supermanifolds: Existence and relation with Affine Connections
The notion of an odd quasi-connection on a supermanifold, which is loosely an
affine connection that carries non-zero Grassmann parity, is examined. Their
torsion and curvature are defined, however, in general, they are not tensors. A
special class of such generalised connections, referred to as odd connections
in this paper, have torsion and curvature tensors. Part of the structure is an
odd involution of the tangent bundle of the supermanifold and this puts drastic
restrictions on the supermanifolds that admit odd connections. In particular,
they must have equal number of even and odd dimensions. Amongst other results,
we show that an odd connection is defined, up to an odd tensor field of type
, by an affine connection and an odd endomorphism of the tangent bundle.
Thus, the theory of odd connections and affine connections are not completely
separate theories. As an example relevant to physics, it is shown that
super-Minkowski spacetime admits a natural odd connection.Comment: 17 pages including one Appendix. Parts of the exposition have been
rewritte and further references added. Accepted for publication in the
Journal of Physics A: Mathematical and Theoretica
Ab initio vibrational free energies including anharmonicity for multicomponent alloys
A density-functional-theory based approach to efficiently compute numerically
exact vibrational free energies - including anharmonicity - for chemically
complex multicomponent alloys is developed. It is based on a combination of
thermodynamic integration and a machine-learning potential. We demonstrate the
performance of the approach by computing the anharmonic free energy of the
prototypical five-component VNbMoTaW refractory high entropy alloy
Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids
We show how to extend the construction of Tulczyjew triples to Lie algebroids
via graded manifolds. We also provide a generalisation of triangular Lie
bialgebroids as higher Poisson and Schouten structures on Lie algebroids.Comment: 28 pages. Completely rewritten and improved. Typos corrected. A
version is to appear in Reports on Mathematical Physics, Vol.66, No. 2, 2010.
Further minor typos correcte
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