Related papers: AV-differential geometry and Newtonian mechanics
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie…
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.
We define a universal deformation formula (UDF) for the actions of the affine group on Frechet algebras. More precisely, starting with any associative Frechet algebra which the affine group acts on in a strongly continuous and isometrical…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…
The fundamental theorem of affine geometry says that a self-bijection $f$ of a finite-dimensional affine space over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected…
Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate…
In the comparison of nonholonomic mechanics and constrained variational mechanics, invariant affine subbundles arise in the determination of the initial conditions where the two methods yield the same trajectories. Motivated by this,…
We study the category of modules admitting compatible actions of the Lie algebra $\mathcal{V}$ of vector fields on an affine space and the algebra $\mathcal{A}$ of polynomial functions. We show that modules in this category which are…
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract…
In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This…
Let M be a manifold endowed with a symmetric affine connection $\Gamma.$ The aim of this paper is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T^{*} M and the space of second-order…
Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $\Lambda $ on $C$ satisfying a weak condition of regularity, are shown to admit an…
A covariant reformulation of General Relativity is briefly considered from three points of view: geometrodynamics, Lagrange-Euler field theory, and gauge field theory. From a geometrodynamics perspective, a definition of the reference frame…
We determine those maps between affine or projective spaces that are linear in the abstract sense of transforming collinear points into collinear points and whose restriction to any line is constant or injective. Our results are extensions…
Gauge fields frequently used as an independent construction additional to so-called wave fields of matter. This artificial separation is of course useful in some applications (like Berry's interactions between the "heavy" and "light"…
We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical…
We describe the main ingredients needed to create, from the smooth lagrangian density, a variational principle for discrete motions of a discrete rod, with corresponding conserved Noether currents. We describe all geometrical objects in…
Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational…
We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal…
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view,…