Searcharxiv⌕ Search

arXiv subjects

Josef Janyška

Publications and source records attributed to Josef Janyška.

17 recordsLinked to original sources
arXiv2020

Smooth and F--smooth systems

We review the geometric theory of \emp{smooth systems of smooth maps}, of \emp{smooth systems of smooth sections} of a smooth double fibred manifold and of \emp{smooth systems of smooth connections} of a smooth fibred manifold. Moreover, after reviewing the concept of \emp{F-smooth space} due to A. Frölicher, we discuss the \emp{F-smooth systems of smooth maps}, the \emp{F-smooth systems of fibrewisely smooth sections} of a smooth double fibred manifold, the \emp{F-smooth systems of fibrewisely smooth connections} of a smooth fibred manifold and of \emp{F-smooth connections} of an F-smooth system of fibrewisely smooth sections.

math.DG↗
arXiv2019

Remarks on natural differential operators with tensor fields

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.

math.DG↗
arXiv2016

On Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures

We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by pairs generating infinitesimal symmetries of basic tensor fields given by the almost-cosymplectic-contact structure.

math.DG↗
arXiv2013

On the characterization of infinitesimal symmetries of the relativistic phase space

The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal symmetries of the phase structures are characterized. More precisely, it is proved that all phase infinitesimal symmetries are special Hamiltonian lifts of distinguished conserved quantities on the phase space. It is proved that generators of infinitesimal symmetries constitute a Lie algebra with respect to a special bracket. A momentum map for groups of symmetries of the geometric structures is provided.

math-ph↗
arXiv2011

Combinatorial differential geometry and ideal Bianchi-Ricci identities II - the torsion case

This paper is a continuation of arXiv:0809.1158, dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without corrections.

math.DG↗
arXiv2008

Geometric structures of the classical general relativistic phase space

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1--dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialise these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

math-ph↗
arXiv2008

Generalized geometrical structures of odd dimensional manifolds

We define an almost--cosymplectic--contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost--coPoisson--Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated algebras of functions. As examples of the above structures, we present geometrical dynamical structures of the phase space of a general relativistic particle, regarded as the 1st jet space of motions in a spacetime. We describe geometric conditions by which a metric and a connection of the phase space yield cosymplectic and dual coPoisson structures, in case of a spacetime with absolute time (a Galilei spacetime), or almost--cosymplectic--contact and dual almost--coPoisson--Jacobi structures, in case of a spacetime without absolute time (an Einstein spacetime).

math.DG↗
arXiv2007

Semi--vector spaces and units of measurement

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of ``positive space'' and its rational powers. Positive spaces are 1-dimensional ``semi-vector spaces'' without the zero vector. A direct approach to this subject might be sufficient. On the other hand, a broader mathematical understanding requires the notions of sesqui- and semi-tensor products between semi-vector spaces and vector spaces. So, the paper is devoted to an original contribution to the algebraic theory of semi-vector spaces, to the algebraic analysis of positive spaces and, eventually, to the algebraic model of physical scales and units of measurement in terms of positive spaces.

math.AC↗
arXiv2005

Hermitian vector fields and special phase functions

We start by analysing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

math-ph↗
arXiv2005

Graded Lie algebra of Hermitian tangent valued forms

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the classification of Hermitian vector fields and to their representation via special phase functions. In view of a possible covariant quantisation of a larger class of "observables" it is natural to consider the Hermitian tangent valued forms. Thus, this paper is devoted to a self--contained analysis of the graded Lie algebra of Hermitian tangent valued forms of a complex line bundle and to their classification in terms of tangent valued forms and forms of the base space. The local classification is obtained in coordinates. For the global classification we need a Hermitian connection: indeed, this is just the connection required in gauge theories.

math-ph↗
arXiv2004

Natural connections given by general linear and classical connections

We assume a vector bundle $p: E\to M$ with a general linear connection $K$ and a classical linear connection $\Lam$ on $M$. We prove that all classical linear connections on the total space $E$ naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on $J^1 E$ naturally given by $(\Lam, K)$ form a 14-parameter family. Both families of connections are described geometrically.

math.DG↗
arXiv2004

Functorial prolongations of some functional bundles

We discuss two kinds of functorial prolongations of the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. We study the prolongation of vector fields in both cases and we prove that the bracket is preserved. Our proof is based on several new results concerning the finite dimensional Weil bundles.

math.DG↗
arXiv2004

On the geometry of variational calculus on some functional bundles

We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise $(k,r)$-jet. The relations to the Euler-Lagrange morphism of the variational calculus are described.

math.DG↗
arXiv2004

Higher order reduction theorems for general linear connections

The reduction theorems for general linear and classical connections are generalized for operators with values in higher order gauge-natural bundles. We prove that natural operators depending on the $s_1$-jets of classical connections, on the $s_2$-jets of general linear connections and on the $r$-jets of tensor fields with values in gauge-natural bundles of order $k\ge 1$, $s_1+2\ge s_2$, $s_1,s_2\ge r-1\ge k-2$, can be factorized through the $(k-2)$-jets of both connections, the $(k-1)$-jets of the tensor fields and sufficiently high covariant differentials of the curvature tensors and the tensor fields.

math.DG↗