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Second order connections on some functional bundles

Antonella Cabras, Ivan Kolář (1999)

Archivum Mathematicum

We study the second order connections in the sense of C. Ehresmann. On a fibered manifold Y , such a connection is a section from Y into the second non-holonomic jet prolongation of Y . Our main aim is to extend the classical theory to the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. This requires several new geometric results about the second order connections on Y , which are deduced in the first part of the paper.

Smooth bundles of generalized half-densities

Daniel Canarutto (2000)

Archivum Mathematicum

Smooth bundles, whose fibres are distribution spaces, are introduced according to the notion of smoothness due to Frölicher. Some fundamental notions of differential geometry, such as tangent and jet spaces, Frölicher-Nijenhuis bracket, connections and curvature, are suitably generalized. It is also shown that a classical connection on a finite-dimensional bundle naturally determines a connection on an associated distributional bundle.

Soldered double linear morphisms

Alena Vanžurová (1992)

Mathematica Bohemica

Our aim is to show a method of finding all natural transformations of a functor T T * into itself. We use here the terminology introduced in [4,5]. The notion of a soldered double linear morphism of soldered double vector spaces (fibrations) is defined. Differentiable maps f : C 0 → C 0 commuting with T T * -soldered automorphisms of a double vector space C 0 = V * × V × V * are investigated. On the set Z s ( C 0 ) of such mappings, appropriate partial operations are introduced. The natural transformations T T * → T T * are bijectively related with the elements...

Some aspects of the homogeneous formalism in field theory and gauge invariance

Marcella Palese, Ekkehart Winterroth (2006)

Archivum Mathematicum

We propose a suitable formulation of the Hamiltonian formalism for Field Theory in terms of Hamiltonian connections and multisymplectic forms where a composite fibered bundle, involving a line bundle, plays the role of an extended configuration bundle. This new approach can be interpreted as a suitable generalization to Field Theory of the homogeneous formalism for Hamiltonian Mechanics. As an example of application, we obtain the expression of a formal energy for a parametrized version of the Hilbert–Einstein...

Some functorial prolongations of general connections

Ivan Kolář (2018)

Archivum Mathematicum

We consider the problem of prolongating general connections on arbitrary fibered manifolds with respect to a product preserving bundle functor. Our main tools are the theory of Weil algebras and the Frölicher-Nijenhuis bracket.

Special tangent valued forms and the Frölicher-Nijenhuis bracket

Antonella Cabras, Ivan Kolář (1993)

Archivum Mathematicum

We define the tangent valued C -forms for a large class of differential geometric categories. We deduce that the Frölicher-Nijenhuis bracket of two tangent valued C -forms is a C -form as well. Then we discuss several concrete cases and we outline the relations to the theory of special connections.

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