M. Castrillón López

The canonical eight-form on manifolds with holonomy group Spin(9)

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Bergers list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained.

Gauge invariance and variational trivial problems on the bundle of connections

AbstractGiven a principal bundle P → M we classify all first order Lagrangian densities on the bundle of connectionsassociated to P that are invariant under the Lie algebra of infinitesimal automorphisms. These are shown to bevariationally trivial and to give constant actions that equal the characteristic numbers of P if dim M is even andzero if dim M is odd. In addition, we show that variationally trivial Lagrangians are characterized by the de Rhamcohomology of the base manifold M and the characteristic classes of P of arbitrary degree.  2003 Elsevier B.V. All rights reserved. MSC: primary 58E30; secondary 17B66, 22E65, 53C07, 81R10Keywords: Characteristic classes; Connections on a principal bundle; Euler–Lagrange equations; Gauge invariance;Infinitesimal symmetries; Jet bundles; Lagrangians; Lie algebra representations; Weil polynomials 1. IntroductionLet p : C → M be the bundle of connections of a principal G -bundle π : P → M . The goal of thispaper is twofold. First, we classify the Lagrangians on

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L∇ variationally equivalent to the second-order Einstein-Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇.

Gauge-invariant variationally trivial problems on T *M

A classification of variationally trivial Lagrangians on T*M which are invariant under the Lie algebra of infinitesimal gauge transformations of the principal bundle π:M×U(1)→M, is given. A characterization of Lagrangian densities on T*M which are invariant under the Lie algebra of all infinitesimal automorphisms of M×U(1) is also obtained.

Constraints in Euler-Poincaré Reduction of Field Theories

The goal of this short note is to show the geometric structure of the Euler-Poincaré reduction procedure in Field Theories with special emphasis on the nature of the set of variations and the set of admissible sections. The method of Lagrange multipliers is also applied for a deeper study of these constraints.

A REPORT ON GAUGE INVARIANT FORMS AND VARIATIONAL PROBLEMS ON THE BUNDLE OF CONNECTIONS OF A PRINCIPAL U (1)-BUNDLE AND ON ASSOCIATED VECTOR BUNDLES

Basic results on gauge invariance of differential forms on the bundle of connections of an arbitrary principal U(1)-bundle and its associated bundles, are reviewed in terms of the underlying geometry to such bundles, within the framework of classical electromagnetism.

U(1)-invariant current forms

Lagrangians are investigated which are defined on the product of the cotangent bundle of a manifold and a vector space, where the vector space is the representation space of a linear representation of the group U(1) as the gauge group. It is shown that Lagrangians with gauge-invariant universal currents are described in terms of gauge-invariant Lagrangians and invariant functions on the vector space.

Killing Vector Fields of Generic Semi-Riemannian Metrics

Let M be a smooth oriented connected n-dimensional manifold and let

Current forms and gauge invariance

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