Abstract:
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 affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for $\Lambda $. This structure is also studied for the Jacobi fields and for the moduli space of extremals.

Abstract:
We give an exposition of the parametrization method of Kuchar [1973] in the context of the multisymplectic approach to field theory, as presented in Gotay and Marsden [2008a]. The purpose of the formalism developed herein is to make any classical field theory, containing a metric as a sole background field, generally covariant (that is, "parametrized," with the spacetime diffeomorphism group as a symmetry group) as well as fully dynamic. This is accomplished by introducing certain "covariance fields" as genuine dynamic fields. As we shall see, the multimomenta conjugate to these new fields form the Piola-Kirchhoff version of the stress-energy-momentum tensor field, and their Euler-Lagrange equations are vacuously satisfied. Thus, these fields have no additional physical content; they serve only to provide an efficient means of parametrizing the theory. Our results are illustrated with two examples, namely an electromagnetic field and a Klein-Gordon vector field, both on a background spacetime.

Abstract:
We show how to "concatenate" variational principles over different bases into one over a single base, thereby providing a unified Lagrangian treatment of interacting systems. As an example we study a Klein-Gordon field interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle.

Abstract:
Let $\pi:P\to M^n$ be a principal G-bundle, and let ${\mathcal{L}}: J^1P \to\Lambda^n(M)$ be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on ${\mathcal C}(P)$, the bundle of connections on P.

Abstract:
A first-order Lagrangian $L^\nabla $ 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^\nabla $ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to $\nabla $.

Abstract:
The classification of 4-dimensional naturally reductive pseudo-Riemannian spaces is given. This classification comprises symmetric spaces, the product of 3-dimensional naturally reductive spaces with the real line and new families of indecomposable manifolds which are studied at the end of the article. The oscillator group is also analyzed from the point of view of this classification.

Abstract:
The un-reduction procedure introduced previously in the context of Mechanics is extended to covariant Field Theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter). Other possibilities are also explored: non-linear $\sigma$-models and the hyperbolic flows of curves.

Abstract:
We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components.

Abstract:
The process of un-reduction, a sort of reversal of reduction by the Lie group symmetries of a variational problem, is explored in the setting of field theories. This process is applied to the problem of curve matching in the plane, when the curves depend on more than one independent variable. This situation occurs in a variety of instances such as matching of surfaces or comparison of evolution between species. A discussion of the appropriate Lagrangian involved in the variational principle is given, as well as some initial numerical investigations.

Abstract:
The goal of this article is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group $H$ of isometries. In a first result, $H$ is a normal subgroup of the group of symmetries associated to the reducing tensor $\bar{S}$. The situation when $H$ is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fiberings of almost contact manifolds over almost Hermitian manifolds. If the structure is moreover Sasakian, the obtained reduced tensor is homogeneous K\"ahler.