  Motivated by the study of physical models associated with General Relativity,
we review some finite-dimensional, geometric and covariant formulations that
allow us to characterize in a simple manner the symmetries for classical field
theory by implementing an appropriate fibre-bundle structure, either at the
Lagrangian, the multisymplectic or the polysymplectic levels. In particular, we
are able to formulate Noether's theorems by means of the covariant momentum
maps and to systematically introduce a covariant Poisson-Hamiltonian framework.
Also, by focusing on the space plus time decomposition for a generic classical
field theory and its relation to these geometric formulations, we are able to
successfully recover the gauge content and the true local degrees of freedom
for the theory. In order to illustrate the relevance of these geometric
frameworks, we center our attention to the analysis of a model for
$3$-dimensional theory of General Relativity that involves an arbitrary
Immirzi-like parameter. At the Lagrangian level, we reproduce the field
equations of the system which for this model turn out to be equivalent to the
vanishing torsion condition and the Einstein equations. We also concentrate on
the analysis of the gauge symmetries of the system in order to obtain the
Lagrangian covariant momentum map associated with the theory and, consequently,
its corresponding Noether currents. Next, we aim our attention to describing
how the gauge symmetries of the model yield covariant canonical transformations
on the covariant multimomenta phase-space, thus giving rise to the existence of
a covariant momentum map. Besides, we analyze the physical system under
consideration within the De Donder-Weyl canonical theory implemented at the
polysymplectic level, thus establishing a relation from the covariant momentum
map to the conserved currents of the theory.
