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

A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity