The mathematical and geometrical structure of the spacetime and the concept of unification, matter and energy

Abstract: Geometrical analysis of a new type of Unified Field Theoretical models follow the guidelines of previous works of the authors is presented. These new unified theoretical models are characterized by an underlying hypercomplex structure, zero non-metricity and the geometrical action is determined fundamentally by the curvature provenient of the breaking of symmetry of a group manifold in higher dimensions. This mechanism of Cartan-MacDowell-Mansouri type, permits us to construct geometrical actions of determinantal type leading a non topological physical Lagrangian due the splitting of a reductive geometry. Our goal is to take advantage of the geometrical and topological properties of this theory in order to determine the minimal group structure of the resultant spacetime Manifold able to support a fermionic structure. From this fact, the relation between antisymmetric torsion and Dirac structure\ of the spacetime is determined and the existence of an important contribution of the torsion to the giromagnetic factor of the fermions, shown. Also we resume and analyze previous cosmological solutions in this new UFT where, as in our work [Class. Quantum Grav. 22 (2005) 4987--5004] for the non abelian Born-Infeld model, the Hosoya and Ogura ansatz is introduced for the important cases of tratorial, totally antisymmetric and general torsion fields. In the case of spacetimes with torsion the real meaning of the spin-frame alignment is find and the question of the minimal coupling is discussed.