In this paper we study the differential structure of a spinor bundle in spaces where the metric tensor g μν(x, ξ, \\ ̄ gx) of the base manifold depends on the position x as well as on the spinor variables ξ and \\ ̄ gx. Notions such as: gauge covariant derivatives of tensors, connections, curvatures, torsions and Bianchi identities are presented in the context of a gauge approach, different than the one proposed in [11, 13], due to the introduction of a Poincaré group and the use of d-connections [6, 8] in the spinor bundle S (2) M. The introduction of basic 1-form fields ϱ μ and spinors ζ a , \\ ̄ gz a with values in the Lie algebra of the Poincaré group is also essential in our study. The gauge field equations are derived by the authors [12]. Finally, we give the Yang-Mills and the Yang-Mills-Higgs equations in a form sufficiently generalized for our approach.