[en] We deal with the geometrical foundations of a covariant scheme, which is developed to give a 'co-ordinate independent' perturbation expansion of SU(2) x SU(2) chiral theories with pions and nucleons. The pion fields play the role of local co-ordinates on a 3-D manifold with constant curvature in isospace. The presence of the nucleon isospinor field forces us to deal with the problem of endowing the manifold with a spin structure. In this way the nucleon isospinor is accommodated in the fiber space of the principal fiber bundle of the tangent bundle of the manifold. (orig.)