The geometry of a covariant expansion in chiral theories with nucleons

We deal with the geometrical foundations of a covariant scheme, which is developed to give a co-ordinate independent perturbation expansion ofSU(2)×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.