[en] Fibre bundle formulations of field theory are considered approximately by the appropriate vector bundle. The fibre vierbein (frame) is considered as dependent indirectly on an element f of the fibre group, which we choose as a harmonic mapping from the base manifold. We do the same for the other fundamental geometrical quantity called the connection, but choose another independent harmonic mapping h instead of f. Additionally, we generalize the torsion-free condition to this more general vector bundle other than tangent bundle, by using the pullback mapping. A new gauge invariant Lagrangian is proposed. (author)