One can integrate fields (the generalization of vector fields) on the surface that is embedded in the space R^N . But the value of integral for the tenzor field on the Riemannian manifold, that in not embedded in the fixed space R^N, have no meaning. However one can attach a meaning to the term "the integrable tensor field". The investigation of the classes of integrable tensor fields have to pass without complication (that is the authors opinion)