In this paper we introduce the universal r -differential calculus on a r -algebra and we prove the universality of the construction. We also present submanifolds, distributions, linear connections and two different r -differential calculi on S N q : the algebra of forms and the algebra of universal differential forms of S N q . Finally we prove the Frobenius theorem on the N -dimensional quantum hyperplane S N q in a general case, for any r -differential calculus on S N q .