Classification of Differential Calculus on Quantum Complex Grassmann Manifolds

A method for the classification of covariant first order differential calculus on a certain class of quantum spaces is devised. The main tool is a notion of tangent space in analogy to Woronowicz' tangent space for differential calculi on quantum groups. The method is applied to quantum complex Grassmann manifolds Oq(Gr(r,N)). It is shown that there exists a unique covariant first order differential calculus of dimension 2r(N-r)=dim(Gr(r,N)) over Oq(Gr(r,N)).