Quantum orthogonal planes: ISOq,r(N) and SOq,r(N) bicovariant calculi and differential geometry on quantum Minkowski space

We construct differential calculi on multiparametric quantum orthogonal pla nes in any dimension N. These calculi are bicovariant under the action of t he full inhomogeneous (multiparametric) quantum group ISOq,r(N), and do con tain dilatations. If we require bicovariance only under the quantum orthogo nal group SOq,r(N), the calculus on the q-plane can be expressed in terms o f its coordinates x(a), differentials dx(a) and partial derivatives partial derivative(a) without the need of dilatations, thus generalizing known res ults: to the multiparametric case. Using real forms that lead to the signat ure (n + 1, m) with m = n - 1, n, n + 1, we find ISOq,r(n + 1, m) and SOq,r (n + 1, m) bicovariant calculi on the multiparametric quantum spaces. The p articular case of the quantum Minkowski space ISOq,r(3, 1)/SOq,r(3, 1) is t reated in detail. The conjugated partial derivatives partial derivative(a)* can be expressed as linear combinations of the partial derivative(a). This allows a deformation of the phase-space where no additional operators (bes ides x(a) and p(a)) are needed.