Geometrical tools for quantum Euclidean spaces

We apply one of the formalisms of noncommutative geometry to R-q(N), the qu antum space covariant under the quantum group SOq(N). Over R-q(N) there are two SOq(N)-covariant differential calculi. For each we find a frame, a met ric and two torsionfree covariant derivatives which are metric compatible u p to a conformal factor and which have a vanishing linear curvature. This g eneralizes results found in a previous article for the case of R-q(3). As i n the case N = 3, one has to slightly enlarge the algebra R-q(N); for N odd one needs only one new generator whereas for N even one needs two. As in t he particular case N = 3 there is a conformal ambiguity in the natural metr ics on the differential calculi over R-q(N). While in our previous article the frame was found "by hand", here we disclose the crucial role of the qua ntum group covariance and exploit it in the construction. As an intermediat e step, we find a homomorphism from the cross product of R-q(N) with U(q)so (N) into R-q(N), an interesting result in itself.