Geometrical Tools for Quantum Euclidean Spaces

We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of $R^3_q$. As in the case N=3, one has to slightly enlarge the algebra $R^N_q$; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over $R^N_q$. While in our previous article the frame was found `by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of $R^N_q$ with $U_qso(N)$ into $R^N_q$, an interesting result in itself.