  We briefly report our application of a version of noncommutative geometry to
the quantum Euclidean space $R^N_q$, for any $N \ge 3$; this space is covariant
under the action of the quantum group $SO_q(N)$, and two covariant differential
calculi are known on it. More precisely, we describe how to construct in a
Cartan `moving-frame formalism' the metric, two covariant derivatives, the
Dirac operator, the frame, the inner derivations dual to the frame elements,
for both of these calculi. The components of the frame elements in the basis of
differentials provide a `local realization' of the
Faddeev-Reshetikhin-Takhtadjan generators of $U_q^{\pm}(so(N))$.
