The geometry of the quantum Euclidean space

A detailed study is made of the noncommutative geometry of R-q(3), the quan tum space covariant under the quantum group SOq(3). For each of its two SOq (3)-covariant differential calculi we find its metric, the corresponding fr ame and two torsion-free covariant derivatives that are metric compatible u p to a conformal factor and both which yield a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The d elicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested, yieldi ng S-2 x R and R-3, respectively, as the limit Riemannian manifolds. (C) 20 00 Elsevier Science B.V. All rights reserved.