The geometry of the quantum Euclidean space

Abstract

A detailed study is made of the noncommutative geometry of R 3 q , the quantum space covariant under the quantum group SO q(3). For each of its two SO q(3)-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and both which yield a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The delicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested, yielding S 2× R and R 3 , respectively, as the limit Riemannian manifolds.