The Geometry of the Quantum Euclidean Space

A detailed study is made of the noncommutative geometry of Rq, the quantum space covariant under the quantum group SOq(3). For each of its two SOq(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 which yield both 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 respectively S2 × R and R3 as the limit Riemannian manifold. Preprint LMU-TPW 97-23 Preprint 98-63, Dip. Matematica e Applicazioni, Università di Napoli