The Geometry of the Quantum Euclidean Space

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 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 $S^2\times R$ and $R^3$ as the limit Riemannian manifold.Comment: 29 pages, latex fil

The Geometry of a qq-Deformed Phase Space

The geometry of the $q$-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable.Comment: latex file, 26 pp, a typing error correcte