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We construct noncommutative `Riemannian manifold' structures on dualquasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariantdifferential calculus, using the quantum frame bundle formalism introducedpreviously. The metric is provided by the braided-Killing form on thebraided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartanform. We also apply the theory to finite sets and in particular to finite groupfunction algebras $C[G]$ with differential calculi and Killing forms determinedby a conjugacy class. The case of the permutation group $C[S_3]$ is worked outin full detail and a unique torsion free and cotorsion free or `Levi-Civita'connection is obtained with noncommutative Ricci curvature essentiallyproportional to the metric (an Einstein space). We also construct Diracoperators in the metric background, including on finite groups such as $S_3$.In the process we clarify the construction of connections from gauge fieldswith nonuniversal calculi on quantum principal bundles of tensor product form.Comment: 43 pages, 1 figure. Revised August 2001 to cut page length (eg  deleted appendix) for publication in CMP. Also fleshed out ex. of q-Killing  metric for q-SU_2 previously mentioned (no significant additions

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials