  We construct noncommutative `Riemannian manifold' structures on dual
quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant
differential calculus, using the quantum frame bundle formalism introduced
previously. The metric is provided by the braided-Killing form on the
braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan
form. We also apply the theory to finite sets and in particular to finite group
function algebras $C[G]$ with differential calculi and Killing forms determined
by a conjugacy class. The case of the permutation group $C[S_3]$ is worked out
in full detail and a unique torsion free and cotorsion free or `Levi-Civita'
connection is obtained with noncommutative Ricci curvature essentially
proportional to the metric (an Einstein space). We also construct Dirac
operators in the metric background, including on finite groups such as $S_3$.
In the process we clarify the construction of connections from gauge fields
with nonuniversal calculi on quantum principal bundles of tensor product form.
